%------------------------------------------------------------------------------
% File     : ITP236^1 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_MinMax 00214_010433
% 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    : 0068_VEBT_MinMax_00214_010433 [Des22]

% Status   : Theorem
% Rating   : 0.40 v8.2.0, 0.23 v8.1.0
% Syntax   : Number of formulae    : 11120 (5066 unt;1289 typ;   0 def)
%            Number of atoms       : 28552 (11652 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 105320 (3164   ~; 537   |;2070   &;88288   @)
%                                         (   0 <=>;11261  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   6 avg)
%            Number of types       :  174 ( 173 usr)
%            Number of type conns  : 4900 (4900   >;   0   *;   0   +;   0  <<)
%            Number of symbols     : 1119 (1116 usr;  75 con; 0-5 aty)
%            Number of variables   : 25757 (2081   ^;22827   !; 849   ?;25757   :)
% 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 22:12:25.702
%------------------------------------------------------------------------------
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_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__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_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__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_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__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__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_It__Real__Oreal_M_Eo_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
    list_P3795440434834930179_o_int: $tType ).

thf(ty_n_t__Set__Oset_It__List__Olist_It__VEBT____Definitions__OVEBT_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    produc334124729049499915VEBT_o: $tType ).

thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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thf(ty_n_t__Set__Oset_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Real__Oreal_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Num__Onum_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Nat__Onat_Mt__Int__Oint_J,type,
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thf(ty_n_t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
    product_prod_int_int: $tType ).

thf(ty_n_t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    list_P4002435161011370285od_o_o: $tType ).

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__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
    set_Product_prod_o_o: $tType ).

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

thf(ty_n_t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
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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__Product____Type__Oprod_It__Nat__Onat_M_Eo_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_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__Set__Oset_It__Code____Numeral__Ointeger_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__List__Olist_I_Eo_J_J,type,
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thf(ty_n_t__Product____Type__Oprod_I_Eo_M_Eo_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,
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thf(ty_n_t__Option__Ooption_It__Int__Oint_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
    filter_nat: $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,
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thf(ty_n_t__VEBT____Definitions__OVEBT,type,
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thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
    set_rat: $tType ).

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

thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Code____Numeral__Ointeger,type,
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thf(ty_n_t__Option__Ooption_I_Eo_J,type,
    option_o: $tType ).

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

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

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

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

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

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

% Explicit typings (1116)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
    archim2889992004027027881ng_rat: rat > int ).

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

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

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

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

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

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

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

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLeq,type,
    bNF_Ca8665028551170535155natLeq: set_Pr1261947904930325089at_nat ).

thf(sy_c_BNF__Cardinal__Order__Relation_OnatLess,type,
    bNF_Ca8459412986667044542atLess: set_Pr1261947904930325089at_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
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    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__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    minus_2270307095948843157_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > product_prod_nat_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__Code____Numeral__Ointeger,type,
    minus_8373710615458151222nteger: code_integer > code_integer > code_integer ).

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

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

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

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

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

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

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

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J_J,type,
    minus_3314409938677909166at_nat: set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Groups_Ominus__class_Ominus_001t__Set__Oset_It__Set__Oset_It__Nat__Onat_J_J,type,
    minus_2163939370556025621et_nat: set_set_nat > set_set_nat > set_set_nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Oone__class_Oone_001t__Extended____Nat__Oenat,type,
    one_on7984719198319812577d_enat: extended_enat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Int__Oint,type,
    one_one_int: int ).

thf(sy_c_Groups_Oone__class_Oone_001t__Nat__Onat,type,
    one_one_nat: nat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Rat__Orat,type,
    one_one_rat: rat ).

thf(sy_c_Groups_Oone__class_Oone_001t__Real__Oreal,type,
    one_one_real: real ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Code____Numeral__Ointeger,type,
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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__Extended____Nat__Oenat,type,
    plus_p3455044024723400733d_enat: extended_enat > extended_enat > extended_enat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Int__Oint,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Nat__Onat,type,
    plus_plus_nat: nat > nat > nat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Num__Onum,type,
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thf(sy_c_Groups_Oplus__class_Oplus_001t__Rat__Orat,type,
    plus_plus_rat: rat > rat > rat ).

thf(sy_c_Groups_Oplus__class_Oplus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Osgn__class_Osgn_001t__Code____Numeral__Ointeger,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,
    sgn_sgn_int: int > int ).

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

thf(sy_c_Groups_Osgn__class_Osgn_001t__Real__Oreal,type,
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thf(sy_c_Groups_Otimes__class_Otimes_001t__Code____Numeral__Ointeger,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__Extended____Nat__Oenat,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__Nat__Onat,type,
    times_times_nat: nat > nat > nat ).

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

thf(sy_c_Groups_Otimes__class_Otimes_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Int__Oint_M_Eo_J,type,
    uminus_uminus_int_o: ( int > $o ) > int > $o ).

thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__List__Olist_It__Nat__Onat_J_M_Eo_J,type,
    uminus5770388063884162150_nat_o: ( list_nat > $o ) > list_nat > $o ).

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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Real__Oreal_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001_062_It__Set__Oset_It__Nat__Onat_J_M_Eo_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Complex__Ocomplex,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Int__Oint,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Rat__Orat,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Real__Oreal,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Groups_Ouminus__class_Ouminus_001t__Set__Oset_It__List__Olist_It__Nat__Onat_J_J,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Nat__Onat,type,
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thf(sy_c_Groups_Ozero__class_Ozero_001t__Rat__Orat,type,
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thf(sy_c_List_Olist_Oset_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Osize__list_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Olist_Otl_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__update_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
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thf(sy_c_List_Olist__update_001t__Int__Oint,type,
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thf(sy_c_List_Olist__update_001t__Nat__Onat,type,
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thf(sy_c_List_Olist__update_001t__Real__Oreal,type,
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thf(sy_c_List_Olist__update_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_List_Olistrel1_001t__Int__Oint,type,
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thf(sy_c_List_Onth_001t__Nat__Onat,type,
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thf(sy_c_List_Oproduct_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_List_Oupt,type,
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thf(sy_c_List_Oupto,type,
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thf(sy_c_List_Oupto__rel,type,
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thf(sy_c_List_Ozip_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_List_Ozip_001_Eo_001_Eo,type,
    zip_o_o: list_o > list_o > list_P4002435161011370285od_o_o ).

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

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

thf(sy_c_List_Ozip_001_Eo_001t__Real__Oreal,type,
    zip_o_real: list_o > list_real > list_P5232166724548748803o_real ).

thf(sy_c_List_Ozip_001_Eo_001t__VEBT____Definitions__OVEBT,type,
    zip_o_VEBT_VEBT: list_o > list_VEBT_VEBT > list_P7495141550334521929T_VEBT ).

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

thf(sy_c_List_Ozip_001t__Int__Oint_001t__Real__Oreal,type,
    zip_int_real: list_int > list_real > list_P6863124054624500543t_real ).

thf(sy_c_List_Ozip_001t__Int__Oint_001t__VEBT____Definitions__OVEBT,type,
    zip_int_VEBT_VEBT: list_int > list_VEBT_VEBT > list_P7524865323317820941T_VEBT ).

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

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

thf(sy_c_List_Ozip_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
    zip_nat_VEBT_VEBT: list_nat > list_VEBT_VEBT > list_P5647936690300460905T_VEBT ).

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

thf(sy_c_List_Ozip_001t__Real__Oreal_001_Eo,type,
    zip_real_o: list_real > list_o > list_P3595434254542482545real_o ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Int__Oint,type,
    zip_real_int: list_real > list_int > list_P4344331454722006975al_int ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Nat__Onat,type,
    zip_real_nat: list_real > list_nat > list_P6834414599653733731al_nat ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__Real__Oreal,type,
    zip_real_real: list_real > list_real > list_P8689742595348180415l_real ).

thf(sy_c_List_Ozip_001t__Real__Oreal_001t__VEBT____Definitions__OVEBT,type,
    zip_real_VEBT_VEBT: list_real > list_VEBT_VEBT > list_P877281246627933069T_VEBT ).

thf(sy_c_List_Ozip_001t__Set__Oset_It__Nat__Onat_J_001t__Set__Oset_It__Nat__Onat_J,type,
    zip_set_nat_set_nat: list_set_nat > list_set_nat > list_P6254988961118846195et_nat ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001_Eo,type,
    zip_VEBT_VEBT_o: list_VEBT_VEBT > list_o > list_P3126845725202233233VEBT_o ).

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
    zip_VEBT_VEBT_int: list_VEBT_VEBT > list_int > list_P4547456442757143711BT_int ).

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

thf(sy_c_List_Ozip_001t__VEBT____Definitions__OVEBT_001t__Real__Oreal,type,
    zip_VEBT_VEBT_real: list_VEBT_VEBT > list_real > list_P2623026923184700063T_real ).

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

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

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

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

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

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

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

thf(sy_c_Nat_Osemiring__1__class_ONats_001t__Int__Oint,type,
    semiring_1_Nats_int: set_int ).

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

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

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

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

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

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

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

thf(sy_c_Nat_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__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__Rat__Orat,type,
    semiri7787848453975740701ux_rat: ( rat > rat ) > nat > rat > rat ).

thf(sy_c_Nat_Osemiring__1__class_Oof__nat__aux_001t__Real__Oreal,type,
    semiri7260567687927622513x_real: ( real > real ) > nat > real > real ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__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_M_Eo_J_J,type,
    size_s9168528473962070013VEBT_o: list_P3126845725202233233VEBT_o > nat ).

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

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

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

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

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

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

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

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

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

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

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

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__aux,type,
    nat_prod_decode_aux: nat > nat > product_prod_nat_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

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

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

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

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

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

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

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

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

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

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

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

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_001_Eo,type,
    some_o: $o > option_o ).

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_Omap__option_001t__Num__Onum_001t__Num__Onum,type,
    map_option_num_num: ( num > num ) > option_num > option_num ).

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

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

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

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

thf(sy_c_Option_Ooption_Othe_001t__Num__Onum,type,
    the_num: option_num > num ).

thf(sy_c_Option_Ooption_Othe_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    the_Pr8591224930841456533at_nat: option4927543243414619207at_nat > product_prod_nat_nat ).

thf(sy_c_Order__Relation_OunderS_001t__Nat__Onat,type,
    order_underS_nat: set_Pr1261947904930325089at_nat > nat > set_nat ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_M_Eo_J_J,type,
    bot_bo5358457235160185703eger_o: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o ).

thf(sy_c_Orderings_Obot__class_Obot_001_062_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_M_062_It__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_M_Eo_J_J,type,
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thf(sy_c_Orderings_Oord__class_Omax_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Int__Oint,type,
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thf(sy_c_Orderings_Oord__class_Omin_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_Omono_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Oorder__class_Ostrict__mono_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Int__Oint,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Nat__Onat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Rat__Orat,type,
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thf(sy_c_Power_Opower__class_Opower_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Product__Type_OPair_001t__Int__Oint_001t__Real__Oreal,type,
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thf(sy_c_Product__Type_OPair_001t__List__Olist_I_Eo_J_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Int__Oint,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__Num__Onum,type,
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thf(sy_c_Product__Type_OPair_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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    collec6321179662152712658at_nat: ( produc3843707927480180839at_nat > $o ) > set_Pr4329608150637261639at_nat ).

thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
    collect_rat: ( rat > $o ) > set_rat ).

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__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_OCollect_001t__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
    collec5514110066124741708at_nat: ( set_Pr1261947904930325089at_nat > $o ) > set_se7855581050983116737at_nat ).

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

thf(sy_c_Set_OPow_001t__Nat__Onat,type,
    pow_nat: set_nat > set_set_nat ).

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__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
    image_2178119161166701260l_real: ( real > filter2146258269922977983l_real ) > set_real > set_fi7789364187291644575l_real ).

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

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

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

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

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

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

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

thf(sy_c_Set_Oinsert_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    insert9069300056098147895at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > set_Pr4329608150637261639at_nat ).

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

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

thf(sy_c_Set_Oinsert_001t__Set__Oset_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_Othe__elem_001t__Int__Oint,type,
    the_elem_int: set_int > int ).

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

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

thf(sy_c_Set_Othe__elem_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    the_el221668144340439132at_nat: set_Pr4329608150637261639at_nat > produc3843707927480180839at_nat ).

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_OatLeastAtMost_001t__Set__Oset_It__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__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__Num__Onum,type,
    set_ord_atMost_num: num > set_num ).

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

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_OgreaterThan_001t__Real__Oreal,type,
    set_or5849166863359141190n_real: real > set_real ).

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__Num__Onum,type,
    set_ord_lessThan_num: num > set_num ).

thf(sy_c_Set__Interval_Oord__class_OlessThan_001t__Rat__Orat,type,
    set_ord_lessThan_rat: rat > set_rat ).

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_Topological__Spaces_Ocontinuous_001t__Real__Oreal_001t__Real__Oreal,type,
    topolo4422821103128117721l_real: filter_real > ( real > real ) > $o ).

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

thf(sy_c_Topological__Spaces_Omonoseq_001t__Int__Oint,type,
    topolo4899668324122417113eq_int: ( nat > int ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Nat__Onat,type,
    topolo4902158794631467389eq_nat: ( nat > nat ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Num__Onum,type,
    topolo1459490580787246023eq_num: ( nat > num ) > $o ).

thf(sy_c_Topological__Spaces_Omonoseq_001t__Rat__Orat,type,
    topolo4267028734544971653eq_rat: ( nat > rat ) > $o ).

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

thf(sy_c_Topological__Spaces_Omonoseq_001t__Set__Oset_It__Int__Oint_J,type,
    topolo3100542954746470799et_int: ( nat > set_int ) > $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_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Ocot_001t__Real__Oreal,type,
    cot_real: real > 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__Rat__Orat,type,
    diffs_rat: ( nat > rat ) > nat > rat ).

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

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

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

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

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

thf(sy_c_Transcendental_Opi,type,
    pi: real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__Definitions_OVEBT__internal_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__Member_OVEBT__internal_Obit__concat,type,
    vEBT_VEBT_bit_concat: nat > nat > nat > nat ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Num__Onum,type,
    vEBT_V819420779217536731ft_num: ( num > num > num ) > option_num > option_num > option_num ).

thf(sy_c_VEBT__MinMax_OVEBT__internal_Ooption__shift_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    vEBT_V1502963449132264192at_nat: ( product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat ) > option4927543243414619207at_nat > option4927543243414619207at_nat > option4927543243414619207at_nat ).

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

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

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

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

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

thf(sy_c_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_Oaccp_001t__VEBT____Definitions__OVEBT,type,
    accp_VEBT_VEBT: ( vEBT_VEBT > vEBT_VEBT > $o ) > vEBT_VEBT > $o ).

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

thf(sy_c_Wellfounded_Omin__ext_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    min_ex6901939911449802026at_nat: set_Pr8693737435421807431at_nat > set_Pr4329608150637261639at_nat ).

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

thf(sy_c_Wellfounded_Owf_001t__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_fChoice_001t__Real__Oreal,type,
    fChoice_real: ( real > $o ) > real ).

thf(sy_c_member_001_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member1535805642427569193e_term: ( code_integer > option6357759511663192854e_term ) > set_Co9149898834107579976e_term > $o ).

thf(sy_c_member_001_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member8845023287901829240e_term: ( int > option6357759511663192854e_term ) > set_in3461395444621081367e_term > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member4242434998011752849e_term: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > set_Pr7604974323444597168e_term > $o ).

thf(sy_c_member_001_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J,type,
    member3222579708246209666e_term: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > set_Pr3642885161833720865e_term > $o ).

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

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

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

thf(sy_c_member_001t__List__Olist_I_Eo_J,type,
    member_list_o: list_o > set_list_o > $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_I_062_It__Code____Numeral__Ointeger_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member3068662437193594005nteger: produc8763457246119570046nteger > set_Pr8056137968301705908nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Int__Oint_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member7034335876925520548nt_int: produc7773217078559923341nt_int > set_Pr1872883991513573699nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_J,type,
    member4164122664394876845nteger: produc1908205239877642774nteger > set_Pr1281608226676607948nteger > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_062_It__Product____Type__Oprod_It__Int__Oint_M_062_It__Product____Type__Ounit_Mt__Code____Evaluation__Oterm_J_J_Mt__Option__Ooption_It__Product____Type__Oprod_I_Eo_Mt__List__Olist_It__Code____Evaluation__Oterm_J_J_J_J_Mt__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_J,type,
    member7618704894036264090nt_int: produc2285326912895808259nt_int > set_Pr9222295170931077689nt_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
    member7466972457876170832od_o_o: product_prod_o_o > set_Product_prod_o_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
    member7847949116333733898_o_int: product_prod_o_int > set_Pr8834758594704517033_o_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
    member2802428098988154798_o_nat: product_prod_o_nat > set_Pr2101469702781467981_o_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__Real__Oreal_J,type,
    member7400031367953476362o_real: product_prod_o_real > set_Pr6573716822653411497o_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
    member5477980866518848620T_VEBT: produc2504756804600209347T_VEBT > set_Pr7543698050874017315T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    member157494554546826820nteger: produc8923325533196201883nteger > set_Pr4811707699266497531nteger > $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__Int__Oint_Mt__Real__Oreal_J,type,
    member2744130022092475746t_real: produc679980390762269497t_real > set_Pr3538720872664544793t_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Int__Oint_Mt__VEBT____Definitions__OVEBT_J,type,
    member2056185340421749780T_VEBT: produc1531783533982839933T_VEBT > set_Pr8044002425091019955T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__List__Olist_I_Eo_J_Mt__List__Olist_I_Eo_J_J,type,
    member4159035015898711888list_o: produc7102631898165422375list_o > set_Pr6227168374412355847list_o > $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__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__Real__Oreal_M_Eo_J,type,
    member772602641336174712real_o: product_prod_real_o > set_Pr4936984352647145239real_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Int__Oint_J,type,
    member1627681773268152802al_int: produc8786904178792722361al_int > set_Pr1019928272762051225al_int > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Nat__Onat_J,type,
    member5805532792777349510al_nat: produc3741383161447143261al_nat > set_Pr3510011417693777981al_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
    member7849222048561428706l_real: produc2422161461964618553l_real > set_Pr6218003697084177305l_real > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Real__Oreal_Mt__VEBT____Definitions__OVEBT_J,type,
    member7262085504369356948T_VEBT: produc3757001726724277373T_VEBT > set_Pr6019664923565264691T_VEBT > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Nat__Onat_J_Mt__Set__Oset_It__Nat__Onat_J_J,type,
    member8277197624267554838et_nat: produc7819656566062154093et_nat > set_Pr5488025237498180813et_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_Mt__Set__Oset_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J_J,type,
    member8757157785044589968at_nat: produc3843707927480180839at_nat > set_Pr4329608150637261639at_nat > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
    member3307348790968139188VEBT_o: produc334124729049499915VEBT_o > set_Pr3175402225741728619VEBT_o > $o ).

thf(sy_c_member_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
    member5419026705395827622BT_int: produc4894624898956917775BT_int > set_Pr5066593544530342725BT_int > $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__Real__Oreal_J,type,
    member8675245146396747942T_real: produc5170161368751668367T_real > set_Pr7765410600122031685T_real > $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__Rat__Orat,type,
    member_rat: rat > set_rat > $o ).

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

thf(sy_c_member_001t__Set__Oset_It__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_m____,type,
    m: nat ).

thf(sy_v_ma,type,
    ma: nat ).

thf(sy_v_maxs____,type,
    maxs: nat ).

thf(sy_v_mi,type,
    mi: nat ).

thf(sy_v_n____,type,
    n: nat ).

thf(sy_v_summary,type,
    summary: vEBT_VEBT ).

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

thf(sy_v_treeList,type,
    treeList: list_VEBT_VEBT ).

% Relevant facts (9795)
thf(fact_0__092_060open_062_092_060exists_062x_O_Aboth__member__options_A_ItreeList_A_B_Amaxs_J_Ax_092_060close_062,axiom,
    ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X_1 ) ).

% \<open>\<exists>x. both_member_options (treeList ! maxs) x\<close>
thf(fact_1_not__min__Null__member,axiom,
    ! [T: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ T )
     => ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ T @ X_1 ) ) ).

% not_min_Null_member
thf(fact_2__092_060open_062Some_Amaxs_A_061_Avebt__maxt_Asummary_092_060close_062,axiom,
    ( ( some_nat @ maxs )
    = ( vEBT_vebt_maxt @ summary ) ) ).

% \<open>Some maxs = vebt_maxt summary\<close>
thf(fact_3_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T2: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T2 @ X )
          | ( vEBT_VEBT_membermima @ T2 @ X ) ) ) ) ).

% both_member_options_def
thf(fact_4__C5_C_I7_J,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ).

% "5"(7)
thf(fact_5__C5_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( vEBT_invar_vebt @ X2 @ n ) ) ).

% "5"(1)
thf(fact_6_in__children__def,axiom,
    ( vEBT_V5917875025757280293ildren
    = ( ^ [N: nat,TreeList: list_VEBT_VEBT,X: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ ( vEBT_VEBT_high @ X @ N ) ) @ ( vEBT_VEBT_low @ X @ N ) ) ) ) ).

% in_children_def
thf(fact_7_both__member__options__equiv__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
        = ( vEBT_vebt_member @ T @ X3 ) ) ) ).

% both_member_options_equiv_member
thf(fact_8_valid__member__both__member__options,axiom,
    ! [T: vEBT_VEBT,N2: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_V8194947554948674370ptions @ T @ X3 )
       => ( vEBT_vebt_member @ T @ X3 ) ) ) ).

% valid_member_both_member_options
thf(fact_9_intind,axiom,
    ! [I: nat,N2: nat,P: nat > $o,X3: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X3 )
       => ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X3 ) @ I ) ) ) ) ).

% intind
thf(fact_10_intind,axiom,
    ! [I: nat,N2: nat,P: int > $o,X3: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X3 )
       => ( P @ ( nth_int @ ( replicate_int @ N2 @ X3 ) @ I ) ) ) ) ).

% intind
thf(fact_11_intind,axiom,
    ! [I: nat,N2: nat,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( P @ X3 )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X3 ) @ I ) ) ) ) ).

% intind
thf(fact_12_assms_I2_J,axiom,
    mi != ma ).

% assms(2)
thf(fact_13_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_14_list__update__id,axiom,
    ! [Xs: list_nat,I: nat] :
      ( ( list_update_nat @ Xs @ I @ ( nth_nat @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_15_list__update__id,axiom,
    ! [Xs: list_int,I: nat] :
      ( ( list_update_int @ Xs @ I @ ( nth_int @ Xs @ I ) )
      = Xs ) ).

% list_update_id
thf(fact_16_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( I != J )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
        = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_17_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_nat,X3: nat] :
      ( ( I != J )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
        = ( nth_nat @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_18_nth__list__update__neq,axiom,
    ! [I: nat,J: nat,Xs: list_int,X3: int] :
      ( ( I != J )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
        = ( nth_int @ Xs @ J ) ) ) ).

% nth_list_update_neq
thf(fact_19_min__Null__member,axiom,
    ! [T: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_minNull @ T )
     => ~ ( vEBT_vebt_member @ T @ X3 ) ) ).

% min_Null_member
thf(fact_20__C5_C_I2_J,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "5"(2)
thf(fact_21_bit__split__inv,axiom,
    ! [X3: nat,D: nat] :
      ( ( vEBT_VEBT_bit_concat @ ( vEBT_VEBT_high @ X3 @ D ) @ ( vEBT_VEBT_low @ X3 @ D ) @ D )
      = X3 ) ).

% bit_split_inv
thf(fact_22__092_060open_062_092_060And_062thesis_O_A_I_092_060And_062maxs_O_ASome_Amaxs_A_061_Avebt__maxt_Asummary_A_092_060Longrightarrow_062_Athesis_J_A_092_060Longrightarrow_062_Athesis_092_060close_062,axiom,
    ~ ! [Maxs: nat] :
        ( ( some_nat @ Maxs )
       != ( vEBT_vebt_maxt @ summary ) ) ).

% \<open>\<And>thesis. (\<And>maxs. Some maxs = vebt_maxt summary \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close>
thf(fact_23_member__valid__both__member__options,axiom,
    ! [Tree: vEBT_VEBT,N2: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ Tree @ N2 )
     => ( ( vEBT_vebt_member @ Tree @ X3 )
       => ( ( vEBT_V5719532721284313246member @ Tree @ X3 )
          | ( vEBT_VEBT_membermima @ Tree @ X3 ) ) ) ) ).

% member_valid_both_member_options
thf(fact_24_maxt__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% maxt_member
thf(fact_25_member__correct,axiom,
    ! [T: vEBT_VEBT,N2: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_vebt_member @ T @ X3 )
        = ( member_nat @ X3 @ ( vEBT_set_vebt @ T ) ) ) ) ).

% member_correct
thf(fact_26__092_060open_062both__member__options_Asummary_A_Ihigh_Ama_An_J_092_060close_062,axiom,
    vEBT_V8194947554948674370ptions @ summary @ ( vEBT_VEBT_high @ ma @ n ) ).

% \<open>both_member_options summary (high ma n)\<close>
thf(fact_27_nth__replicate,axiom,
    ! [I: nat,N2: nat,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X3 ) @ I )
        = X3 ) ) ).

% nth_replicate
thf(fact_28_nth__replicate,axiom,
    ! [I: nat,N2: nat,X3: int] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_int @ ( replicate_int @ N2 @ X3 ) @ I )
        = X3 ) ) ).

% nth_replicate
thf(fact_29_nth__replicate,axiom,
    ! [I: nat,N2: nat,X3: nat] :
      ( ( ord_less_nat @ I @ N2 )
     => ( ( nth_nat @ ( replicate_nat @ N2 @ X3 ) @ I )
        = X3 ) ) ).

% nth_replicate
thf(fact_30__C5_C_I8_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "5"(8)
thf(fact_31_greater__shift,axiom,
    ( ord_less_nat
    = ( ^ [Y: nat,X: nat] : ( vEBT_VEBT_greater @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% greater_shift
thf(fact_32_less__shift,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_less @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% less_shift
thf(fact_33__092_060open_062high_Ama_An_A_060_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
    ord_less_nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the_nat @ ( vEBT_vebt_maxt @ summary ) ) ).

% \<open>high ma n < the (vebt_maxt summary)\<close>
thf(fact_34_mint__member,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Maxi ) )
       => ( vEBT_vebt_member @ T @ Maxi ) ) ) ).

% mint_member
thf(fact_35_maxt__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Maxi: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ Maxi ) )
       => ( ( vEBT_vebt_member @ T @ X3 )
         => ( ord_less_eq_nat @ X3 @ Maxi ) ) ) ) ).

% maxt_corr_help
thf(fact_36_maxt__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X3 ) )
       => ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).

% maxt_corr
thf(fact_37_maxt__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_max_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
       => ( ( vEBT_vebt_maxt @ T )
          = ( some_nat @ X3 ) ) ) ) ).

% maxt_sound
thf(fact_38_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X3: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ).

% buildup_nothing_in_leaf
thf(fact_39_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X3: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X3 ) ).

% buildup_nothing_in_min_max
thf(fact_40_option_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( some_nat @ X22 )
        = ( some_nat @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_41_option_Oinject,axiom,
    ! [X22: product_prod_nat_nat,Y2: product_prod_nat_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X22 )
        = ( some_P7363390416028606310at_nat @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_42_option_Oinject,axiom,
    ! [X22: num,Y2: num] :
      ( ( ( some_num @ X22 )
        = ( some_num @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% option.inject
thf(fact_43_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_44_valid__eq1,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T @ D )
     => ( vEBT_VEBT_valid @ T @ D ) ) ).

% valid_eq1
thf(fact_45_valid__eq2,axiom,
    ! [T: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T @ D )
     => ( vEBT_invar_vebt @ T @ D ) ) ).

% valid_eq2
thf(fact_46_inthall,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N2: nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
       => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_47_inthall,axiom,
    ! [Xs: list_real,P: real > $o,N2: nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
       => ( P @ ( nth_real @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_48_inthall,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,N2: nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
       => ( P @ ( nth_set_nat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_49_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_50_inthall,axiom,
    ! [Xs: list_o,P: $o > $o,N2: nat] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
       => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_51_inthall,axiom,
    ! [Xs: list_nat,P: nat > $o,N2: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_52_inthall,axiom,
    ! [Xs: list_int,P: int > $o,N2: nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
         => ( P @ X4 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
       => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_53_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_54__092_060open_062high_Ama_An_A_092_060le_062_Athe_A_Ivebt__maxt_Asummary_J_092_060close_062,axiom,
    ord_less_eq_nat @ ( vEBT_VEBT_high @ ma @ n ) @ ( the_nat @ ( vEBT_vebt_maxt @ summary ) ) ).

% \<open>high ma n \<le> the (vebt_maxt summary)\<close>
thf(fact_55_valid__tree__deg__neq__0,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_tree_deg_neq_0
thf(fact_56_valid__0__not,axiom,
    ! [T: vEBT_VEBT] :
      ~ ( vEBT_invar_vebt @ T @ zero_zero_nat ) ).

% valid_0_not
thf(fact_57_min__in__set__def,axiom,
    ( vEBT_VEBT_min_in_set
    = ( ^ [Xs2: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs2 )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs2 )
             => ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% min_in_set_def
thf(fact_58_max__in__set__def,axiom,
    ( vEBT_VEBT_max_in_set
    = ( ^ [Xs2: set_nat,X: nat] :
          ( ( member_nat @ X @ Xs2 )
          & ! [Y: nat] :
              ( ( member_nat @ Y @ Xs2 )
             => ( ord_less_eq_nat @ Y @ X ) ) ) ) ) ).

% max_in_set_def
thf(fact_59_mem__Collect__eq,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_60_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_61_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_62_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_63_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_64_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_65_Collect__mem__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_66_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_67_Collect__mem__eq,axiom,
    ! [A2: set_list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : ( member_list_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_68_Collect__mem__eq,axiom,
    ! [A2: set_set_nat] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] : ( member_set_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_69_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_70_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_71_Collect__cong,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X4: real] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_real @ P )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_72_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X4: list_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_73_Collect__cong,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X4: set_nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_set_nat @ P )
        = ( collect_set_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_74_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_75_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ( P @ X4 )
          = ( Q @ X4 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_76_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_77_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_78_mint__corr__help,axiom,
    ! [T: vEBT_VEBT,N2: nat,Mini: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ Mini ) )
       => ( ( vEBT_vebt_member @ T @ X3 )
         => ( ord_less_eq_nat @ Mini @ X3 ) ) ) ) ).

% mint_corr_help
thf(fact_79_replicate__eq__replicate,axiom,
    ! [M: nat,X3: vEBT_VEBT,N2: nat,Y3: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X3 )
        = ( replicate_VEBT_VEBT @ N2 @ Y3 ) )
      = ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X3 = Y3 ) ) ) ) ).

% replicate_eq_replicate
thf(fact_80_replicate__eq__replicate,axiom,
    ! [M: nat,X3: int,N2: nat,Y3: int] :
      ( ( ( replicate_int @ M @ X3 )
        = ( replicate_int @ N2 @ Y3 ) )
      = ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X3 = Y3 ) ) ) ) ).

% replicate_eq_replicate
thf(fact_81_replicate__eq__replicate,axiom,
    ! [M: nat,X3: nat,N2: nat,Y3: nat] :
      ( ( ( replicate_nat @ M @ X3 )
        = ( replicate_nat @ N2 @ Y3 ) )
      = ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X3 = Y3 ) ) ) ) ).

% replicate_eq_replicate
thf(fact_82_length__replicate,axiom,
    ! [N2: nat,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
      = N2 ) ).

% length_replicate
thf(fact_83_length__replicate,axiom,
    ! [N2: nat,X3: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N2 @ X3 ) )
      = N2 ) ).

% length_replicate
thf(fact_84_length__replicate,axiom,
    ! [N2: nat,X3: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N2 @ X3 ) )
      = N2 ) ).

% length_replicate
thf(fact_85_length__replicate,axiom,
    ! [N2: nat,X3: int] :
      ( ( size_size_list_int @ ( replicate_int @ N2 @ X3 ) )
      = N2 ) ).

% length_replicate
thf(fact_86_length__list__update,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) )
      = ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% length_list_update
thf(fact_87_length__list__update,axiom,
    ! [Xs: list_o,I: nat,X3: $o] :
      ( ( size_size_list_o @ ( list_update_o @ Xs @ I @ X3 ) )
      = ( size_size_list_o @ Xs ) ) ).

% length_list_update
thf(fact_88_length__list__update,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] :
      ( ( size_size_list_nat @ ( list_update_nat @ Xs @ I @ X3 ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_list_update
thf(fact_89_length__list__update,axiom,
    ! [Xs: list_int,I: nat,X3: int] :
      ( ( size_size_list_int @ ( list_update_int @ Xs @ I @ X3 ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_list_update
thf(fact_90_mint__sound,axiom,
    ! [T: vEBT_VEBT,N2: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 )
       => ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X3 ) ) ) ) ).

% mint_sound
thf(fact_91_mint__corr,axiom,
    ! [T: vEBT_VEBT,N2: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = ( some_nat @ X3 ) )
       => ( vEBT_VEBT_min_in_set @ ( vEBT_VEBT_set_vebt @ T ) @ X3 ) ) ) ).

% mint_corr
thf(fact_92_Ball__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_93_Ball__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_94_Ball__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
           => ( P @ X ) ) )
      = ( ( P @ A )
        | ( N2 = zero_zero_nat ) ) ) ).

% Ball_set_replicate
thf(fact_95_Bex__set__replicate,axiom,
    ! [N2: nat,A: vEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ? [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_96_Bex__set__replicate,axiom,
    ! [N2: nat,A: int,P: int > $o] :
      ( ( ? [X: int] :
            ( ( member_int @ X @ ( set_int2 @ ( replicate_int @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_97_Bex__set__replicate,axiom,
    ! [N2: nat,A: nat,P: nat > $o] :
      ( ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ ( replicate_nat @ N2 @ A ) ) )
            & ( P @ X ) ) )
      = ( ( P @ A )
        & ( N2 != zero_zero_nat ) ) ) ).

% Bex_set_replicate
thf(fact_98_in__set__replicate,axiom,
    ! [X3: product_prod_nat_nat,N2: nat,Y3: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_99_in__set__replicate,axiom,
    ! [X3: real,N2: nat,Y3: real] :
      ( ( member_real @ X3 @ ( set_real2 @ ( replicate_real @ N2 @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_100_in__set__replicate,axiom,
    ! [X3: set_nat,N2: nat,Y3: set_nat] :
      ( ( member_set_nat @ X3 @ ( set_set_nat2 @ ( replicate_set_nat @ N2 @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_101_in__set__replicate,axiom,
    ! [X3: vEBT_VEBT,N2: nat,Y3: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_102_in__set__replicate,axiom,
    ! [X3: int,N2: nat,Y3: int] :
      ( ( member_int @ X3 @ ( set_int2 @ ( replicate_int @ N2 @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_103_in__set__replicate,axiom,
    ! [X3: nat,N2: nat,Y3: nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ ( replicate_nat @ N2 @ Y3 ) ) )
      = ( ( X3 = Y3 )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_104_list__update__beyond,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] :
      ( ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ I )
     => ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_105_list__update__beyond,axiom,
    ! [Xs: list_o,I: nat,X3: $o] :
      ( ( ord_less_eq_nat @ ( size_size_list_o @ Xs ) @ I )
     => ( ( list_update_o @ Xs @ I @ X3 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_106_list__update__beyond,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( size_size_list_nat @ Xs ) @ I )
     => ( ( list_update_nat @ Xs @ I @ X3 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_107_list__update__beyond,axiom,
    ! [Xs: list_int,I: nat,X3: int] :
      ( ( ord_less_eq_nat @ ( size_size_list_int @ Xs ) @ I )
     => ( ( list_update_int @ Xs @ I @ X3 )
        = Xs ) ) ).

% list_update_beyond
thf(fact_108__C5_C_I4_J,axiom,
    ( m
    = ( suc @ n ) ) ).

% "5"(4)
thf(fact_109_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_110_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_o,X3: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_111_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_112_nth__list__update__eq,axiom,
    ! [I: nat,Xs: list_int,X3: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ I )
        = X3 ) ) ).

% nth_list_update_eq
thf(fact_113_lesseq__shift,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] : ( vEBT_VEBT_lesseq @ ( some_nat @ X ) @ ( some_nat @ Y ) ) ) ) ).

% lesseq_shift
thf(fact_114_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_115_set__swap,axiom,
    ! [I: nat,Xs: list_o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ J @ ( size_size_list_o @ Xs ) )
       => ( ( set_o2 @ ( list_update_o @ ( list_update_o @ Xs @ I @ ( nth_o @ Xs @ J ) ) @ J @ ( nth_o @ Xs @ I ) ) )
          = ( set_o2 @ Xs ) ) ) ) ).

% set_swap
thf(fact_116_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_117_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_118_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_119_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_o] :
      ( ( size_size_list_o @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_120_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_nat] :
      ( ( size_size_list_nat @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_121_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs3: list_int] :
      ( ( size_size_list_int @ Xs3 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_122_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_123_neq__if__length__neq,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs )
       != ( size_size_list_o @ Ys ) )
     => ( Xs != Ys ) ) ).

% neq_if_length_neq
thf(fact_124_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_125_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_126_option_Osel,axiom,
    ! [X22: nat] :
      ( ( the_nat @ ( some_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_127_option_Osel,axiom,
    ! [X22: product_prod_nat_nat] :
      ( ( the_Pr8591224930841456533at_nat @ ( some_P7363390416028606310at_nat @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_128_option_Osel,axiom,
    ! [X22: num] :
      ( ( the_num @ ( some_num @ X22 ) )
      = X22 ) ).

% option.sel
thf(fact_129_length__pos__if__in__set,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_130_length__pos__if__in__set,axiom,
    ! [X3: real,Xs: list_real] :
      ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_131_length__pos__if__in__set,axiom,
    ! [X3: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3254054031482475050et_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_132_length__pos__if__in__set,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_133_length__pos__if__in__set,axiom,
    ! [X3: $o,Xs: list_o] :
      ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_134_length__pos__if__in__set,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_135_length__pos__if__in__set,axiom,
    ! [X3: int,Xs: list_int] :
      ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_136_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ! [Xs3: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_137_length__induct,axiom,
    ! [P: list_o > $o,Xs: list_o] :
      ( ! [Xs3: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_138_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs3: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_139_length__induct,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ! [Xs3: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs3 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs3 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_140_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_VEBT_VEBT,Z: list_VEBT_VEBT] : Y4 = Z )
    = ( ^ [Xs2: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( ( nth_VEBT_VEBT @ Xs2 @ I2 )
                = ( nth_VEBT_VEBT @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_141_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_o,Z: list_o] : Y4 = Z )
    = ( ^ [Xs2: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs2 )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) )
             => ( ( nth_o @ Xs2 @ I2 )
                = ( nth_o @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_142_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_nat,Z: list_nat] : Y4 = Z )
    = ( ^ [Xs2: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs2 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) )
             => ( ( nth_nat @ Xs2 @ I2 )
                = ( nth_nat @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_143_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y4: list_int,Z: list_int] : Y4 = Z )
    = ( ^ [Xs2: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs2 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I2: nat] :
              ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) )
             => ( ( nth_int @ Xs2 @ I2 )
                = ( nth_int @ Ys3 @ I2 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_144_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: vEBT_VEBT] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs2: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_VEBT_VEBT @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_145_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: $o] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs2: list_o] :
            ( ( ( size_size_list_o @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_o @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_146_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: nat] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs2: list_nat] :
            ( ( ( size_size_list_nat @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_nat @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_147_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ K )
           => ? [X5: int] : ( P @ I2 @ X5 ) ) )
      = ( ? [Xs2: list_int] :
            ( ( ( size_size_list_int @ Xs2 )
              = K )
            & ! [I2: nat] :
                ( ( ord_less_nat @ I2 @ K )
               => ( P @ I2 @ ( nth_int @ Xs2 @ I2 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_148_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( nth_VEBT_VEBT @ Xs @ I3 )
              = ( nth_VEBT_VEBT @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_149_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
           => ( ( nth_o @ Xs @ I3 )
              = ( nth_o @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_150_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I3 )
              = ( nth_nat @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_151_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I3 )
              = ( nth_int @ Ys @ I3 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_152_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( X4 = X3 ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X3 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_153_replicate__length__same,axiom,
    ! [Xs: list_o,X3: $o] :
      ( ! [X4: $o] :
          ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
         => ( X4 = X3 ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X3 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_154_replicate__length__same,axiom,
    ! [Xs: list_nat,X3: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
         => ( X4 = X3 ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X3 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_155_replicate__length__same,axiom,
    ! [Xs: list_int,X3: int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
         => ( X4 = X3 ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X3 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_156_replicate__eqI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N2: nat,X3: product_prod_nat_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = N2 )
     => ( ! [Y5: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ Y5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( Y5 = X3 ) )
       => ( Xs
          = ( replic4235873036481779905at_nat @ N2 @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_157_replicate__eqI,axiom,
    ! [Xs: list_real,N2: nat,X3: real] :
      ( ( ( size_size_list_real @ Xs )
        = N2 )
     => ( ! [Y5: real] :
            ( ( member_real @ Y5 @ ( set_real2 @ Xs ) )
           => ( Y5 = X3 ) )
       => ( Xs
          = ( replicate_real @ N2 @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_158_replicate__eqI,axiom,
    ! [Xs: list_set_nat,N2: nat,X3: set_nat] :
      ( ( ( size_s3254054031482475050et_nat @ Xs )
        = N2 )
     => ( ! [Y5: set_nat] :
            ( ( member_set_nat @ Y5 @ ( set_set_nat2 @ Xs ) )
           => ( Y5 = X3 ) )
       => ( Xs
          = ( replicate_set_nat @ N2 @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_159_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N2: nat,X3: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = N2 )
     => ( ! [Y5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y5 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( Y5 = X3 ) )
       => ( Xs
          = ( replicate_VEBT_VEBT @ N2 @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_160_replicate__eqI,axiom,
    ! [Xs: list_o,N2: nat,X3: $o] :
      ( ( ( size_size_list_o @ Xs )
        = N2 )
     => ( ! [Y5: $o] :
            ( ( member_o @ Y5 @ ( set_o2 @ Xs ) )
           => ( Y5 = X3 ) )
       => ( Xs
          = ( replicate_o @ N2 @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_161_replicate__eqI,axiom,
    ! [Xs: list_nat,N2: nat,X3: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = N2 )
     => ( ! [Y5: nat] :
            ( ( member_nat @ Y5 @ ( set_nat2 @ Xs ) )
           => ( Y5 = X3 ) )
       => ( Xs
          = ( replicate_nat @ N2 @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_162_replicate__eqI,axiom,
    ! [Xs: list_int,N2: nat,X3: int] :
      ( ( ( size_size_list_int @ Xs )
        = N2 )
     => ( ! [Y5: int] :
            ( ( member_int @ Y5 @ ( set_int2 @ Xs ) )
           => ( Y5 = X3 ) )
       => ( Xs
          = ( replicate_int @ N2 @ X3 ) ) ) ) ).

% replicate_eqI
thf(fact_163_all__set__conv__all__nth,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_164_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
           => ( P @ ( nth_o @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_165_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_166_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
           => ( P @ ( nth_int @ Xs @ I2 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_167_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I3 ) ) )
     => ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_168_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: real > $o,X3: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
         => ( P @ ( nth_real @ Xs @ I3 ) ) )
     => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_169_all__nth__imp__all__set,axiom,
    ! [Xs: list_set_nat,P: set_nat > $o,X3: set_nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s3254054031482475050et_nat @ Xs ) )
         => ( P @ ( nth_set_nat @ Xs @ I3 ) ) )
     => ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_170_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X3: vEBT_VEBT] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) )
     => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_171_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: $o > $o,X3: $o] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
         => ( P @ ( nth_o @ Xs @ I3 ) ) )
     => ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_172_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X3: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I3 ) ) )
     => ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_173_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: int > $o,X3: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I3 ) ) )
     => ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
       => ( P @ X3 ) ) ) ).

% all_nth_imp_all_set
thf(fact_174_in__set__conv__nth,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Xs @ I2 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_175_in__set__conv__nth,axiom,
    ! [X3: real,Xs: list_real] :
      ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ I2 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_176_in__set__conv__nth,axiom,
    ! [X3: set_nat,Xs: list_set_nat] :
      ( ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs ) )
            & ( ( nth_set_nat @ Xs @ I2 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_177_in__set__conv__nth,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ I2 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_178_in__set__conv__nth,axiom,
    ! [X3: $o,Xs: list_o] :
      ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ I2 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_179_in__set__conv__nth,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I2 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_180_in__set__conv__nth,axiom,
    ! [X3: int,Xs: list_int] :
      ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I2 )
              = X3 ) ) ) ) ).

% in_set_conv_nth
thf(fact_181_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_182_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
     => ( ! [X4: $o] :
            ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_183_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_184_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_int,P: int > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
           => ( P @ X4 ) )
       => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_185_nth__mem,axiom,
    ! [N2: nat,Xs: list_P6011104703257516679at_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( member8440522571783428010at_nat @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% nth_mem
thf(fact_186_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_187_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_188_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_189_nth__mem,axiom,
    ! [N2: nat,Xs: list_o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
     => ( member_o @ ( nth_o @ Xs @ N2 ) @ ( set_o2 @ Xs ) ) ) ).

% nth_mem
thf(fact_190_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_191_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_192_set__update__memI,axiom,
    ! [N2: nat,Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
     => ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ N2 @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_193_set__update__memI,axiom,
    ! [N2: nat,Xs: list_real,X3: real] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
     => ( member_real @ X3 @ ( set_real2 @ ( list_update_real @ Xs @ N2 @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_194_set__update__memI,axiom,
    ! [N2: nat,Xs: list_set_nat,X3: set_nat] :
      ( ( ord_less_nat @ N2 @ ( size_s3254054031482475050et_nat @ Xs ) )
     => ( member_set_nat @ X3 @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ N2 @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_195_set__update__memI,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ N2 @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_196_set__update__memI,axiom,
    ! [N2: nat,Xs: list_o,X3: $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
     => ( member_o @ X3 @ ( set_o2 @ ( list_update_o @ Xs @ N2 @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_197_set__update__memI,axiom,
    ! [N2: nat,Xs: list_nat,X3: nat] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( member_nat @ X3 @ ( set_nat2 @ ( list_update_nat @ Xs @ N2 @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_198_set__update__memI,axiom,
    ! [N2: nat,Xs: list_int,X3: int] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( member_int @ X3 @ ( set_int2 @ ( list_update_int @ Xs @ N2 @ X3 ) ) ) ) ).

% set_update_memI
thf(fact_199_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 )
          = Xs )
        = ( ( nth_VEBT_VEBT @ Xs @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_200_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_o,X3: $o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ( list_update_o @ Xs @ I @ X3 )
          = Xs )
        = ( ( nth_o @ Xs @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_201_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( list_update_nat @ Xs @ I @ X3 )
          = Xs )
        = ( ( nth_nat @ Xs @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_202_list__update__same__conv,axiom,
    ! [I: nat,Xs: list_int,X3: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( list_update_int @ Xs @ I @ X3 )
          = Xs )
        = ( ( nth_int @ Xs @ I )
          = X3 ) ) ) ).

% list_update_same_conv
thf(fact_203_nth__list__update,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,J: nat,X3: vEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
            = X3 ) )
        & ( ( I != J )
         => ( ( nth_VEBT_VEBT @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) @ J )
            = ( nth_VEBT_VEBT @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_204_nth__list__update,axiom,
    ! [I: nat,Xs: list_o,X3: $o,J: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( nth_o @ ( list_update_o @ Xs @ I @ X3 ) @ J )
        = ( ( ( I = J )
           => X3 )
          & ( ( I != J )
           => ( nth_o @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_205_nth__list__update,axiom,
    ! [I: nat,Xs: list_nat,J: nat,X3: nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
            = X3 ) )
        & ( ( I != J )
         => ( ( nth_nat @ ( list_update_nat @ Xs @ I @ X3 ) @ J )
            = ( nth_nat @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_206_nth__list__update,axiom,
    ! [I: nat,Xs: list_int,J: nat,X3: int] :
      ( ( ord_less_nat @ I @ ( size_size_list_int @ Xs ) )
     => ( ( ( I = J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
            = X3 ) )
        & ( ( I != J )
         => ( ( nth_int @ ( list_update_int @ Xs @ I @ X3 ) @ J )
            = ( nth_int @ Xs @ J ) ) ) ) ) ).

% nth_list_update
thf(fact_207_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_208_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_209_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_210_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_211_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_212_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_213_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_214__C5_C_I5_J,axiom,
    ( deg
    = ( plus_plus_nat @ n @ m ) ) ).

% "5"(5)
thf(fact_215_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_216_buildup__gives__empty,axiom,
    ! [N2: nat] :
      ( ( vEBT_VEBT_set_vebt @ ( vEBT_vebt_buildup @ N2 ) )
      = bot_bot_set_nat ) ).

% buildup_gives_empty
thf(fact_217_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_218_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_219_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_220_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_221_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_222_even__odd__cases,axiom,
    ! [X3: nat] :
      ( ! [N3: nat] :
          ( X3
         != ( plus_plus_nat @ N3 @ N3 ) )
     => ~ ! [N3: nat] :
            ( X3
           != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).

% even_odd_cases
thf(fact_223_add__shift,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ( plus_plus_nat @ X3 @ Y3 )
        = Z2 )
      = ( ( vEBT_VEBT_add @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z2 ) ) ) ).

% add_shift
thf(fact_224_order__refl,axiom,
    ! [X3: set_int] : ( ord_less_eq_set_int @ X3 @ X3 ) ).

% order_refl
thf(fact_225_order__refl,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ X3 @ X3 ) ).

% order_refl
thf(fact_226_order__refl,axiom,
    ! [X3: num] : ( ord_less_eq_num @ X3 @ X3 ) ).

% order_refl
thf(fact_227_order__refl,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ X3 @ X3 ) ).

% order_refl
thf(fact_228_order__refl,axiom,
    ! [X3: int] : ( ord_less_eq_int @ X3 @ X3 ) ).

% order_refl
thf(fact_229_add__left__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_230_add__left__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_231_add__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_232_add__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add_left_cancel
thf(fact_233_add__right__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_234_add__right__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_235_add__right__cancel,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_236_add__right__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add_right_cancel
thf(fact_237_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% nat.inject
thf(fact_238_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_239_add__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_240_add__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_241_add__le__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_242_add__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_left
thf(fact_243_add__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_244_add__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_245_add__le__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_246_add__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% add_le_cancel_right
thf(fact_247_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_248_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_249_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_250_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_251_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_252_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_253_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_254_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_255_add__cancel__left__left,axiom,
    ! [B: complex,A: complex] :
      ( ( ( plus_plus_complex @ B @ A )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_left
thf(fact_256_add__cancel__left__left,axiom,
    ! [B: real,A: real] :
      ( ( ( plus_plus_real @ B @ A )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_left
thf(fact_257_add__cancel__left__left,axiom,
    ! [B: rat,A: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_left
thf(fact_258_add__cancel__left__left,axiom,
    ! [B: nat,A: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_left
thf(fact_259_add__cancel__left__left,axiom,
    ! [B: int,A: int] :
      ( ( ( plus_plus_int @ B @ A )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_left
thf(fact_260_add__cancel__left__right,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = A )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_left_right
thf(fact_261_add__cancel__left__right,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = A )
      = ( B = zero_zero_real ) ) ).

% add_cancel_left_right
thf(fact_262_add__cancel__left__right,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = A )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_left_right
thf(fact_263_add__cancel__left__right,axiom,
    ! [A: nat,B: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = A )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_left_right
thf(fact_264_add__cancel__left__right,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = A )
      = ( B = zero_zero_int ) ) ).

% add_cancel_left_right
thf(fact_265_add__cancel__right__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ B @ A ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_left
thf(fact_266_add__cancel__right__left,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ B @ A ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_left
thf(fact_267_add__cancel__right__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ B @ A ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_left
thf(fact_268_add__cancel__right__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ B @ A ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_left
thf(fact_269_add__cancel__right__left,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ B @ A ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_left
thf(fact_270_add__cancel__right__right,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( plus_plus_complex @ A @ B ) )
      = ( B = zero_zero_complex ) ) ).

% add_cancel_right_right
thf(fact_271_add__cancel__right__right,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( plus_plus_real @ A @ B ) )
      = ( B = zero_zero_real ) ) ).

% add_cancel_right_right
thf(fact_272_add__cancel__right__right,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( plus_plus_rat @ A @ B ) )
      = ( B = zero_zero_rat ) ) ).

% add_cancel_right_right
thf(fact_273_add__cancel__right__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( plus_plus_nat @ A @ B ) )
      = ( B = zero_zero_nat ) ) ).

% add_cancel_right_right
thf(fact_274_add__cancel__right__right,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( plus_plus_int @ A @ B ) )
      = ( B = zero_zero_int ) ) ).

% add_cancel_right_right
thf(fact_275_add__eq__0__iff__both__eq__0,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ( plus_plus_nat @ X3 @ Y3 )
        = zero_zero_nat )
      = ( ( X3 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_276_zero__eq__add__iff__both__eq__0,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X3 @ Y3 ) )
      = ( ( X3 = zero_zero_nat )
        & ( Y3 = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_277_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_278_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_279_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_280_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_281_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_282_add__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_283_add__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_284_add__less__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_285_add__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_left
thf(fact_286_add__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( ord_less_real @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_287_add__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( ord_less_rat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_288_add__less__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( ord_less_nat @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_289_add__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( ord_less_int @ A @ B ) ) ).

% add_less_cancel_right
thf(fact_290_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_291_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_292_Suc__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_eq
thf(fact_293_Suc__le__mono,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ ( suc @ M ) )
      = ( ord_less_eq_nat @ N2 @ M ) ) ).

% Suc_le_mono
thf(fact_294_add__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ M @ ( suc @ N2 ) )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc_right
thf(fact_295_add__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        & ( N2 = zero_zero_nat ) ) ) ).

% add_is_0
thf(fact_296_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_297_nat__add__left__cancel__less,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_less
thf(fact_298_nat__add__left__cancel__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% nat_add_left_cancel_le
thf(fact_299_add__le__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel1
thf(fact_300_add__le__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel1
thf(fact_301_add__le__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel1
thf(fact_302_add__le__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel1
thf(fact_303_add__le__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% add_le_same_cancel2
thf(fact_304_add__le__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% add_le_same_cancel2
thf(fact_305_add__le__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_eq_nat @ A @ zero_zero_nat ) ) ).

% add_le_same_cancel2
thf(fact_306_add__le__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_eq_int @ A @ zero_zero_int ) ) ).

% add_le_same_cancel2
thf(fact_307_le__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel1
thf(fact_308_le__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel1
thf(fact_309_le__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel1
thf(fact_310_le__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel1
thf(fact_311_le__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ B ) ) ).

% le_add_same_cancel2
thf(fact_312_le__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ).

% le_add_same_cancel2
thf(fact_313_le__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_eq_nat @ zero_zero_nat @ B ) ) ).

% le_add_same_cancel2
thf(fact_314_le__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_eq_int @ zero_zero_int @ B ) ) ).

% le_add_same_cancel2
thf(fact_315_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_316_double__add__le__zero__iff__single__add__le__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% double_add_le_zero_iff_single_add_le_zero
thf(fact_317_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_318_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_319_zero__le__double__add__iff__zero__le__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_double_add_iff_zero_le_single_add
thf(fact_320_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_321_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_322_zero__less__double__add__iff__zero__less__single__add,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_double_add_iff_zero_less_single_add
thf(fact_323_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_324_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_325_double__add__less__zero__iff__single__add__less__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% double_add_less_zero_iff_single_add_less_zero
thf(fact_326_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_327_less__add__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel2
thf(fact_328_less__add__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel2
thf(fact_329_less__add__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel2
thf(fact_330_less__add__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel2
thf(fact_331_less__add__same__cancel1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( ord_less_real @ zero_zero_real @ B ) ) ).

% less_add_same_cancel1
thf(fact_332_less__add__same__cancel1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( ord_less_rat @ zero_zero_rat @ B ) ) ).

% less_add_same_cancel1
thf(fact_333_less__add__same__cancel1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( ord_less_nat @ zero_zero_nat @ B ) ) ).

% less_add_same_cancel1
thf(fact_334_less__add__same__cancel1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ B ) ) ).

% less_add_same_cancel1
thf(fact_335_add__less__same__cancel2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ B ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel2
thf(fact_336_add__less__same__cancel2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel2
thf(fact_337_add__less__same__cancel2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel2
thf(fact_338_add__less__same__cancel2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel2
thf(fact_339_add__less__same__cancel1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( plus_plus_real @ B @ A ) @ B )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% add_less_same_cancel1
thf(fact_340_add__less__same__cancel1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ B @ A ) @ B )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% add_less_same_cancel1
thf(fact_341_add__less__same__cancel1,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( ord_less_nat @ A @ zero_zero_nat ) ) ).

% add_less_same_cancel1
thf(fact_342_add__less__same__cancel1,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% add_less_same_cancel1
thf(fact_343_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_344_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_345_add__gr__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        | ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% add_gr_0
thf(fact_346_one__is__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( plus_plus_nat @ M @ N2 ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% one_is_add
thf(fact_347_add__is__1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( ( M
            = ( suc @ zero_zero_nat ) )
          & ( N2 = zero_zero_nat ) )
        | ( ( M = zero_zero_nat )
          & ( N2
            = ( suc @ zero_zero_nat ) ) ) ) ) ).

% add_is_1
thf(fact_348_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_349_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_350_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_351_ab__semigroup__add__class_Oadd__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% ab_semigroup_add_class.add_ac(1)
thf(fact_352_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_353_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_354_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_355_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_356_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_357_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_358_group__cancel_Oadd1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( plus_plus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_359_group__cancel_Oadd1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( plus_plus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_360_group__cancel_Oadd1,axiom,
    ! [A2: nat,K: nat,A: nat,B: nat] :
      ( ( A2
        = ( plus_plus_nat @ K @ A ) )
     => ( ( plus_plus_nat @ A2 @ B )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_361_group__cancel_Oadd1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( plus_plus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add1
thf(fact_362_group__cancel_Oadd2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B2 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_363_group__cancel_Oadd2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B2 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_364_group__cancel_Oadd2,axiom,
    ! [B2: nat,K: nat,B: nat,A: nat] :
      ( ( B2
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B2 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_365_group__cancel_Oadd2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B2 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_366_add_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.assoc
thf(fact_367_add_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.assoc
thf(fact_368_add_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.assoc
thf(fact_369_add_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.assoc
thf(fact_370_add_Oleft__cancel,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_371_add_Oleft__cancel,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_372_add_Oleft__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
      = ( B = C ) ) ).

% add.left_cancel
thf(fact_373_add_Oright__cancel,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_374_add_Oright__cancel,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_375_add_Oright__cancel,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
      = ( B = C ) ) ).

% add.right_cancel
thf(fact_376_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A3: real,B3: real] : ( plus_plus_real @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_377_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_378_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B3: nat] : ( plus_plus_nat @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_379_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B3: int] : ( plus_plus_int @ B3 @ A3 ) ) ) ).

% add.commute
thf(fact_380_add_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( plus_plus_real @ B @ ( plus_plus_real @ A @ C ) )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% add.left_commute
thf(fact_381_add_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( plus_plus_rat @ B @ ( plus_plus_rat @ A @ C ) )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_382_add_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ B @ ( plus_plus_nat @ A @ C ) )
      = ( plus_plus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% add.left_commute
thf(fact_383_add_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ B @ ( plus_plus_int @ A @ C ) )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% add.left_commute
thf(fact_384_add__left__imp__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( plus_plus_real @ A @ B )
        = ( plus_plus_real @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_385_add__left__imp__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = ( plus_plus_rat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_386_add__left__imp__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( plus_plus_nat @ A @ B )
        = ( plus_plus_nat @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_387_add__left__imp__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( plus_plus_int @ A @ B )
        = ( plus_plus_int @ A @ C ) )
     => ( B = C ) ) ).

% add_left_imp_eq
thf(fact_388_add__right__imp__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ( plus_plus_real @ B @ A )
        = ( plus_plus_real @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_389_add__right__imp__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ( plus_plus_rat @ B @ A )
        = ( plus_plus_rat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_390_add__right__imp__eq,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ( plus_plus_nat @ B @ A )
        = ( plus_plus_nat @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_391_add__right__imp__eq,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ( plus_plus_int @ B @ A )
        = ( plus_plus_int @ C @ A ) )
     => ( B = C ) ) ).

% add_right_imp_eq
thf(fact_392_Suc__inject,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ( suc @ X3 )
        = ( suc @ Y3 ) )
     => ( X3 = Y3 ) ) ).

% Suc_inject
thf(fact_393_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_394_add__Suc__shift,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ M @ ( suc @ N2 ) ) ) ).

% add_Suc_shift
thf(fact_395_less__imp__Suc__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ? [K2: nat] :
          ( N2
          = ( suc @ ( plus_plus_nat @ M @ K2 ) ) ) ) ).

% less_imp_Suc_add
thf(fact_396_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M2 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_397_less__add__Suc2,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ M @ I ) ) ) ).

% less_add_Suc2
thf(fact_398_less__add__Suc1,axiom,
    ! [I: nat,M: nat] : ( ord_less_nat @ I @ ( suc @ ( plus_plus_nat @ I @ M ) ) ) ).

% less_add_Suc1
thf(fact_399_less__natE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ~ ! [Q2: nat] :
            ( N2
           != ( suc @ ( plus_plus_nat @ M @ Q2 ) ) ) ) ).

% less_natE
thf(fact_400_bot_Oextremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A ) ).

% bot.extremum
thf(fact_401_bot_Oextremum,axiom,
    ! [A: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A ) ).

% bot.extremum
thf(fact_402_bot_Oextremum,axiom,
    ! [A: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A ) ).

% bot.extremum
thf(fact_403_bot_Oextremum,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A ) ).

% bot.extremum
thf(fact_404_bot_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ bot_bot_nat @ A ) ).

% bot.extremum
thf(fact_405_bot_Oextremum__unique,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
      = ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_unique
thf(fact_406_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_407_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_408_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_409_bot_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
      = ( A = bot_bot_nat ) ) ).

% bot.extremum_unique
thf(fact_410_bot_Oextremum__uniqueI,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A @ bot_bo2099793752762293965at_nat )
     => ( A = bot_bo2099793752762293965at_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_411_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_412_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_413_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_414_bot_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( ord_less_eq_nat @ A @ bot_bot_nat )
     => ( A = bot_bot_nat ) ) ).

% bot.extremum_uniqueI
thf(fact_415_bot_Oextremum__strict,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% bot.extremum_strict
thf(fact_416_bot_Oextremum__strict,axiom,
    ! [A: set_real] :
      ~ ( ord_less_set_real @ A @ bot_bot_set_real ) ).

% bot.extremum_strict
thf(fact_417_bot_Oextremum__strict,axiom,
    ! [A: set_nat] :
      ~ ( ord_less_set_nat @ A @ bot_bot_set_nat ) ).

% bot.extremum_strict
thf(fact_418_bot_Oextremum__strict,axiom,
    ! [A: set_int] :
      ~ ( ord_less_set_int @ A @ bot_bot_set_int ) ).

% bot.extremum_strict
thf(fact_419_bot_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ bot_bot_nat ) ).

% bot.extremum_strict
thf(fact_420_bot_Onot__eq__extremum,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( A != bot_bo2099793752762293965at_nat )
      = ( ord_le7866589430770878221at_nat @ bot_bo2099793752762293965at_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_421_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_422_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_423_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_424_bot_Onot__eq__extremum,axiom,
    ! [A: nat] :
      ( ( A != bot_bot_nat )
      = ( ord_less_nat @ bot_bot_nat @ A ) ) ).

% bot.not_eq_extremum
thf(fact_425_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_426_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_427_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_428_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_429_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_430_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_431_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_432_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_433_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_434_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_435_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_436_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_437_add__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_mono
thf(fact_438_add__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_439_add__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_mono
thf(fact_440_add__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_mono
thf(fact_441_add__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_left_mono
thf(fact_442_add__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_443_add__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_left_mono
thf(fact_444_add__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_left_mono
thf(fact_445_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( B
           != ( plus_plus_nat @ A @ C2 ) ) ) ).

% less_eqE
thf(fact_446_add__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_right_mono
thf(fact_447_add__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_448_add__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_right_mono
thf(fact_449_add__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_right_mono
thf(fact_450_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_451_add__le__imp__le__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_452_add__le__imp__le__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_453_add__le__imp__le__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_454_add__le__imp__le__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_left
thf(fact_455_add__le__imp__le__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_eq_real @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_456_add__le__imp__le__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_457_add__le__imp__le__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_458_add__le__imp__le__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_eq_int @ A @ B ) ) ).

% add_le_imp_le_right
thf(fact_459_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_460_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_461_comm__monoid__add__class_Oadd__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% comm_monoid_add_class.add_0
thf(fact_462_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_463_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_464_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_465_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_466_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_467_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_468_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_469_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_470_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_471_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_472_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_473_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_474_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_475_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_476_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_477_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_478_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( I = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_479_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_480_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_481_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_482_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_483_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_484_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_485_add__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_486_add__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_487_add__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_488_add__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_strict_mono
thf(fact_489_add__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_490_add__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_491_add__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_492_add__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) ) ) ).

% add_strict_left_mono
thf(fact_493_add__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_494_add__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_495_add__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_496_add__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) ) ) ).

% add_strict_right_mono
thf(fact_497_add__less__imp__less__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_498_add__less__imp__less__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_499_add__less__imp__less__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_500_add__less__imp__less__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_left
thf(fact_501_add__less__imp__less__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
     => ( ord_less_real @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_502_add__less__imp__less__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
     => ( ord_less_rat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_503_add__less__imp__less__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
     => ( ord_less_nat @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_504_add__less__imp__less__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
     => ( ord_less_int @ A @ B ) ) ).

% add_less_imp_less_right
thf(fact_505_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_506_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_507_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_508_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_509_old_Onat_Oexhaust,axiom,
    ! [Y3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y3
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_510_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_511_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X4: nat] : ( P @ X4 @ zero_zero_nat )
     => ( ! [Y5: nat] : ( P @ zero_zero_nat @ ( suc @ Y5 ) )
       => ( ! [X4: nat,Y5: nat] :
              ( ( P @ X4 @ Y5 )
             => ( P @ ( suc @ X4 ) @ ( suc @ Y5 ) ) )
         => ( P @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_512_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_513_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_514_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_515_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_516_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% not0_implies_Suc
thf(fact_517_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_518_add__eq__self__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( plus_plus_nat @ M @ N2 )
        = M )
     => ( N2 = zero_zero_nat ) ) ).

% add_eq_self_zero
thf(fact_519_Nat_OlessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ I @ K )
     => ( ( K
         != ( suc @ I ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_520_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_521_Suc__lessE,axiom,
    ! [I: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_522_Suc__lessI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ( suc @ M )
         != N2 )
       => ( ord_less_nat @ ( suc @ M ) @ N2 ) ) ) ).

% Suc_lessI
thf(fact_523_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_524_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_525_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
            & ( P @ I2 ) ) )
      = ( ( P @ N2 )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N2 )
            & ( P @ I2 ) ) ) ) ).

% Ex_less_Suc
thf(fact_526_less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) ) ) ).

% less_Suc_eq
thf(fact_527_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_528_All__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ N2 )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N2 )
           => ( P @ I2 ) ) ) ) ).

% All_less_Suc
thf(fact_529_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M )
      = ( ? [M4: nat] :
            ( ( M
              = ( suc @ M4 ) )
            & ( ord_less_nat @ N2 @ M4 ) ) ) ) ).

% Suc_less_eq2
thf(fact_530_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_531_Suc__less__SucD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_less_SucD
thf(fact_532_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_533_less__Suc__induct,axiom,
    ! [I: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] : ( P @ I3 @ ( suc @ I3 ) )
       => ( ! [I3: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I3 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I3 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I3 @ K2 ) ) ) ) )
         => ( P @ I @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_534_strict__inc__induct,axiom,
    ! [I: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I @ J )
     => ( ! [I3: nat] :
            ( ( J
              = ( suc @ I3 ) )
           => ( P @ I3 ) )
       => ( ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ J )
             => ( ( P @ ( suc @ I3 ) )
               => ( P @ I3 ) ) )
         => ( P @ I ) ) ) ) ).

% strict_inc_induct
thf(fact_535_not__less__less__Suc__eq,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% not_less_less_Suc_eq
thf(fact_536_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_537_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_538_not__add__less1,axiom,
    ! [I: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I @ J ) @ I ) ).

% not_add_less1
thf(fact_539_not__add__less2,axiom,
    ! [J: nat,I: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I ) @ I ) ).

% not_add_less2
thf(fact_540_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_541_trans__less__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_542_trans__less__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ord_less_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_543_less__add__eq__less,axiom,
    ! [K: nat,L: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ K @ L )
     => ( ( ( plus_plus_nat @ M @ L )
          = ( plus_plus_nat @ K @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% less_add_eq_less
thf(fact_544_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_545_le__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( M
          = ( suc @ N2 ) ) ) ) ).

% le_SucE
thf(fact_546_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_547_Suc__le__D,axiom,
    ! [N2: nat,M5: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M5 )
     => ? [M3: nat] :
          ( M5
          = ( suc @ M3 ) ) ) ).

% Suc_le_D
thf(fact_548_le__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
      = ( ( ord_less_eq_nat @ M @ N2 )
        | ( M
          = ( suc @ N2 ) ) ) ) ).

% le_Suc_eq
thf(fact_549_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_550_not__less__eq__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_eq_nat @ M @ N2 ) )
      = ( ord_less_eq_nat @ ( suc @ N2 ) @ M ) ) ).

% not_less_eq_eq
thf(fact_551_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M6: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M6 ) @ N3 )
             => ( P @ M6 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_552_nat__induct__at__least,axiom,
    ! [M: nat,N2: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( P @ M )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ M @ N3 )
             => ( ( P @ N3 )
               => ( P @ ( suc @ N3 ) ) ) )
         => ( P @ N2 ) ) ) ) ).

% nat_induct_at_least
thf(fact_553_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X4: nat] : ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y5: nat,Z3: nat] :
              ( ( R @ X4 @ Y5 )
             => ( ( R @ Y5 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
           => ( R @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_554_add__leE,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ~ ( ( ord_less_eq_nat @ M @ N2 )
         => ~ ( ord_less_eq_nat @ K @ N2 ) ) ) ).

% add_leE
thf(fact_555_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_556_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_557_add__leD1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% add_leD1
thf(fact_558_add__leD2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ M @ K ) @ N2 )
     => ( ord_less_eq_nat @ K @ N2 ) ) ).

% add_leD2
thf(fact_559_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_560_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_561_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_562_trans__le__add1,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_563_trans__le__add2,axiom,
    ! [I: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_nat @ I @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_564_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( plus_plus_nat @ M2 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_565_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B2 )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_566_subset__code_I1_J,axiom,
    ! [Xs: list_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B2 )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs ) )
           => ( member_real @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_567_subset__code_I1_J,axiom,
    ! [Xs: list_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ B2 )
      = ( ! [X: set_nat] :
            ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
           => ( member_set_nat @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_568_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B2: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B2 )
      = ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( member_VEBT_VEBT @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_569_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B2 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_570_subset__code_I1_J,axiom,
    ! [Xs: list_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B2 )
      = ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs ) )
           => ( member_int @ X @ B2 ) ) ) ) ).

% subset_code(1)
thf(fact_571_add__decreasing,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_572_add__decreasing,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_573_add__decreasing,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_574_add__decreasing,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing
thf(fact_575_add__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_576_add__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_577_add__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_578_add__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing
thf(fact_579_add__decreasing2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_580_add__decreasing2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_581_add__decreasing2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ C @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_582_add__decreasing2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ C ) @ B ) ) ) ).

% add_decreasing2
thf(fact_583_add__increasing2,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ B @ A )
       => ( ord_less_eq_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_584_add__increasing2,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( ord_less_eq_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_585_add__increasing2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_eq_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_586_add__increasing2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_eq_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_increasing2
thf(fact_587_add__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_588_add__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_589_add__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_590_add__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_nonneg
thf(fact_591_add__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_nonpos
thf(fact_592_add__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_nonpos
thf(fact_593_add__nonpos__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_nonpos
thf(fact_594_add__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_nonpos
thf(fact_595_add__nonneg__eq__0__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( plus_plus_real @ X3 @ Y3 )
            = zero_zero_real )
          = ( ( X3 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_596_add__nonneg__eq__0__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( plus_plus_rat @ X3 @ Y3 )
            = zero_zero_rat )
          = ( ( X3 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_597_add__nonneg__eq__0__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( plus_plus_nat @ X3 @ Y3 )
            = zero_zero_nat )
          = ( ( X3 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_598_add__nonneg__eq__0__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( plus_plus_int @ X3 @ Y3 )
            = zero_zero_int )
          = ( ( X3 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_599_add__nonpos__eq__0__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ( plus_plus_real @ X3 @ Y3 )
            = zero_zero_real )
          = ( ( X3 = zero_zero_real )
            & ( Y3 = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_600_add__nonpos__eq__0__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X3 @ Y3 )
            = zero_zero_rat )
          = ( ( X3 = zero_zero_rat )
            & ( Y3 = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_601_add__nonpos__eq__0__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y3 @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X3 @ Y3 )
            = zero_zero_nat )
          = ( ( X3 = zero_zero_nat )
            & ( Y3 = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_602_add__nonpos__eq__0__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
       => ( ( ( plus_plus_int @ X3 @ Y3 )
            = zero_zero_int )
          = ( ( X3 = zero_zero_int )
            & ( Y3 = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_603_add__less__le__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_604_add__less__le__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_605_add__less__le__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_606_add__less__le__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_less_le_mono
thf(fact_607_add__le__less__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ord_less_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_608_add__le__less__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_609_add__le__less__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_610_add__le__less__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ord_less_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) ) ) ) ).

% add_le_less_mono
thf(fact_611_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_612_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_613_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_614_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_615_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_616_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_617_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_618_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_619_pos__add__strict,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_620_pos__add__strict,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_621_pos__add__strict,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_622_pos__add__strict,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% pos_add_strict
thf(fact_623_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C2: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C2 ) )
           => ( C2 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_624_add__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_625_add__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_626_add__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_627_add__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_pos
thf(fact_628_add__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_neg
thf(fact_629_add__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_neg
thf(fact_630_add__neg__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_neg
thf(fact_631_add__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_neg
thf(fact_632_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_633_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_634_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ord_less_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_635_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_636_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_637_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > real,N2: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_real @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_638_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > rat,N2: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_rat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_639_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > num,N2: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_num @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_640_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > nat,N2: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_641_lift__Suc__mono__less__iff,axiom,
    ! [F: nat > int,N2: nat,M: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_int @ ( F @ N2 ) @ ( F @ M ) )
        = ( ord_less_nat @ N2 @ M ) ) ) ).

% lift_Suc_mono_less_iff
thf(fact_642_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_643_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_644_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N4 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_645_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_646_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_647_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_648_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_rat @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_649_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N2: nat,N4: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ord_less_eq_num @ ( F @ N4 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_650_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_651_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_652_less__Suc__eq__0__disj,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ( M = zero_zero_nat )
        | ? [J3: nat] :
            ( ( M
              = ( suc @ J3 ) )
            & ( ord_less_nat @ J3 @ N2 ) ) ) ) ).

% less_Suc_eq_0_disj
thf(fact_653_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M3: nat] :
          ( N2
          = ( suc @ M3 ) ) ) ).

% gr0_implies_Suc
thf(fact_654_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
           => ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I2: nat] :
            ( ( ord_less_nat @ I2 @ N2 )
           => ( P @ ( suc @ I2 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_655_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M2: nat] :
            ( N2
            = ( suc @ M2 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_656_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( suc @ N2 ) )
            & ( P @ I2 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ N2 )
            & ( P @ ( suc @ I2 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_657_less__imp__add__positive,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_658_le__imp__less__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% le_imp_less_Suc
thf(fact_659_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_660_less__Suc__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_Suc_eq_le
thf(fact_661_le__less__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
        = ( N2 = M ) ) ) ).

% le_less_Suc_eq
thf(fact_662_Suc__le__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_lessD
thf(fact_663_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_664_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_665_Suc__le__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_le_eq
thf(fact_666_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_667_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: 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 @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_668_add__strict__increasing2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_669_add__strict__increasing2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_670_add__strict__increasing2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_671_add__strict__increasing2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing2
thf(fact_672_add__strict__increasing,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ B @ ( plus_plus_real @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_673_add__strict__increasing,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ B @ ( plus_plus_rat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_674_add__strict__increasing,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ B @ ( plus_plus_nat @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_675_add__strict__increasing,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ B @ ( plus_plus_int @ A @ C ) ) ) ) ).

% add_strict_increasing
thf(fact_676_add__pos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_677_add__pos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_678_add__pos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_679_add__pos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_pos_nonneg
thf(fact_680_add__nonpos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_nonpos_neg
thf(fact_681_add__nonpos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_nonpos_neg
thf(fact_682_add__nonpos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_nonpos_neg
thf(fact_683_add__nonpos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_nonpos_neg
thf(fact_684_add__nonneg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_685_add__nonneg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_686_add__nonneg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_687_add__nonneg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ A @ B ) ) ) ) ).

% add_nonneg_pos
thf(fact_688_add__neg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( plus_plus_real @ A @ B ) @ zero_zero_real ) ) ) ).

% add_neg_nonpos
thf(fact_689_add__neg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% add_neg_nonpos
thf(fact_690_add__neg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( plus_plus_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% add_neg_nonpos
thf(fact_691_add__neg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( plus_plus_int @ A @ B ) @ zero_zero_int ) ) ) ).

% add_neg_nonpos
thf(fact_692_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ! [I4: nat] :
                ( ( ord_less_eq_nat @ I4 @ K2 )
               => ~ ( P @ I4 ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_693_set__update__subsetI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,I: nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ A2 )
     => ( ( member8440522571783428010at_nat @ X3 @ A2 )
       => ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_694_set__update__subsetI,axiom,
    ! [Xs: list_real,A2: set_real,X3: real,I: nat] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ A2 )
     => ( ( member_real @ X3 @ A2 )
       => ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_695_set__update__subsetI,axiom,
    ! [Xs: list_set_nat,A2: set_set_nat,X3: set_nat,I: nat] :
      ( ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ Xs ) @ A2 )
     => ( ( member_set_nat @ X3 @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( set_set_nat2 @ ( list_update_set_nat @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_696_set__update__subsetI,axiom,
    ! [Xs: list_VEBT_VEBT,A2: set_VEBT_VEBT,X3: vEBT_VEBT,I: nat] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ A2 )
     => ( ( member_VEBT_VEBT @ X3 @ A2 )
       => ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_697_set__update__subsetI,axiom,
    ! [Xs: list_nat,A2: set_nat,X3: nat,I: nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ A2 )
     => ( ( member_nat @ X3 @ A2 )
       => ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_698_set__update__subsetI,axiom,
    ! [Xs: list_int,A2: set_int,X3: int,I: nat] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ A2 )
     => ( ( member_int @ X3 @ A2 )
       => ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X3 ) ) @ A2 ) ) ) ).

% set_update_subsetI
thf(fact_699_nle__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_eq_rat @ A @ B ) )
      = ( ( ord_less_eq_rat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_700_nle__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_eq_num @ A @ B ) )
      = ( ( ord_less_eq_num @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_701_nle__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_eq_nat @ A @ B ) )
      = ( ( ord_less_eq_nat @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_702_nle__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_eq_int @ A @ B ) )
      = ( ( ord_less_eq_int @ B @ A )
        & ( B != A ) ) ) ).

% nle_le
thf(fact_703_le__cases3,axiom,
    ! [X3: rat,Y3: rat,Z2: rat] :
      ( ( ( ord_less_eq_rat @ X3 @ Y3 )
       => ~ ( ord_less_eq_rat @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_rat @ Y3 @ X3 )
         => ~ ( ord_less_eq_rat @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_rat @ X3 @ Z2 )
           => ~ ( ord_less_eq_rat @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_rat @ Z2 @ Y3 )
             => ~ ( ord_less_eq_rat @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_rat @ Y3 @ Z2 )
               => ~ ( ord_less_eq_rat @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_rat @ Z2 @ X3 )
                 => ~ ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_704_le__cases3,axiom,
    ! [X3: num,Y3: num,Z2: num] :
      ( ( ( ord_less_eq_num @ X3 @ Y3 )
       => ~ ( ord_less_eq_num @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_num @ Y3 @ X3 )
         => ~ ( ord_less_eq_num @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_num @ X3 @ Z2 )
           => ~ ( ord_less_eq_num @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_num @ Z2 @ Y3 )
             => ~ ( ord_less_eq_num @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_num @ Y3 @ Z2 )
               => ~ ( ord_less_eq_num @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_num @ Z2 @ X3 )
                 => ~ ( ord_less_eq_num @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_705_le__cases3,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ( ord_less_eq_nat @ X3 @ Y3 )
       => ~ ( ord_less_eq_nat @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_nat @ Y3 @ X3 )
         => ~ ( ord_less_eq_nat @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_nat @ X3 @ Z2 )
           => ~ ( ord_less_eq_nat @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_nat @ Z2 @ Y3 )
             => ~ ( ord_less_eq_nat @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_nat @ Y3 @ Z2 )
               => ~ ( ord_less_eq_nat @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_nat @ Z2 @ X3 )
                 => ~ ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_706_le__cases3,axiom,
    ! [X3: int,Y3: int,Z2: int] :
      ( ( ( ord_less_eq_int @ X3 @ Y3 )
       => ~ ( ord_less_eq_int @ Y3 @ Z2 ) )
     => ( ( ( ord_less_eq_int @ Y3 @ X3 )
         => ~ ( ord_less_eq_int @ X3 @ Z2 ) )
       => ( ( ( ord_less_eq_int @ X3 @ Z2 )
           => ~ ( ord_less_eq_int @ Z2 @ Y3 ) )
         => ( ( ( ord_less_eq_int @ Z2 @ Y3 )
             => ~ ( ord_less_eq_int @ Y3 @ X3 ) )
           => ( ( ( ord_less_eq_int @ Y3 @ Z2 )
               => ~ ( ord_less_eq_int @ Z2 @ X3 ) )
             => ~ ( ( ord_less_eq_int @ Z2 @ X3 )
                 => ~ ( ord_less_eq_int @ X3 @ Y3 ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_707_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_int,Z: set_int] : Y4 = Z )
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y )
          & ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_708_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: rat,Z: rat] : Y4 = Z )
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_709_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: num,Z: num] : Y4 = Z )
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_710_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_711_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z: int] : Y4 = Z )
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_712_ord__eq__le__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( A = B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_713_ord__eq__le__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_714_ord__eq__le__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_715_ord__eq__le__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_716_ord__eq__le__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_eq_le_trans
thf(fact_717_ord__le__eq__trans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_718_ord__le__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_719_ord__le__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_720_ord__le__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_721_ord__le__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% ord_le_eq_trans
thf(fact_722_order__antisym,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ord_less_eq_set_int @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_723_order__antisym,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_724_order__antisym,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_725_order__antisym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_726_order__antisym,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ X3 )
       => ( X3 = Y3 ) ) ) ).

% order_antisym
thf(fact_727_order_Otrans,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_eq_set_int @ A @ C ) ) ) ).

% order.trans
thf(fact_728_order_Otrans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_eq_rat @ A @ C ) ) ) ).

% order.trans
thf(fact_729_order_Otrans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_eq_num @ A @ C ) ) ) ).

% order.trans
thf(fact_730_order_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_eq_nat @ A @ C ) ) ) ).

% order.trans
thf(fact_731_order_Otrans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_eq_int @ A @ C ) ) ) ).

% order.trans
thf(fact_732_order__trans,axiom,
    ! [X3: set_int,Y3: set_int,Z2: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ord_less_eq_set_int @ Y3 @ Z2 )
       => ( ord_less_eq_set_int @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_733_order__trans,axiom,
    ! [X3: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z2 )
       => ( ord_less_eq_rat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_734_order__trans,axiom,
    ! [X3: num,Y3: num,Z2: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z2 )
       => ( ord_less_eq_num @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_735_order__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_eq_nat @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_736_order__trans,axiom,
    ! [X3: int,Y3: int,Z2: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z2 )
       => ( ord_less_eq_int @ X3 @ Z2 ) ) ) ).

% order_trans
thf(fact_737_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A4: rat,B4: rat] :
          ( ( ord_less_eq_rat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: rat,B4: rat] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_738_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A4: num,B4: num] :
          ( ( ord_less_eq_num @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: num,B4: num] :
            ( ( P @ B4 @ A4 )
           => ( P @ A4 @ B4 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_739_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: 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 @ B ) ) ) ).

% linorder_wlog
thf(fact_740_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: 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 @ B ) ) ) ).

% linorder_wlog
thf(fact_741_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: set_int,Z: set_int] : Y4 = Z )
    = ( ^ [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_742_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: rat,Z: rat] : Y4 = Z )
    = ( ^ [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ B3 @ A3 )
          & ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_743_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: num,Z: num] : Y4 = Z )
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_eq_num @ B3 @ A3 )
          & ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_744_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ B3 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_745_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y4: int,Z: int] : Y4 = Z )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ B3 @ A3 )
          & ( ord_less_eq_int @ A3 @ B3 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_746_dual__order_Oantisym,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_747_dual__order_Oantisym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_748_dual__order_Oantisym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_749_dual__order_Oantisym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_750_dual__order_Oantisym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ A @ B )
       => ( A = B ) ) ) ).

% dual_order.antisym
thf(fact_751_dual__order_Otrans,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_eq_set_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_752_dual__order_Otrans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_eq_rat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_753_dual__order_Otrans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_eq_num @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_754_dual__order_Otrans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_eq_nat @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_755_dual__order_Otrans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_eq_int @ C @ A ) ) ) ).

% dual_order.trans
thf(fact_756_antisym,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_757_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_758_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_759_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_760_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_761_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: set_int,Z: set_int] : Y4 = Z )
    = ( ^ [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_762_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: rat,Z: rat] : Y4 = Z )
    = ( ^ [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
          & ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_763_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: num,Z: num] : Y4 = Z )
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
          & ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_764_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
    = ( ^ [A3: nat,B3: nat] :
          ( ( ord_less_eq_nat @ A3 @ B3 )
          & ( ord_less_eq_nat @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_765_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y4: int,Z: int] : Y4 = Z )
    = ( ^ [A3: int,B3: int] :
          ( ( ord_less_eq_int @ A3 @ B3 )
          & ( ord_less_eq_int @ B3 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_766_order__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_767_order__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_768_order__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_769_order__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X4: int,Y5: int] :
              ( ( ord_less_eq_int @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_770_order__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_771_order__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_772_order__subst1,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_773_order__subst1,axiom,
    ! [A: num,F: int > num,B: int,C: int] :
      ( ( ord_less_eq_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ! [X4: int,Y5: int] :
              ( ( ord_less_eq_int @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_774_order__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_775_order__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_eq_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_776_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_777_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_778_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_779_order__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_780_order__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_781_order__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_782_order__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_nat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_783_order__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_int @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_784_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_785_order__subst2,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_num @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_786_order__eq__refl,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_787_order__eq__refl,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_788_order__eq__refl,axiom,
    ! [X3: num,Y3: num] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_num @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_789_order__eq__refl,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_790_order__eq__refl,axiom,
    ! [X3: int,Y3: int] :
      ( ( X3 = Y3 )
     => ( ord_less_eq_int @ X3 @ Y3 ) ) ).

% order_eq_refl
thf(fact_791_linorder__linear,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
      | ( ord_less_eq_rat @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_792_linorder__linear,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
      | ( ord_less_eq_num @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_793_linorder__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_794_linorder__linear,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
      | ( ord_less_eq_int @ Y3 @ X3 ) ) ).

% linorder_linear
thf(fact_795_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_796_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_797_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_798_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_799_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_800_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_801_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_802_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_803_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_804_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_805_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_806_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_807_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_808_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_809_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_810_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_811_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_812_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_813_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_814_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_eq_nat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_815_linorder__le__cases,axiom,
    ! [X3: rat,Y3: rat] :
      ( ~ ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_816_linorder__le__cases,axiom,
    ! [X3: num,Y3: num] :
      ( ~ ( ord_less_eq_num @ X3 @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_817_linorder__le__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_818_linorder__le__cases,axiom,
    ! [X3: int,Y3: int] :
      ( ~ ( ord_less_eq_int @ X3 @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X3 ) ) ).

% linorder_le_cases
thf(fact_819_order__antisym__conv,axiom,
    ! [Y3: set_int,X3: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X3 )
     => ( ( ord_less_eq_set_int @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_820_order__antisym__conv,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_821_order__antisym__conv,axiom,
    ! [Y3: num,X3: num] :
      ( ( ord_less_eq_num @ Y3 @ X3 )
     => ( ( ord_less_eq_num @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_822_order__antisym__conv,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_823_order__antisym__conv,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_eq_int @ Y3 @ X3 )
     => ( ( ord_less_eq_int @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% order_antisym_conv
thf(fact_824_zero__reorient,axiom,
    ! [X3: complex] :
      ( ( zero_zero_complex = X3 )
      = ( X3 = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_825_zero__reorient,axiom,
    ! [X3: real] :
      ( ( zero_zero_real = X3 )
      = ( X3 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_826_zero__reorient,axiom,
    ! [X3: rat] :
      ( ( zero_zero_rat = X3 )
      = ( X3 = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_827_zero__reorient,axiom,
    ! [X3: nat] :
      ( ( zero_zero_nat = X3 )
      = ( X3 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_828_zero__reorient,axiom,
    ! [X3: int] :
      ( ( zero_zero_int = X3 )
      = ( X3 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_829_lt__ex,axiom,
    ! [X3: real] :
    ? [Y5: real] : ( ord_less_real @ Y5 @ X3 ) ).

% lt_ex
thf(fact_830_lt__ex,axiom,
    ! [X3: rat] :
    ? [Y5: rat] : ( ord_less_rat @ Y5 @ X3 ) ).

% lt_ex
thf(fact_831_lt__ex,axiom,
    ! [X3: int] :
    ? [Y5: int] : ( ord_less_int @ Y5 @ X3 ) ).

% lt_ex
thf(fact_832_gt__ex,axiom,
    ! [X3: real] :
    ? [X_1: real] : ( ord_less_real @ X3 @ X_1 ) ).

% gt_ex
thf(fact_833_gt__ex,axiom,
    ! [X3: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_834_gt__ex,axiom,
    ! [X3: nat] :
    ? [X_1: nat] : ( ord_less_nat @ X3 @ X_1 ) ).

% gt_ex
thf(fact_835_gt__ex,axiom,
    ! [X3: int] :
    ? [X_1: int] : ( ord_less_int @ X3 @ X_1 ) ).

% gt_ex
thf(fact_836_dense,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ? [Z3: real] :
          ( ( ord_less_real @ X3 @ Z3 )
          & ( ord_less_real @ Z3 @ Y3 ) ) ) ).

% dense
thf(fact_837_dense,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ? [Z3: rat] :
          ( ( ord_less_rat @ X3 @ Z3 )
          & ( ord_less_rat @ Z3 @ Y3 ) ) ) ).

% dense
thf(fact_838_less__imp__neq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_839_less__imp__neq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_840_less__imp__neq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_841_less__imp__neq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_842_less__imp__neq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% less_imp_neq
thf(fact_843_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_844_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_845_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_846_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_847_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_848_ord__eq__less__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A = B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_849_ord__eq__less__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A = B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_850_ord__eq__less__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( A = B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_851_ord__eq__less__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A = B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_852_ord__eq__less__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A = B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_eq_less_trans
thf(fact_853_ord__less__eq__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( B = C )
       => ( ord_less_real @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_854_ord__less__eq__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( B = C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_855_ord__less__eq__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( B = C )
       => ( ord_less_num @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_856_ord__less__eq__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( B = C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_857_ord__less__eq__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( B = C )
       => ( ord_less_int @ A @ C ) ) ) ).

% ord_less_eq_trans
thf(fact_858_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X4: nat] :
          ( ! [Y6: nat] :
              ( ( ord_less_nat @ Y6 @ X4 )
             => ( P @ Y6 ) )
         => ( P @ X4 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_859_antisym__conv3,axiom,
    ! [Y3: real,X3: real] :
      ( ~ ( ord_less_real @ Y3 @ X3 )
     => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_860_antisym__conv3,axiom,
    ! [Y3: rat,X3: rat] :
      ( ~ ( ord_less_rat @ Y3 @ X3 )
     => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_861_antisym__conv3,axiom,
    ! [Y3: num,X3: num] :
      ( ~ ( ord_less_num @ Y3 @ X3 )
     => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_862_antisym__conv3,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_nat @ Y3 @ X3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_863_antisym__conv3,axiom,
    ! [Y3: int,X3: int] :
      ( ~ ( ord_less_int @ Y3 @ X3 )
     => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv3
thf(fact_864_linorder__cases,axiom,
    ! [X3: real,Y3: real] :
      ( ~ ( ord_less_real @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_real @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_865_linorder__cases,axiom,
    ! [X3: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_rat @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_866_linorder__cases,axiom,
    ! [X3: num,Y3: num] :
      ( ~ ( ord_less_num @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_num @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_867_linorder__cases,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_868_linorder__cases,axiom,
    ! [X3: int,Y3: int] :
      ( ~ ( ord_less_int @ X3 @ Y3 )
     => ( ( X3 != Y3 )
       => ( ord_less_int @ Y3 @ X3 ) ) ) ).

% linorder_cases
thf(fact_869_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_870_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_871_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_872_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_873_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_874_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_875_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_876_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_877_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_878_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_879_exists__least__iff,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X6: nat] : ( P2 @ X6 ) )
    = ( ^ [P3: nat > $o] :
        ? [N: nat] :
          ( ( P3 @ N )
          & ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N )
             => ~ ( P3 @ M2 ) ) ) ) ) ).

% exists_least_iff
thf(fact_880_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: 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 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_881_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A4: rat,B4: rat] :
          ( ( ord_less_rat @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: rat] : ( P @ A4 @ A4 )
       => ( ! [A4: rat,B4: rat] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_882_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A4: num,B4: num] :
          ( ( ord_less_num @ A4 @ B4 )
         => ( P @ A4 @ B4 ) )
     => ( ! [A4: num] : ( P @ A4 @ A4 )
       => ( ! [A4: num,B4: num] :
              ( ( P @ B4 @ A4 )
             => ( P @ A4 @ B4 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_883_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: 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 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_884_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: 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 @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_885_order_Ostrict__trans,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_886_order_Ostrict__trans,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_887_order_Ostrict__trans,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_888_order_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_889_order_Ostrict__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans
thf(fact_890_not__less__iff__gr__or__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
      = ( ( ord_less_real @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_891_not__less__iff__gr__or__eq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
      = ( ( ord_less_rat @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_892_not__less__iff__gr__or__eq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
      = ( ( ord_less_num @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_893_not__less__iff__gr__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ( ord_less_nat @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_894_not__less__iff__gr__or__eq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
      = ( ( ord_less_int @ Y3 @ X3 )
        | ( X3 = Y3 ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_895_dual__order_Ostrict__trans,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_896_dual__order_Ostrict__trans,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_897_dual__order_Ostrict__trans,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_898_dual__order_Ostrict__trans,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_899_dual__order_Ostrict__trans,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans
thf(fact_900_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_901_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_902_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_903_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_904_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_905_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_906_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_907_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_908_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_909_dual__order_Ostrict__implies__not__eq,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( A != B ) ) ).

% dual_order.strict_implies_not_eq
thf(fact_910_linorder__neqE,axiom,
    ! [X3: real,Y3: real] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_real @ X3 @ Y3 )
       => ( ord_less_real @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_911_linorder__neqE,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_rat @ X3 @ Y3 )
       => ( ord_less_rat @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_912_linorder__neqE,axiom,
    ! [X3: num,Y3: num] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_num @ X3 @ Y3 )
       => ( ord_less_num @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_913_linorder__neqE,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_914_linorder__neqE,axiom,
    ! [X3: int,Y3: int] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_int @ X3 @ Y3 )
       => ( ord_less_int @ Y3 @ X3 ) ) ) ).

% linorder_neqE
thf(fact_915_order__less__asym,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_916_order__less__asym,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_917_order__less__asym,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_918_order__less__asym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_919_order__less__asym,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X3 ) ) ).

% order_less_asym
thf(fact_920_linorder__neq__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( X3 != Y3 )
      = ( ( ord_less_real @ X3 @ Y3 )
        | ( ord_less_real @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_921_linorder__neq__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( X3 != Y3 )
      = ( ( ord_less_rat @ X3 @ Y3 )
        | ( ord_less_rat @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_922_linorder__neq__iff,axiom,
    ! [X3: num,Y3: num] :
      ( ( X3 != Y3 )
      = ( ( ord_less_num @ X3 @ Y3 )
        | ( ord_less_num @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_923_linorder__neq__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
      = ( ( ord_less_nat @ X3 @ Y3 )
        | ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_924_linorder__neq__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( X3 != Y3 )
      = ( ( ord_less_int @ X3 @ Y3 )
        | ( ord_less_int @ Y3 @ X3 ) ) ) ).

% linorder_neq_iff
thf(fact_925_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_926_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_927_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_928_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_929_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_930_order__less__trans,axiom,
    ! [X3: real,Y3: real,Z2: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ( ord_less_real @ Y3 @ Z2 )
       => ( ord_less_real @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_931_order__less__trans,axiom,
    ! [X3: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z2 )
       => ( ord_less_rat @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_932_order__less__trans,axiom,
    ! [X3: num,Y3: num,Z2: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z2 )
       => ( ord_less_num @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_933_order__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_934_order__less__trans,axiom,
    ! [X3: int,Y3: int,Z2: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z2 )
       => ( ord_less_int @ X3 @ Z2 ) ) ) ).

% order_less_trans
thf(fact_935_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_936_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_937_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_938_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_939_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_940_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_941_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_942_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_943_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_944_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_945_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_946_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_947_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_948_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_949_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_950_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_951_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_952_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_953_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_954_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_955_order__less__irrefl,axiom,
    ! [X3: real] :
      ~ ( ord_less_real @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_956_order__less__irrefl,axiom,
    ! [X3: rat] :
      ~ ( ord_less_rat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_957_order__less__irrefl,axiom,
    ! [X3: num] :
      ~ ( ord_less_num @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_958_order__less__irrefl,axiom,
    ! [X3: nat] :
      ~ ( ord_less_nat @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_959_order__less__irrefl,axiom,
    ! [X3: int] :
      ~ ( ord_less_int @ X3 @ X3 ) ).

% order_less_irrefl
thf(fact_960_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_961_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_962_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_num @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_963_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_964_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y5: int] :
              ( ( ord_less_int @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_965_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_966_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_967_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_num @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_968_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_969_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y5: int] :
              ( ( ord_less_int @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_970_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_971_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_972_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_973_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_974_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_975_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_976_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_977_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_978_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_979_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_980_order__less__not__sym,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_981_order__less__not__sym,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_982_order__less__not__sym,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_983_order__less__not__sym,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_984_order__less__not__sym,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X3 ) ) ).

% order_less_not_sym
thf(fact_985_order__less__imp__triv,axiom,
    ! [X3: real,Y3: real,P: $o] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_986_order__less__imp__triv,axiom,
    ! [X3: rat,Y3: rat,P: $o] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_987_order__less__imp__triv,axiom,
    ! [X3: num,Y3: num,P: $o] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( ( ord_less_num @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_988_order__less__imp__triv,axiom,
    ! [X3: nat,Y3: nat,P: $o] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_989_order__less__imp__triv,axiom,
    ! [X3: int,Y3: int,P: $o] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( ( ord_less_int @ Y3 @ X3 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_990_linorder__less__linear,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_real @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_991_linorder__less__linear,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_rat @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_992_linorder__less__linear,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_num @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_993_linorder__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_994_linorder__less__linear,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
      | ( X3 = Y3 )
      | ( ord_less_int @ Y3 @ X3 ) ) ).

% linorder_less_linear
thf(fact_995_order__less__imp__not__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_996_order__less__imp__not__eq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_997_order__less__imp__not__eq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_998_order__less__imp__not__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_999_order__less__imp__not__eq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( X3 != Y3 ) ) ).

% order_less_imp_not_eq
thf(fact_1000_order__less__imp__not__eq2,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1001_order__less__imp__not__eq2,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1002_order__less__imp__not__eq2,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1003_order__less__imp__not__eq2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1004_order__less__imp__not__eq2,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( Y3 != X3 ) ) ).

% order_less_imp_not_eq2
thf(fact_1005_order__less__imp__not__less,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ~ ( ord_less_real @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1006_order__less__imp__not__less,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ~ ( ord_less_rat @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1007_order__less__imp__not__less,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ~ ( ord_less_num @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1008_order__less__imp__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ~ ( ord_less_nat @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1009_order__less__imp__not__less,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ~ ( ord_less_int @ Y3 @ X3 ) ) ).

% order_less_imp_not_less
thf(fact_1010_nat__neq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( M != N2 )
      = ( ( ord_less_nat @ M @ N2 )
        | ( ord_less_nat @ N2 @ M ) ) ) ).

% nat_neq_iff
thf(fact_1011_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_1012_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_1013_less__not__refl3,axiom,
    ! [S: nat,T: nat] :
      ( ( ord_less_nat @ S @ T )
     => ( S != T ) ) ).

% less_not_refl3
thf(fact_1014_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_1015_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M6: nat] :
              ( ( ord_less_nat @ M6 @ N3 )
             => ( P @ M6 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_1016_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M6: nat] :
              ( ( ord_less_nat @ M6 @ N3 )
              & ~ ( P @ M6 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_1017_linorder__neqE__nat,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_nat @ X3 @ Y3 )
       => ( ord_less_nat @ Y3 @ X3 ) ) ) ).

% linorder_neqE_nat
thf(fact_1018_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_1019_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_1020_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_1021_le__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% le_antisym
thf(fact_1022_nat__le__linear,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
      | ( ord_less_eq_nat @ N2 @ M ) ) ).

% nat_le_linear
thf(fact_1023_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ? [X4: nat] :
            ( ( P @ X4 )
            & ! [Y6: nat] :
                ( ( P @ Y6 )
               => ( ord_less_eq_nat @ Y6 @ X4 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_1024_size__neq__size__imp__neq,axiom,
    ! [X3: list_VEBT_VEBT,Y3: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X3 )
       != ( size_s6755466524823107622T_VEBT @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1025_size__neq__size__imp__neq,axiom,
    ! [X3: list_o,Y3: list_o] :
      ( ( ( size_size_list_o @ X3 )
       != ( size_size_list_o @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1026_size__neq__size__imp__neq,axiom,
    ! [X3: list_nat,Y3: list_nat] :
      ( ( ( size_size_list_nat @ X3 )
       != ( size_size_list_nat @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1027_size__neq__size__imp__neq,axiom,
    ! [X3: list_int,Y3: list_int] :
      ( ( ( size_size_list_int @ X3 )
       != ( size_size_list_int @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1028_size__neq__size__imp__neq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ( size_size_num @ X3 )
       != ( size_size_num @ Y3 ) )
     => ( X3 != Y3 ) ) ).

% size_neq_size_imp_neq
thf(fact_1029_zero__le,axiom,
    ! [X3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X3 ) ).

% zero_le
thf(fact_1030_leD,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ Y3 @ X3 )
     => ~ ( ord_less_real @ X3 @ Y3 ) ) ).

% leD
thf(fact_1031_leD,axiom,
    ! [Y3: set_int,X3: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ X3 )
     => ~ ( ord_less_set_int @ X3 @ Y3 ) ) ).

% leD
thf(fact_1032_leD,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X3 )
     => ~ ( ord_less_rat @ X3 @ Y3 ) ) ).

% leD
thf(fact_1033_leD,axiom,
    ! [Y3: num,X3: num] :
      ( ( ord_less_eq_num @ Y3 @ X3 )
     => ~ ( ord_less_num @ X3 @ Y3 ) ) ).

% leD
thf(fact_1034_leD,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_eq_nat @ Y3 @ X3 )
     => ~ ( ord_less_nat @ X3 @ Y3 ) ) ).

% leD
thf(fact_1035_leD,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_eq_int @ Y3 @ X3 )
     => ~ ( ord_less_int @ X3 @ Y3 ) ) ).

% leD
thf(fact_1036_leI,axiom,
    ! [X3: real,Y3: real] :
      ( ~ ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_eq_real @ Y3 @ X3 ) ) ).

% leI
thf(fact_1037_leI,axiom,
    ! [X3: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X3 @ Y3 )
     => ( ord_less_eq_rat @ Y3 @ X3 ) ) ).

% leI
thf(fact_1038_leI,axiom,
    ! [X3: num,Y3: num] :
      ( ~ ( ord_less_num @ X3 @ Y3 )
     => ( ord_less_eq_num @ Y3 @ X3 ) ) ).

% leI
thf(fact_1039_leI,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% leI
thf(fact_1040_leI,axiom,
    ! [X3: int,Y3: int] :
      ( ~ ( ord_less_int @ X3 @ Y3 )
     => ( ord_less_eq_int @ Y3 @ X3 ) ) ).

% leI
thf(fact_1041_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1042_nless__le,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ~ ( ord_less_set_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1043_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1044_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1045_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1046_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_1047_antisym__conv1,axiom,
    ! [X3: real,Y3: real] :
      ( ~ ( ord_less_real @ X3 @ Y3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1048_antisym__conv1,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ~ ( ord_less_set_int @ X3 @ Y3 )
     => ( ( ord_less_eq_set_int @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1049_antisym__conv1,axiom,
    ! [X3: rat,Y3: rat] :
      ( ~ ( ord_less_rat @ X3 @ Y3 )
     => ( ( ord_less_eq_rat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1050_antisym__conv1,axiom,
    ! [X3: num,Y3: num] :
      ( ~ ( ord_less_num @ X3 @ Y3 )
     => ( ( ord_less_eq_num @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1051_antisym__conv1,axiom,
    ! [X3: nat,Y3: nat] :
      ( ~ ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1052_antisym__conv1,axiom,
    ! [X3: int,Y3: int] :
      ( ~ ( ord_less_int @ X3 @ Y3 )
     => ( ( ord_less_eq_int @ X3 @ Y3 )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv1
thf(fact_1053_antisym__conv2,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
     => ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1054_antisym__conv2,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ~ ( ord_less_set_int @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1055_antisym__conv2,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1056_antisym__conv2,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1057_antisym__conv2,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1058_antisym__conv2,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% antisym_conv2
thf(fact_1059_dense__ge,axiom,
    ! [Z2: real,Y3: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ Z2 @ X4 )
         => ( ord_less_eq_real @ Y3 @ X4 ) )
     => ( ord_less_eq_real @ Y3 @ Z2 ) ) ).

% dense_ge
thf(fact_1060_dense__ge,axiom,
    ! [Z2: rat,Y3: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_rat @ Z2 @ X4 )
         => ( ord_less_eq_rat @ Y3 @ X4 ) )
     => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ).

% dense_ge
thf(fact_1061_dense__le,axiom,
    ! [Y3: real,Z2: real] :
      ( ! [X4: real] :
          ( ( ord_less_real @ X4 @ Y3 )
         => ( ord_less_eq_real @ X4 @ Z2 ) )
     => ( ord_less_eq_real @ Y3 @ Z2 ) ) ).

% dense_le
thf(fact_1062_dense__le,axiom,
    ! [Y3: rat,Z2: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Y3 )
         => ( ord_less_eq_rat @ X4 @ Z2 ) )
     => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ).

% dense_le
thf(fact_1063_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ~ ( ord_less_eq_real @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1064_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y )
          & ~ ( ord_less_eq_set_int @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1065_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ~ ( ord_less_eq_rat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1066_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ~ ( ord_less_eq_num @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1067_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ~ ( ord_less_eq_nat @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1068_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ~ ( ord_less_eq_int @ Y @ X ) ) ) ) ).

% less_le_not_le
thf(fact_1069_not__le__imp__less,axiom,
    ! [Y3: real,X3: real] :
      ( ~ ( ord_less_eq_real @ Y3 @ X3 )
     => ( ord_less_real @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1070_not__le__imp__less,axiom,
    ! [Y3: rat,X3: rat] :
      ( ~ ( ord_less_eq_rat @ Y3 @ X3 )
     => ( ord_less_rat @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1071_not__le__imp__less,axiom,
    ! [Y3: num,X3: num] :
      ( ~ ( ord_less_eq_num @ Y3 @ X3 )
     => ( ord_less_num @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1072_not__le__imp__less,axiom,
    ! [Y3: nat,X3: nat] :
      ( ~ ( ord_less_eq_nat @ Y3 @ X3 )
     => ( ord_less_nat @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1073_not__le__imp__less,axiom,
    ! [Y3: int,X3: int] :
      ( ~ ( ord_less_eq_int @ Y3 @ X3 )
     => ( ord_less_int @ X3 @ Y3 ) ) ).

% not_le_imp_less
thf(fact_1074_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_1075_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_1076_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B3: rat] :
          ( ( ord_less_rat @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1077_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_num @ A3 @ B3 )
          | ( A3 = B3 ) ) ) ) ).

% order.order_iff_strict
thf(fact_1078_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_1079_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_1080_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_1081_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_1082_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1083_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
          & ( A3 != B3 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1084_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_1085_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_1086_order_Ostrict__trans1,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1087_order_Ostrict__trans1,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( ord_less_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1088_order_Ostrict__trans1,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1089_order_Ostrict__trans1,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1090_order_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1091_order_Ostrict__trans1,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans1
thf(fact_1092_order_Ostrict__trans2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ B @ C )
       => ( ord_less_real @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1093_order_Ostrict__trans2,axiom,
    ! [A: set_int,B: set_int,C: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ( ord_less_eq_set_int @ B @ C )
       => ( ord_less_set_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1094_order_Ostrict__trans2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ord_less_rat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1095_order_Ostrict__trans2,axiom,
    ! [A: num,B: num,C: num] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ord_less_num @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1096_order_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ord_less_nat @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1097_order_Ostrict__trans2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ C )
       => ( ord_less_int @ A @ C ) ) ) ).

% order.strict_trans2
thf(fact_1098_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_1099_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_1100_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B3: rat] :
          ( ( ord_less_eq_rat @ A3 @ B3 )
          & ~ ( ord_less_eq_rat @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1101_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B3: num] :
          ( ( ord_less_eq_num @ A3 @ B3 )
          & ~ ( ord_less_eq_num @ B3 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1102_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_1103_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_1104_dense__ge__bounded,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ Z2 @ X3 )
     => ( ! [W: real] :
            ( ( ord_less_real @ Z2 @ W )
           => ( ( ord_less_real @ W @ X3 )
             => ( ord_less_eq_real @ Y3 @ W ) ) )
       => ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_1105_dense__ge__bounded,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( ord_less_rat @ Z2 @ X3 )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ Z2 @ W )
           => ( ( ord_less_rat @ W @ X3 )
             => ( ord_less_eq_rat @ Y3 @ W ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ) ).

% dense_ge_bounded
thf(fact_1106_dense__le__bounded,axiom,
    ! [X3: real,Y3: real,Z2: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ! [W: real] :
            ( ( ord_less_real @ X3 @ W )
           => ( ( ord_less_real @ W @ Y3 )
             => ( ord_less_eq_real @ W @ Z2 ) ) )
       => ( ord_less_eq_real @ Y3 @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_1107_dense__le__bounded,axiom,
    ! [X3: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ! [W: rat] :
            ( ( ord_less_rat @ X3 @ W )
           => ( ( ord_less_rat @ W @ Y3 )
             => ( ord_less_eq_rat @ W @ Z2 ) ) )
       => ( ord_less_eq_rat @ Y3 @ Z2 ) ) ) ).

% dense_le_bounded
thf(fact_1108_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_1109_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_1110_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B3: rat,A3: rat] :
          ( ( ord_less_rat @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1111_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B3: num,A3: num] :
          ( ( ord_less_num @ B3 @ A3 )
          | ( A3 = B3 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1112_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_1113_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_1114_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_1115_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_1116_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B3: rat,A3: rat] :
          ( ( ord_less_eq_rat @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1117_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B3: num,A3: num] :
          ( ( ord_less_eq_num @ B3 @ A3 )
          & ( A3 != B3 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1118_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_1119_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_1120_dual__order_Ostrict__trans1,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1121_dual__order_Ostrict__trans1,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_eq_set_int @ B @ A )
     => ( ( ord_less_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1122_dual__order_Ostrict__trans1,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1123_dual__order_Ostrict__trans1,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_eq_num @ B @ A )
     => ( ( ord_less_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1124_dual__order_Ostrict__trans1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( ord_less_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1125_dual__order_Ostrict__trans1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans1
thf(fact_1126_dual__order_Ostrict__trans2,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ B )
       => ( ord_less_real @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1127_dual__order_Ostrict__trans2,axiom,
    ! [B: set_int,A: set_int,C: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ( ord_less_eq_set_int @ C @ B )
       => ( ord_less_set_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1128_dual__order_Ostrict__trans2,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ B )
       => ( ord_less_rat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1129_dual__order_Ostrict__trans2,axiom,
    ! [B: num,A: num,C: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( ord_less_eq_num @ C @ B )
       => ( ord_less_num @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1130_dual__order_Ostrict__trans2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( ord_less_eq_nat @ C @ B )
       => ( ord_less_nat @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1131_dual__order_Ostrict__trans2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ B )
       => ( ord_less_int @ C @ A ) ) ) ).

% dual_order.strict_trans2
thf(fact_1132_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_1133_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_1134_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B3: rat,A3: rat] :
          ( ( ord_less_eq_rat @ B3 @ A3 )
          & ~ ( ord_less_eq_rat @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1135_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B3: num,A3: num] :
          ( ( ord_less_eq_num @ B3 @ A3 )
          & ~ ( ord_less_eq_num @ A3 @ B3 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1136_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_1137_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_1138_order_Ostrict__implies__order,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1139_order_Ostrict__implies__order,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_set_int @ A @ B )
     => ( ord_less_eq_set_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1140_order_Ostrict__implies__order,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1141_order_Ostrict__implies__order,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( ord_less_eq_num @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1142_order_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_eq_nat @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1143_order_Ostrict__implies__order,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% order.strict_implies_order
thf(fact_1144_dual__order_Ostrict__implies__order,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_eq_real @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1145_dual__order_Ostrict__implies__order,axiom,
    ! [B: set_int,A: set_int] :
      ( ( ord_less_set_int @ B @ A )
     => ( ord_less_eq_set_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1146_dual__order_Ostrict__implies__order,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_eq_rat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1147_dual__order_Ostrict__implies__order,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ord_less_eq_num @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1148_dual__order_Ostrict__implies__order,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ord_less_eq_nat @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1149_dual__order_Ostrict__implies__order,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_eq_int @ B @ A ) ) ).

% dual_order.strict_implies_order
thf(fact_1150_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_1151_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_set_int @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_1152_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_1153_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_num @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_1154_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_nat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_1155_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_int @ X @ Y )
          | ( X = Y ) ) ) ) ).

% order_le_less
thf(fact_1156_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_eq_real @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_1157_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_1158_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_eq_rat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_1159_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y: num] :
          ( ( ord_less_eq_num @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_1160_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y: nat] :
          ( ( ord_less_eq_nat @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_1161_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y: int] :
          ( ( ord_less_eq_int @ X @ Y )
          & ( X != Y ) ) ) ) ).

% order_less_le
thf(fact_1162_linorder__not__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ~ ( ord_less_eq_real @ X3 @ Y3 ) )
      = ( ord_less_real @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_1163_linorder__not__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ~ ( ord_less_eq_rat @ X3 @ Y3 ) )
      = ( ord_less_rat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_1164_linorder__not__le,axiom,
    ! [X3: num,Y3: num] :
      ( ( ~ ( ord_less_eq_num @ X3 @ Y3 ) )
      = ( ord_less_num @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_1165_linorder__not__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_eq_nat @ X3 @ Y3 ) )
      = ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_1166_linorder__not__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ~ ( ord_less_eq_int @ X3 @ Y3 ) )
      = ( ord_less_int @ Y3 @ X3 ) ) ).

% linorder_not_le
thf(fact_1167_linorder__not__less,axiom,
    ! [X3: real,Y3: real] :
      ( ( ~ ( ord_less_real @ X3 @ Y3 ) )
      = ( ord_less_eq_real @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_1168_linorder__not__less,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ~ ( ord_less_rat @ X3 @ Y3 ) )
      = ( ord_less_eq_rat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_1169_linorder__not__less,axiom,
    ! [X3: num,Y3: num] :
      ( ( ~ ( ord_less_num @ X3 @ Y3 ) )
      = ( ord_less_eq_num @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_1170_linorder__not__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ~ ( ord_less_nat @ X3 @ Y3 ) )
      = ( ord_less_eq_nat @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_1171_linorder__not__less,axiom,
    ! [X3: int,Y3: int] :
      ( ( ~ ( ord_less_int @ X3 @ Y3 ) )
      = ( ord_less_eq_int @ Y3 @ X3 ) ) ).

% linorder_not_less
thf(fact_1172_order__less__imp__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1173_order__less__imp__le,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_set_int @ X3 @ Y3 )
     => ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1174_order__less__imp__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1175_order__less__imp__le,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( ord_less_eq_num @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1176_order__less__imp__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1177_order__less__imp__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( ord_less_eq_int @ X3 @ Y3 ) ) ).

% order_less_imp_le
thf(fact_1178_order__le__neq__trans,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( A != B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1179_order__le__neq__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ord_less_eq_set_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1180_order__le__neq__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( A != B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1181_order__le__neq__trans,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( A != B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1182_order__le__neq__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( A != B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1183_order__le__neq__trans,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( A != B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_le_neq_trans
thf(fact_1184_order__neq__le__trans,axiom,
    ! [A: real,B: real] :
      ( ( A != B )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1185_order__neq__le__trans,axiom,
    ! [A: set_int,B: set_int] :
      ( ( A != B )
     => ( ( ord_less_eq_set_int @ A @ B )
       => ( ord_less_set_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1186_order__neq__le__trans,axiom,
    ! [A: rat,B: rat] :
      ( ( A != B )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1187_order__neq__le__trans,axiom,
    ! [A: num,B: num] :
      ( ( A != B )
     => ( ( ord_less_eq_num @ A @ B )
       => ( ord_less_num @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1188_order__neq__le__trans,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1189_order__neq__le__trans,axiom,
    ! [A: int,B: int] :
      ( ( A != B )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% order_neq_le_trans
thf(fact_1190_order__le__less__trans,axiom,
    ! [X3: real,Y3: real,Z2: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
     => ( ( ord_less_real @ Y3 @ Z2 )
       => ( ord_less_real @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1191_order__le__less__trans,axiom,
    ! [X3: set_int,Y3: set_int,Z2: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ord_less_set_int @ Y3 @ Z2 )
       => ( ord_less_set_int @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1192_order__le__less__trans,axiom,
    ! [X3: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ord_less_rat @ Y3 @ Z2 )
       => ( ord_less_rat @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1193_order__le__less__trans,axiom,
    ! [X3: num,Y3: num,Z2: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ord_less_num @ Y3 @ Z2 )
       => ( ord_less_num @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1194_order__le__less__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1195_order__le__less__trans,axiom,
    ! [X3: int,Y3: int,Z2: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z2 )
       => ( ord_less_int @ X3 @ Z2 ) ) ) ).

% order_le_less_trans
thf(fact_1196_order__less__le__trans,axiom,
    ! [X3: real,Y3: real,Z2: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ( ord_less_eq_real @ Y3 @ Z2 )
       => ( ord_less_real @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1197_order__less__le__trans,axiom,
    ! [X3: set_int,Y3: set_int,Z2: set_int] :
      ( ( ord_less_set_int @ X3 @ Y3 )
     => ( ( ord_less_eq_set_int @ Y3 @ Z2 )
       => ( ord_less_set_int @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1198_order__less__le__trans,axiom,
    ! [X3: rat,Y3: rat,Z2: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ( ord_less_eq_rat @ Y3 @ Z2 )
       => ( ord_less_rat @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1199_order__less__le__trans,axiom,
    ! [X3: num,Y3: num,Z2: num] :
      ( ( ord_less_num @ X3 @ Y3 )
     => ( ( ord_less_eq_num @ Y3 @ Z2 )
       => ( ord_less_num @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1200_order__less__le__trans,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ord_less_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ Y3 @ Z2 )
       => ( ord_less_nat @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1201_order__less__le__trans,axiom,
    ! [X3: int,Y3: int,Z2: int] :
      ( ( ord_less_int @ X3 @ Y3 )
     => ( ( ord_less_eq_int @ Y3 @ Z2 )
       => ( ord_less_int @ X3 @ Z2 ) ) ) ).

% order_less_le_trans
thf(fact_1202_order__le__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1203_order__le__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1204_order__le__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_num @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1205_order__le__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1206_order__le__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_eq_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y5: int] :
              ( ( ord_less_int @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1207_order__le__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1208_order__le__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1209_order__le__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_num @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1210_order__le__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1211_order__le__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_eq_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X4: int,Y5: int] :
              ( ( ord_less_int @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1212_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1213_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1214_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1215_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1216_order__le__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1217_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1218_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1219_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1220_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1221_order__le__less__subst2,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1222_order__less__le__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1223_order__less__le__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1224_order__less__le__subst1,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1225_order__less__le__subst1,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1226_order__less__le__subst1,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_eq_rat @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1227_order__less__le__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1228_order__less__le__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1229_order__less__le__subst1,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( ord_less_num @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_num @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1230_order__less__le__subst1,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( ord_less_nat @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_nat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1231_order__less__le__subst1,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( ord_less_int @ A @ ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_eq_num @ X4 @ Y5 )
             => ( ord_less_eq_int @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1232_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1233_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1234_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > real,C: real] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_num @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1235_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > real,C: real] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1236_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > real,C: real] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_real @ ( F @ B ) @ C )
       => ( ! [X4: int,Y5: int] :
              ( ( ord_less_int @ X4 @ Y5 )
             => ( ord_less_real @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1237_order__less__le__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: real,Y5: real] :
              ( ( ord_less_real @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1238_order__less__le__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: rat,Y5: rat] :
              ( ( ord_less_rat @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1239_order__less__le__subst2,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_num @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: num,Y5: num] :
              ( ( ord_less_num @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1240_order__less__le__subst2,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: nat,Y5: nat] :
              ( ( ord_less_nat @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1241_order__less__le__subst2,axiom,
    ! [A: int,B: int,F: int > rat,C: rat] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_rat @ ( F @ B ) @ C )
       => ( ! [X4: int,Y5: int] :
              ( ( ord_less_int @ X4 @ Y5 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( F @ Y5 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1242_linorder__le__less__linear,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
      | ( ord_less_real @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1243_linorder__le__less__linear,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
      | ( ord_less_rat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1244_linorder__le__less__linear,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
      | ( ord_less_num @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1245_linorder__le__less__linear,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
      | ( ord_less_nat @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1246_linorder__le__less__linear,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
      | ( ord_less_int @ Y3 @ X3 ) ) ).

% linorder_le_less_linear
thf(fact_1247_order__le__imp__less__or__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
     => ( ( ord_less_real @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1248_order__le__imp__less__or__eq,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ( ord_less_set_int @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1249_order__le__imp__less__or__eq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ( ord_less_rat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1250_order__le__imp__less__or__eq,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_num @ X3 @ Y3 )
     => ( ( ord_less_num @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1251_order__le__imp__less__or__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ X3 @ Y3 )
     => ( ( ord_less_nat @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1252_order__le__imp__less__or__eq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ( ord_less_int @ X3 @ Y3 )
        | ( X3 = Y3 ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1253_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_1254_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1255_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1256_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_1257_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1258_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_1259_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1260_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_1261_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1262_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1263_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M6: nat] :
                  ( ( ord_less_nat @ M6 @ N3 )
                  & ~ ( P @ M6 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_1264_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_1265_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_1266_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_1267_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1268_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M2: nat,N: nat] :
          ( ( ord_less_eq_nat @ M2 @ N )
          & ( M2 != N ) ) ) ) ).

% nat_less_le
thf(fact_1269_less__imp__le__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_imp_le_nat
thf(fact_1270_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M2: nat,N: nat] :
          ( ( ord_less_nat @ M2 @ N )
          | ( M2 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_1271_less__or__eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_nat @ M @ N2 )
        | ( M = N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% less_or_eq_imp_le
thf(fact_1272_le__neq__implies__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( M != N2 )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% le_neq_implies_less
thf(fact_1273_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I: nat,J: nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_nat @ I3 @ J2 )
         => ( ord_less_nat @ ( F @ I3 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I @ J )
       => ( ord_less_eq_nat @ ( F @ I ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_1274_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_1275_double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( plus_plus_rat @ A @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% double_eq_0_iff
thf(fact_1276_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_1277_assms_I1_J,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) ) @ deg @ treeList @ summary ) @ deg ).

% assms(1)
thf(fact_1278_mint__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_mint @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% mint_corr_help_empty
thf(fact_1279_maxt__corr__help__empty,axiom,
    ! [T: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T @ N2 )
     => ( ( ( vEBT_vebt_maxt @ T )
          = none_nat )
       => ( ( vEBT_VEBT_set_vebt @ T )
          = bot_bot_set_nat ) ) ) ).

% maxt_corr_help_empty
thf(fact_1280_field__le__epsilon,axiom,
    ! [X3: real,Y3: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X3 @ ( plus_plus_real @ Y3 @ E ) ) )
     => ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% field_le_epsilon
thf(fact_1281_field__le__epsilon,axiom,
    ! [X3: rat,Y3: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ Y3 @ E ) ) )
     => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% field_le_epsilon
thf(fact_1282_option_Osize__gen_I2_J,axiom,
    ! [X3: nat > nat,X22: nat] :
      ( ( size_option_nat @ X3 @ ( some_nat @ X22 ) )
      = ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1283_option_Osize__gen_I2_J,axiom,
    ! [X3: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X3 @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1284_option_Osize__gen_I2_J,axiom,
    ! [X3: num > nat,X22: num] :
      ( ( size_option_num @ X3 @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X3 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_1285_add__less__zeroD,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
        | ( ord_less_real @ Y3 @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_1286_add__less__zeroD,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X3 @ zero_zero_rat )
        | ( ord_less_rat @ Y3 @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_1287_add__less__zeroD,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X3 @ Y3 ) @ zero_zero_int )
     => ( ( ord_less_int @ X3 @ zero_zero_int )
        | ( ord_less_int @ Y3 @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_1288_empty__subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ bot_bo2099793752762293965at_nat @ A2 ) ).

% empty_subsetI
thf(fact_1289_empty__subsetI,axiom,
    ! [A2: set_real] : ( ord_less_eq_set_real @ bot_bot_set_real @ A2 ) ).

% empty_subsetI
thf(fact_1290_empty__subsetI,axiom,
    ! [A2: set_nat] : ( ord_less_eq_set_nat @ bot_bot_set_nat @ A2 ) ).

% empty_subsetI
thf(fact_1291_empty__subsetI,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ bot_bot_set_int @ A2 ) ).

% empty_subsetI
thf(fact_1292_subset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% subset_empty
thf(fact_1293_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_1294_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_1295_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_1296_empty__iff,axiom,
    ! [C: set_nat] :
      ~ ( member_set_nat @ C @ bot_bot_set_set_nat ) ).

% empty_iff
thf(fact_1297_empty__iff,axiom,
    ! [C: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ C @ bot_bo2099793752762293965at_nat ) ).

% empty_iff
thf(fact_1298_empty__iff,axiom,
    ! [C: real] :
      ~ ( member_real @ C @ bot_bot_set_real ) ).

% empty_iff
thf(fact_1299_empty__iff,axiom,
    ! [C: nat] :
      ~ ( member_nat @ C @ bot_bot_set_nat ) ).

% empty_iff
thf(fact_1300_empty__iff,axiom,
    ! [C: int] :
      ~ ( member_int @ C @ bot_bot_set_int ) ).

% empty_iff
thf(fact_1301_all__not__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ! [X: set_nat] :
            ~ ( member_set_nat @ X @ A2 ) )
      = ( A2 = bot_bot_set_set_nat ) ) ).

% all_not_in_conv
thf(fact_1302_all__not__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ! [X: product_prod_nat_nat] :
            ~ ( member8440522571783428010at_nat @ X @ A2 ) )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% all_not_in_conv
thf(fact_1303_all__not__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ! [X: real] :
            ~ ( member_real @ X @ A2 ) )
      = ( A2 = bot_bot_set_real ) ) ).

% all_not_in_conv
thf(fact_1304_all__not__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ! [X: nat] :
            ~ ( member_nat @ X @ A2 ) )
      = ( A2 = bot_bot_set_nat ) ) ).

% all_not_in_conv
thf(fact_1305_all__not__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ! [X: int] :
            ~ ( member_int @ X @ A2 ) )
      = ( A2 = bot_bot_set_int ) ) ).

% all_not_in_conv
thf(fact_1306_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_1307_deg__SUcn__Node,axiom,
    ! [Tree: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ Tree @ ( suc @ ( suc @ N2 ) ) )
     => ? [Info2: option4927543243414619207at_nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT] :
          ( Tree
          = ( vEBT_Node @ Info2 @ ( suc @ ( suc @ N2 ) ) @ TreeList3 @ S2 ) ) ) ).

% deg_SUcn_Node
thf(fact_1308_empty__Collect__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( bot_bot_set_list_nat
        = ( collect_list_nat @ P ) )
      = ( ! [X: list_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1309_empty__Collect__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( bot_bot_set_set_nat
        = ( collect_set_nat @ P ) )
      = ( ! [X: set_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1310_empty__Collect__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( bot_bo2099793752762293965at_nat
        = ( collec3392354462482085612at_nat @ P ) )
      = ( ! [X: product_prod_nat_nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1311_empty__Collect__eq,axiom,
    ! [P: real > $o] :
      ( ( bot_bot_set_real
        = ( collect_real @ P ) )
      = ( ! [X: real] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1312_empty__Collect__eq,axiom,
    ! [P: nat > $o] :
      ( ( bot_bot_set_nat
        = ( collect_nat @ P ) )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1313_empty__Collect__eq,axiom,
    ! [P: int > $o] :
      ( ( bot_bot_set_int
        = ( collect_int @ P ) )
      = ( ! [X: int] :
            ~ ( P @ X ) ) ) ).

% empty_Collect_eq
thf(fact_1314_Collect__empty__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( ( collect_list_nat @ P )
        = bot_bot_set_list_nat )
      = ( ! [X: list_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1315_Collect__empty__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( ( collect_set_nat @ P )
        = bot_bot_set_set_nat )
      = ( ! [X: set_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1316_Collect__empty__eq,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( ! [X: product_prod_nat_nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1317_Collect__empty__eq,axiom,
    ! [P: real > $o] :
      ( ( ( collect_real @ P )
        = bot_bot_set_real )
      = ( ! [X: real] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1318_Collect__empty__eq,axiom,
    ! [P: nat > $o] :
      ( ( ( collect_nat @ P )
        = bot_bot_set_nat )
      = ( ! [X: nat] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1319_Collect__empty__eq,axiom,
    ! [P: int > $o] :
      ( ( ( collect_int @ P )
        = bot_bot_set_int )
      = ( ! [X: int] :
            ~ ( P @ X ) ) ) ).

% Collect_empty_eq
thf(fact_1320_psubsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( A2 != B2 )
       => ( ord_less_set_int @ A2 @ B2 ) ) ) ).

% psubsetI
thf(fact_1321_subset__antisym,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( A2 = B2 ) ) ) ).

% subset_antisym
thf(fact_1322_subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A2 )
         => ( member8440522571783428010at_nat @ X4 @ B2 ) )
     => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_1323_subsetI,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( member_real @ X4 @ B2 ) )
     => ( ord_less_eq_set_real @ A2 @ B2 ) ) ).

% subsetI
thf(fact_1324_subsetI,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ A2 )
         => ( member_set_nat @ X4 @ B2 ) )
     => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_1325_subsetI,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( member_nat @ X4 @ B2 ) )
     => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% subsetI
thf(fact_1326_subsetI,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( member_int @ X4 @ B2 ) )
     => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% subsetI
thf(fact_1327_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_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) )
          & ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 ) ) ) ) ).

% mi_eq_ma_no_ch
thf(fact_1328_not__Some__eq,axiom,
    ! [X3: option_nat] :
      ( ( ! [Y: nat] :
            ( X3
           != ( some_nat @ Y ) ) )
      = ( X3 = none_nat ) ) ).

% not_Some_eq
thf(fact_1329_not__Some__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ( ! [Y: product_prod_nat_nat] :
            ( X3
           != ( some_P7363390416028606310at_nat @ Y ) ) )
      = ( X3 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_1330_not__Some__eq,axiom,
    ! [X3: option_num] :
      ( ( ! [Y: num] :
            ( X3
           != ( some_num @ Y ) ) )
      = ( X3 = none_num ) ) ).

% not_Some_eq
thf(fact_1331_not__None__eq,axiom,
    ! [X3: option_nat] :
      ( ( X3 != none_nat )
      = ( ? [Y: nat] :
            ( X3
            = ( some_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_1332_not__None__eq,axiom,
    ! [X3: option4927543243414619207at_nat] :
      ( ( X3 != none_P5556105721700978146at_nat )
      = ( ? [Y: product_prod_nat_nat] :
            ( X3
            = ( some_P7363390416028606310at_nat @ Y ) ) ) ) ).

% not_None_eq
thf(fact_1333_not__None__eq,axiom,
    ! [X3: option_num] :
      ( ( X3 != none_num )
      = ( ? [Y: num] :
            ( X3
            = ( some_num @ Y ) ) ) ) ).

% not_None_eq
thf(fact_1334_option_Ocollapse,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( ( some_nat @ ( the_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_1335_option_Ocollapse,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_1336_option_Ocollapse,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( ( some_num @ ( the_num @ Option ) )
        = Option ) ) ).

% option.collapse
thf(fact_1337_add__def,axiom,
    ( vEBT_VEBT_add
    = ( vEBT_V4262088993061758097ft_nat @ plus_plus_nat ) ) ).

% add_def
thf(fact_1338_psubsetE,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
         => ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).

% psubsetE
thf(fact_1339_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_1340_psubset__imp__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% psubset_imp_subset
thf(fact_1341_psubset__subset__trans,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_1342_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_1343_subset__psubset__trans,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( ord_less_set_int @ B2 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_1344_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_1345_Collect__mono__iff,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) )
      = ( ! [X: real] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_1346_Collect__mono__iff,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) )
      = ( ! [X: list_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_1347_Collect__mono__iff,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) )
      = ( ! [X: set_nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_1348_Collect__mono__iff,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) )
      = ( ! [X: nat] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_1349_Collect__mono__iff,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) )
      = ( ! [X: int] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_1350_set__eq__subset,axiom,
    ( ( ^ [Y4: set_int,Z: set_int] : Y4 = Z )
    = ( ^ [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_1351_subset__trans,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C4 )
       => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_1352_Collect__mono,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X4: real] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_mono
thf(fact_1353_Collect__mono,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X4: list_nat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1354_Collect__mono,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ! [X4: set_nat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_le6893508408891458716et_nat @ ( collect_set_nat @ P ) @ ( collect_set_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1355_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X4: nat] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1356_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X4: int] :
          ( ( P @ X4 )
         => ( Q @ X4 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_1357_subset__refl,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% subset_refl
thf(fact_1358_subset__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
        ! [T2: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ T2 @ A5 )
         => ( member8440522571783428010at_nat @ T2 @ B5 ) ) ) ) ).

% subset_iff
thf(fact_1359_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_1360_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_1361_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_1362_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_1363_equalityD2,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_int @ B2 @ A2 ) ) ).

% equalityD2
thf(fact_1364_equalityD1,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% equalityD1
thf(fact_1365_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
        ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ A5 )
         => ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1366_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
        ! [X: real] :
          ( ( member_real @ X @ A5 )
         => ( member_real @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1367_subset__eq,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
        ! [X: set_nat] :
          ( ( member_set_nat @ X @ A5 )
         => ( member_set_nat @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1368_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A5 )
         => ( member_nat @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1369_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
        ! [X: int] :
          ( ( member_int @ X @ A5 )
         => ( member_int @ X @ B5 ) ) ) ) ).

% subset_eq
thf(fact_1370_equalityE,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( A2 = B2 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B2 )
         => ~ ( ord_less_eq_set_int @ B2 @ A2 ) ) ) ).

% equalityE
thf(fact_1371_subsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ( member8440522571783428010at_nat @ C @ A2 )
       => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1372_subsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1373_subsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1374_subsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1375_subsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B2 ) ) ) ).

% subsetD
thf(fact_1376_in__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ( member8440522571783428010at_nat @ X3 @ A2 )
       => ( member8440522571783428010at_nat @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_1377_in__mono,axiom,
    ! [A2: set_real,B2: set_real,X3: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ( member_real @ X3 @ A2 )
       => ( member_real @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_1378_in__mono,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X3: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ B2 )
     => ( ( member_set_nat @ X3 @ A2 )
       => ( member_set_nat @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_1379_in__mono,axiom,
    ! [A2: set_nat,B2: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( member_nat @ X3 @ A2 )
       => ( member_nat @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_1380_in__mono,axiom,
    ! [A2: set_int,B2: set_int,X3: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( member_int @ X3 @ A2 )
       => ( member_int @ X3 @ B2 ) ) ) ).

% in_mono
thf(fact_1381_bot__set__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat @ bot_bot_list_nat_o ) ) ).

% bot_set_def
thf(fact_1382_bot__set__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat @ bot_bot_set_nat_o ) ) ).

% bot_set_def
thf(fact_1383_bot__set__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat @ bot_bo482883023278783056_nat_o ) ) ).

% bot_set_def
thf(fact_1384_bot__set__def,axiom,
    ( bot_bot_set_real
    = ( collect_real @ bot_bot_real_o ) ) ).

% bot_set_def
thf(fact_1385_bot__set__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat @ bot_bot_nat_o ) ) ).

% bot_set_def
thf(fact_1386_bot__set__def,axiom,
    ( bot_bot_set_int
    = ( collect_int @ bot_bot_int_o ) ) ).

% bot_set_def
thf(fact_1387_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_1388_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option_nat > $o,Y3: option_nat] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A4: nat,B4: nat] :
              ( ( X3
                = ( some_nat @ A4 ) )
             => ( ( Y3
                  = ( some_nat @ B4 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_1389_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A4: nat,B4: product_prod_nat_nat] :
              ( ( X3
                = ( some_nat @ A4 ) )
             => ( ( Y3
                  = ( some_P7363390416028606310at_nat @ B4 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_1390_combine__options__cases,axiom,
    ! [X3: option_nat,P: option_nat > option_num > $o,Y3: option_num] :
      ( ( ( X3 = none_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_num )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A4: nat,B4: num] :
              ( ( X3
                = ( some_nat @ A4 ) )
             => ( ( Y3
                  = ( some_num @ B4 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_1391_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_nat > $o,Y3: option_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A4: product_prod_nat_nat,B4: nat] :
              ( ( X3
                = ( some_P7363390416028606310at_nat @ A4 ) )
             => ( ( Y3
                  = ( some_nat @ B4 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_1392_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( ( X3
                = ( some_P7363390416028606310at_nat @ A4 ) )
             => ( ( Y3
                  = ( some_P7363390416028606310at_nat @ B4 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_1393_combine__options__cases,axiom,
    ! [X3: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y3: option_num] :
      ( ( ( X3 = none_P5556105721700978146at_nat )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_num )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A4: product_prod_nat_nat,B4: num] :
              ( ( X3
                = ( some_P7363390416028606310at_nat @ A4 ) )
             => ( ( Y3
                  = ( some_num @ B4 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_1394_combine__options__cases,axiom,
    ! [X3: option_num,P: option_num > option_nat > $o,Y3: option_nat] :
      ( ( ( X3 = none_num )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A4: num,B4: nat] :
              ( ( X3
                = ( some_num @ A4 ) )
             => ( ( Y3
                  = ( some_nat @ B4 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_1395_combine__options__cases,axiom,
    ! [X3: option_num,P: option_num > option4927543243414619207at_nat > $o,Y3: option4927543243414619207at_nat] :
      ( ( ( X3 = none_num )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_P5556105721700978146at_nat )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A4: num,B4: product_prod_nat_nat] :
              ( ( X3
                = ( some_num @ A4 ) )
             => ( ( Y3
                  = ( some_P7363390416028606310at_nat @ B4 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_1396_combine__options__cases,axiom,
    ! [X3: option_num,P: option_num > option_num > $o,Y3: option_num] :
      ( ( ( X3 = none_num )
       => ( P @ X3 @ Y3 ) )
     => ( ( ( Y3 = none_num )
         => ( P @ X3 @ Y3 ) )
       => ( ! [A4: num,B4: num] :
              ( ( X3
                = ( some_num @ A4 ) )
             => ( ( Y3
                  = ( some_num @ B4 ) )
               => ( P @ X3 @ Y3 ) ) )
         => ( P @ X3 @ Y3 ) ) ) ) ).

% combine_options_cases
thf(fact_1397_split__option__all,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ! [X6: option_nat] : ( P2 @ X6 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          & ! [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_1398_split__option__all,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ! [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          & ! [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_1399_split__option__all,axiom,
    ( ( ^ [P2: option_num > $o] :
        ! [X6: option_num] : ( P2 @ X6 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          & ! [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_all
thf(fact_1400_split__option__ex,axiom,
    ( ( ^ [P2: option_nat > $o] :
        ? [X6: option_nat] : ( P2 @ X6 ) )
    = ( ^ [P3: option_nat > $o] :
          ( ( P3 @ none_nat )
          | ? [X: nat] : ( P3 @ ( some_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_1401_split__option__ex,axiom,
    ( ( ^ [P2: option4927543243414619207at_nat > $o] :
        ? [X6: option4927543243414619207at_nat] : ( P2 @ X6 ) )
    = ( ^ [P3: option4927543243414619207at_nat > $o] :
          ( ( P3 @ none_P5556105721700978146at_nat )
          | ? [X: product_prod_nat_nat] : ( P3 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_1402_split__option__ex,axiom,
    ( ( ^ [P2: option_num > $o] :
        ? [X6: option_num] : ( P2 @ X6 ) )
    = ( ^ [P3: option_num > $o] :
          ( ( P3 @ none_num )
          | ? [X: num] : ( P3 @ ( some_num @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_1403_option_Oexhaust,axiom,
    ! [Y3: option_nat] :
      ( ( Y3 != none_nat )
     => ~ ! [X23: nat] :
            ( Y3
           != ( some_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_1404_option_Oexhaust,axiom,
    ! [Y3: option4927543243414619207at_nat] :
      ( ( Y3 != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y3
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_1405_option_Oexhaust,axiom,
    ! [Y3: option_num] :
      ( ( Y3 != none_num )
     => ~ ! [X23: num] :
            ( Y3
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_1406_option_OdiscI,axiom,
    ! [Option: option_nat,X22: nat] :
      ( ( Option
        = ( some_nat @ X22 ) )
     => ( Option != none_nat ) ) ).

% option.discI
thf(fact_1407_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_1408_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option
        = ( some_num @ X22 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_1409_option_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( none_nat
     != ( some_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_1410_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_1411_option_Odistinct_I1_J,axiom,
    ! [X22: num] :
      ( none_num
     != ( some_num @ X22 ) ) ).

% option.distinct(1)
thf(fact_1412_vebt__maxt_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Ma ) ) ).

% vebt_maxt.simps(3)
thf(fact_1413_vebt__mint_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Ux: nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Ux @ Uy @ Uz ) )
      = ( some_nat @ Mi ) ) ).

% vebt_mint.simps(3)
thf(fact_1414_option_Oexpand,axiom,
    ! [Option: option_nat,Option2: option_nat] :
      ( ( ( Option = none_nat )
        = ( Option2 = none_nat ) )
     => ( ( ( Option != none_nat )
         => ( ( Option2 != none_nat )
           => ( ( the_nat @ Option )
              = ( the_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_1415_option_Oexpand,axiom,
    ! [Option: option4927543243414619207at_nat,Option2: option4927543243414619207at_nat] :
      ( ( ( Option = none_P5556105721700978146at_nat )
        = ( Option2 = none_P5556105721700978146at_nat ) )
     => ( ( ( Option != none_P5556105721700978146at_nat )
         => ( ( Option2 != none_P5556105721700978146at_nat )
           => ( ( the_Pr8591224930841456533at_nat @ Option )
              = ( the_Pr8591224930841456533at_nat @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_1416_option_Oexpand,axiom,
    ! [Option: option_num,Option2: option_num] :
      ( ( ( Option = none_num )
        = ( Option2 = none_num ) )
     => ( ( ( Option != none_num )
         => ( ( Option2 != none_num )
           => ( ( the_num @ Option )
              = ( the_num @ Option2 ) ) ) )
       => ( Option = Option2 ) ) ) ).

% option.expand
thf(fact_1417_option_Osize_I3_J,axiom,
    ( ( size_size_option_nat @ none_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1418_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1419_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_1420_option_Osize__gen_I1_J,axiom,
    ! [X3: nat > nat] :
      ( ( size_option_nat @ X3 @ none_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1421_option_Osize__gen_I1_J,axiom,
    ! [X3: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X3 @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1422_option_Osize__gen_I1_J,axiom,
    ! [X3: num > nat] :
      ( ( size_option_num @ X3 @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_1423_option_Oexhaust__sel,axiom,
    ! [Option: option_nat] :
      ( ( Option != none_nat )
     => ( Option
        = ( some_nat @ ( the_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1424_option_Oexhaust__sel,axiom,
    ! [Option: option4927543243414619207at_nat] :
      ( ( Option != none_P5556105721700978146at_nat )
     => ( Option
        = ( some_P7363390416028606310at_nat @ ( the_Pr8591224930841456533at_nat @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1425_option_Oexhaust__sel,axiom,
    ! [Option: option_num] :
      ( ( Option != none_num )
     => ( Option
        = ( some_num @ ( the_num @ Option ) ) ) ) ).

% option.exhaust_sel
thf(fact_1426_linorder__neqE__linordered__idom,axiom,
    ! [X3: real,Y3: real] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_real @ X3 @ Y3 )
       => ( ord_less_real @ Y3 @ X3 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1427_linorder__neqE__linordered__idom,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_rat @ X3 @ Y3 )
       => ( ord_less_rat @ Y3 @ X3 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1428_linorder__neqE__linordered__idom,axiom,
    ! [X3: int,Y3: int] :
      ( ( X3 != Y3 )
     => ( ~ ( ord_less_int @ X3 @ Y3 )
       => ( ord_less_int @ Y3 @ X3 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_1429_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_1: real] : ( ord_less_real @ X2 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1430_linordered__field__no__ub,axiom,
    ! [X2: rat] :
    ? [X_1: rat] : ( ord_less_rat @ X2 @ X_1 ) ).

% linordered_field_no_ub
thf(fact_1431_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y5: real] : ( ord_less_real @ Y5 @ X2 ) ).

% linordered_field_no_lb
thf(fact_1432_linordered__field__no__lb,axiom,
    ! [X2: rat] :
    ? [Y5: rat] : ( ord_less_rat @ Y5 @ X2 ) ).

% linordered_field_no_lb
thf(fact_1433_not__psubset__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ~ ( ord_le7866589430770878221at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ).

% not_psubset_empty
thf(fact_1434_not__psubset__empty,axiom,
    ! [A2: set_real] :
      ~ ( ord_less_set_real @ A2 @ bot_bot_set_real ) ).

% not_psubset_empty
thf(fact_1435_not__psubset__empty,axiom,
    ! [A2: set_nat] :
      ~ ( ord_less_set_nat @ A2 @ bot_bot_set_nat ) ).

% not_psubset_empty
thf(fact_1436_not__psubset__empty,axiom,
    ! [A2: set_int] :
      ~ ( ord_less_set_int @ A2 @ bot_bot_set_int ) ).

% not_psubset_empty
thf(fact_1437_ex__in__conv,axiom,
    ! [A2: set_set_nat] :
      ( ( ? [X: set_nat] : ( member_set_nat @ X @ A2 ) )
      = ( A2 != bot_bot_set_set_nat ) ) ).

% ex_in_conv
thf(fact_1438_ex__in__conv,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ? [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A2 ) )
      = ( A2 != bot_bo2099793752762293965at_nat ) ) ).

% ex_in_conv
thf(fact_1439_ex__in__conv,axiom,
    ! [A2: set_real] :
      ( ( ? [X: real] : ( member_real @ X @ A2 ) )
      = ( A2 != bot_bot_set_real ) ) ).

% ex_in_conv
thf(fact_1440_ex__in__conv,axiom,
    ! [A2: set_nat] :
      ( ( ? [X: nat] : ( member_nat @ X @ A2 ) )
      = ( A2 != bot_bot_set_nat ) ) ).

% ex_in_conv
thf(fact_1441_ex__in__conv,axiom,
    ! [A2: set_int] :
      ( ( ? [X: int] : ( member_int @ X @ A2 ) )
      = ( A2 != bot_bot_set_int ) ) ).

% ex_in_conv
thf(fact_1442_equals0I,axiom,
    ! [A2: set_set_nat] :
      ( ! [Y5: set_nat] :
          ~ ( member_set_nat @ Y5 @ A2 )
     => ( A2 = bot_bot_set_set_nat ) ) ).

% equals0I
thf(fact_1443_equals0I,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [Y5: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ Y5 @ A2 )
     => ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% equals0I
thf(fact_1444_equals0I,axiom,
    ! [A2: set_real] :
      ( ! [Y5: real] :
          ~ ( member_real @ Y5 @ A2 )
     => ( A2 = bot_bot_set_real ) ) ).

% equals0I
thf(fact_1445_equals0I,axiom,
    ! [A2: set_nat] :
      ( ! [Y5: nat] :
          ~ ( member_nat @ Y5 @ A2 )
     => ( A2 = bot_bot_set_nat ) ) ).

% equals0I
thf(fact_1446_equals0I,axiom,
    ! [A2: set_int] :
      ( ! [Y5: int] :
          ~ ( member_int @ Y5 @ A2 )
     => ( A2 = bot_bot_set_int ) ) ).

% equals0I
thf(fact_1447_equals0D,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( A2 = bot_bot_set_set_nat )
     => ~ ( member_set_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_1448_equals0D,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => ~ ( member8440522571783428010at_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_1449_equals0D,axiom,
    ! [A2: set_real,A: real] :
      ( ( A2 = bot_bot_set_real )
     => ~ ( member_real @ A @ A2 ) ) ).

% equals0D
thf(fact_1450_equals0D,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( A2 = bot_bot_set_nat )
     => ~ ( member_nat @ A @ A2 ) ) ).

% equals0D
thf(fact_1451_equals0D,axiom,
    ! [A2: set_int,A: int] :
      ( ( A2 = bot_bot_set_int )
     => ~ ( member_int @ A @ A2 ) ) ).

% equals0D
thf(fact_1452_emptyE,axiom,
    ! [A: set_nat] :
      ~ ( member_set_nat @ A @ bot_bot_set_set_nat ) ).

% emptyE
thf(fact_1453_emptyE,axiom,
    ! [A: product_prod_nat_nat] :
      ~ ( member8440522571783428010at_nat @ A @ bot_bo2099793752762293965at_nat ) ).

% emptyE
thf(fact_1454_emptyE,axiom,
    ! [A: real] :
      ~ ( member_real @ A @ bot_bot_set_real ) ).

% emptyE
thf(fact_1455_emptyE,axiom,
    ! [A: nat] :
      ~ ( member_nat @ A @ bot_bot_set_nat ) ).

% emptyE
thf(fact_1456_emptyE,axiom,
    ! [A: int] :
      ~ ( member_int @ A @ bot_bot_set_int ) ).

% emptyE
thf(fact_1457_option_Osize_I4_J,axiom,
    ! [X22: nat] :
      ( ( size_size_option_nat @ ( some_nat @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1458_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_1459_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_1460_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X3 )
      = ( ( X3 = Mi )
        | ( X3 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_1461_vebt__member_Osimps_I4_J,axiom,
    ! [V: product_prod_nat_nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ ( suc @ zero_zero_nat ) @ Vb @ Vc ) @ X3 ) ).

% vebt_member.simps(4)
thf(fact_1462_vebt__member_Osimps_I3_J,axiom,
    ! [V: product_prod_nat_nat,Uy: list_VEBT_VEBT,Uz: vEBT_VEBT,X3: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V ) @ zero_zero_nat @ Uy @ Uz ) @ X3 ) ).

% vebt_member.simps(3)
thf(fact_1463_VEBT__internal_OminNull_Osimps_I5_J,axiom,
    ! [Uz: product_prod_nat_nat,Va: nat,Vb: list_VEBT_VEBT,Vc: vEBT_VEBT] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz ) @ Va @ Vb @ Vc ) ) ).

% VEBT_internal.minNull.simps(5)
thf(fact_1464_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_1465_VEBT_Oinject_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,Y11: option4927543243414619207at_nat,Y12: nat,Y13: list_VEBT_VEBT,Y14: vEBT_VEBT] :
      ( ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
        = ( vEBT_Node @ Y11 @ Y12 @ Y13 @ Y14 ) )
      = ( ( X11 = Y11 )
        & ( X12 = Y12 )
        & ( X13 = Y13 )
        & ( X14 = Y14 ) ) ) ).

% VEBT.inject(1)
thf(fact_1466_old_Oprod_Oinject,axiom,
    ! [A: int,B: int,A6: int,B6: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1467_old_Oprod_Oinject,axiom,
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: code_integer > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ A @ B )
        = ( produc6137756002093451184nteger @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1468_old_Oprod_Oinject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1469_old_Oprod_Oinject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1470_old_Oprod_Oinject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A6 @ B6 ) )
      = ( ( A = A6 )
        & ( B = B6 ) ) ) ).

% old.prod.inject
thf(fact_1471_prod_Oinject,axiom,
    ! [X1: int,X22: int,Y1: int,Y2: int] :
      ( ( ( product_Pair_int_int @ X1 @ X22 )
        = ( product_Pair_int_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_1472_prod_Oinject,axiom,
    ! [X1: code_integer > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: code_integer > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ X1 @ X22 )
        = ( produc6137756002093451184nteger @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_1473_prod_Oinject,axiom,
    ! [X1: produc6241069584506657477e_term > option6357759511663192854e_term,X22: produc8923325533196201883nteger,Y1: produc6241069584506657477e_term > option6357759511663192854e_term,Y2: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ X1 @ X22 )
        = ( produc8603105652947943368nteger @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_1474_prod_Oinject,axiom,
    ! [X1: produc8551481072490612790e_term > option6357759511663192854e_term,X22: product_prod_int_int,Y1: produc8551481072490612790e_term > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ X1 @ X22 )
        = ( produc5700946648718959541nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_1475_prod_Oinject,axiom,
    ! [X1: int > option6357759511663192854e_term,X22: product_prod_int_int,Y1: int > option6357759511663192854e_term,Y2: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ X1 @ X22 )
        = ( produc4305682042979456191nt_int @ Y1 @ Y2 ) )
      = ( ( X1 = Y1 )
        & ( X22 = Y2 ) ) ) ).

% prod.inject
thf(fact_1476_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_VEBT_VEBT,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( nth_Pr744662078594809490T_VEBT @ ( enumerate_VEBT_VEBT @ N2 @ Xs ) @ M )
        = ( produc599794634098209291T_VEBT @ ( plus_plus_nat @ N2 @ M ) @ ( nth_VEBT_VEBT @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1477_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_o,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_o @ Xs ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( enumerate_o @ N2 @ Xs ) @ M )
        = ( product_Pair_nat_o @ ( plus_plus_nat @ N2 @ M ) @ ( nth_o @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1478_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_nat @ Xs ) )
     => ( ( nth_Pr7617993195940197384at_nat @ ( enumerate_nat @ N2 @ Xs ) @ M )
        = ( product_Pair_nat_nat @ ( plus_plus_nat @ N2 @ M ) @ ( nth_nat @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1479_nth__enumerate__eq,axiom,
    ! [M: nat,Xs: list_int,N2: nat] :
      ( ( ord_less_nat @ M @ ( size_size_list_int @ Xs ) )
     => ( ( nth_Pr3440142176431000676at_int @ ( enumerate_int @ N2 @ Xs ) @ M )
        = ( product_Pair_nat_int @ ( plus_plus_nat @ N2 @ M ) @ ( nth_int @ Xs @ M ) ) ) ) ).

% nth_enumerate_eq
thf(fact_1480_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_1481_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: 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 @ B ) ) ) ) ).

% Euclid_induct
thf(fact_1482_List_Ofinite__set,axiom,
    ! [Xs: list_VEBT_VEBT] : ( finite5795047828879050333T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) ) ).

% List.finite_set
thf(fact_1483_List_Ofinite__set,axiom,
    ! [Xs: list_nat] : ( finite_finite_nat @ ( set_nat2 @ Xs ) ) ).

% List.finite_set
thf(fact_1484_List_Ofinite__set,axiom,
    ! [Xs: list_int] : ( finite_finite_int @ ( set_int2 @ Xs ) ) ).

% List.finite_set
thf(fact_1485_List_Ofinite__set,axiom,
    ! [Xs: list_complex] : ( finite3207457112153483333omplex @ ( set_complex2 @ Xs ) ) ).

% List.finite_set
thf(fact_1486_List_Ofinite__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat] : ( finite6177210948735845034at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ).

% List.finite_set
thf(fact_1487_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_1488_length__enumerate,axiom,
    ! [N2: nat,Xs: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( enumerate_o @ N2 @ Xs ) )
      = ( size_size_list_o @ Xs ) ) ).

% length_enumerate
thf(fact_1489_length__enumerate,axiom,
    ! [N2: nat,Xs: list_nat] :
      ( ( size_s5460976970255530739at_nat @ ( enumerate_nat @ N2 @ Xs ) )
      = ( size_size_list_nat @ Xs ) ) ).

% length_enumerate
thf(fact_1490_length__enumerate,axiom,
    ! [N2: nat,Xs: list_int] :
      ( ( size_s2970893825323803983at_int @ ( enumerate_int @ N2 @ Xs ) )
      = ( size_size_list_int @ Xs ) ) ).

% length_enumerate
thf(fact_1491_psubsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
     => ( ( member8440522571783428010at_nat @ C @ A2 )
       => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1492_psubsetD,axiom,
    ! [A2: set_real,B2: set_real,C: real] :
      ( ( ord_less_set_real @ A2 @ B2 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1493_psubsetD,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,C: set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B2 )
     => ( ( member_set_nat @ C @ A2 )
       => ( member_set_nat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1494_psubsetD,axiom,
    ! [A2: set_nat,B2: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1495_psubsetD,axiom,
    ! [A2: set_int,B2: set_int,C: int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B2 ) ) ) ).

% psubsetD
thf(fact_1496_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc8306885398267862888on_nat] :
      ( ! [Uu2: nat > nat > nat,Uv2: option_nat] :
          ( X3
         != ( produc8929957630744042906on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > nat,V2: nat] :
            ( X3
           != ( produc8929957630744042906on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > nat,A4: nat,B4: nat] :
              ( X3
             != ( produc8929957630744042906on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ A4 ) @ ( some_nat @ B4 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1497_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc5542196010084753463at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv2: option4927543243414619207at_nat] :
          ( X3
         != ( produc2899441246263362727at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V2: product_prod_nat_nat] :
            ( X3
           != ( produc2899441246263362727at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A4: product_prod_nat_nat,B4: product_prod_nat_nat] :
              ( X3
             != ( produc2899441246263362727at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ A4 ) @ ( some_P7363390416028606310at_nat @ B4 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1498_VEBT__internal_Ooption__shift_Ocases,axiom,
    ! [X3: produc1193250871479095198on_num] :
      ( ! [Uu2: num > num > num,Uv2: option_num] :
          ( X3
         != ( produc5778274026573060048on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > num,V2: num] :
            ( X3
           != ( produc5778274026573060048on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > num,A4: num,B4: num] :
              ( X3
             != ( produc5778274026573060048on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ A4 ) @ ( some_num @ B4 ) ) ) ) ) ) ).

% VEBT_internal.option_shift.cases
thf(fact_1499_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc2233624965454879586on_nat] :
      ( ! [Uu2: nat > nat > $o,Uv2: option_nat] :
          ( X3
         != ( produc4035269172776083154on_nat @ Uu2 @ ( produc5098337634421038937on_nat @ none_nat @ Uv2 ) ) )
     => ( ! [Uw2: nat > nat > $o,V2: nat] :
            ( X3
           != ( produc4035269172776083154on_nat @ Uw2 @ ( produc5098337634421038937on_nat @ ( some_nat @ V2 ) @ none_nat ) ) )
       => ~ ! [F2: nat > nat > $o,X4: nat,Y5: nat] :
              ( X3
             != ( produc4035269172776083154on_nat @ F2 @ ( produc5098337634421038937on_nat @ ( some_nat @ X4 ) @ ( some_nat @ Y5 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1500_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc5491161045314408544at_nat] :
      ( ! [Uu2: product_prod_nat_nat > product_prod_nat_nat > $o,Uv2: option4927543243414619207at_nat] :
          ( X3
         != ( produc3994169339658061776at_nat @ Uu2 @ ( produc488173922507101015at_nat @ none_P5556105721700978146at_nat @ Uv2 ) ) )
     => ( ! [Uw2: product_prod_nat_nat > product_prod_nat_nat > $o,V2: product_prod_nat_nat] :
            ( X3
           != ( produc3994169339658061776at_nat @ Uw2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ V2 ) @ none_P5556105721700978146at_nat ) ) )
       => ~ ! [F2: product_prod_nat_nat > product_prod_nat_nat > $o,X4: product_prod_nat_nat,Y5: product_prod_nat_nat] :
              ( X3
             != ( produc3994169339658061776at_nat @ F2 @ ( produc488173922507101015at_nat @ ( some_P7363390416028606310at_nat @ X4 ) @ ( some_P7363390416028606310at_nat @ Y5 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1501_VEBT__internal_Ooption__comp__shift_Ocases,axiom,
    ! [X3: produc7036089656553540234on_num] :
      ( ! [Uu2: num > num > $o,Uv2: option_num] :
          ( X3
         != ( produc3576312749637752826on_num @ Uu2 @ ( produc8585076106096196333on_num @ none_num @ Uv2 ) ) )
     => ( ! [Uw2: num > num > $o,V2: num] :
            ( X3
           != ( produc3576312749637752826on_num @ Uw2 @ ( produc8585076106096196333on_num @ ( some_num @ V2 ) @ none_num ) ) )
       => ~ ! [F2: num > num > $o,X4: num,Y5: num] :
              ( X3
             != ( produc3576312749637752826on_num @ F2 @ ( produc8585076106096196333on_num @ ( some_num @ X4 ) @ ( some_num @ Y5 ) ) ) ) ) ) ).

% VEBT_internal.option_comp_shift.cases
thf(fact_1502_finite__list,axiom,
    ! [A2: set_VEBT_VEBT] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ? [Xs3: list_VEBT_VEBT] :
          ( ( set_VEBT_VEBT2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1503_finite__list,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ? [Xs3: list_nat] :
          ( ( set_nat2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1504_finite__list,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ? [Xs3: list_int] :
          ( ( set_int2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1505_finite__list,axiom,
    ! [A2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ? [Xs3: list_complex] :
          ( ( set_complex2 @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1506_finite__list,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ? [Xs3: list_P6011104703257516679at_nat] :
          ( ( set_Pr5648618587558075414at_nat @ Xs3 )
          = A2 ) ) ).

% finite_list
thf(fact_1507_vebt__member_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT,X3: nat] :
      ~ ( vEBT_vebt_member @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) @ X3 ) ).

% vebt_member.simps(2)
thf(fact_1508_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_1509_ex__min__if__finite,axiom,
    ! [S3: set_real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ S3 )
            & ~ ? [Xa: real] :
                  ( ( member_real @ Xa @ S3 )
                  & ( ord_less_real @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1510_ex__min__if__finite,axiom,
    ! [S3: set_rat] :
      ( ( finite_finite_rat @ S3 )
     => ( ( S3 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ S3 )
            & ~ ? [Xa: rat] :
                  ( ( member_rat @ Xa @ S3 )
                  & ( ord_less_rat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1511_ex__min__if__finite,axiom,
    ! [S3: set_num] :
      ( ( finite_finite_num @ S3 )
     => ( ( S3 != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ S3 )
            & ~ ? [Xa: num] :
                  ( ( member_num @ Xa @ S3 )
                  & ( ord_less_num @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1512_ex__min__if__finite,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ S3 )
            & ~ ? [Xa: nat] :
                  ( ( member_nat @ Xa @ S3 )
                  & ( ord_less_nat @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1513_ex__min__if__finite,axiom,
    ! [S3: set_int] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ S3 )
            & ~ ? [Xa: int] :
                  ( ( member_int @ Xa @ S3 )
                  & ( ord_less_int @ Xa @ X4 ) ) ) ) ) ).

% ex_min_if_finite
thf(fact_1514_infinite__growing,axiom,
    ! [X7: set_real] :
      ( ( X7 != bot_bot_set_real )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ X7 )
           => ? [Xa: real] :
                ( ( member_real @ Xa @ X7 )
                & ( ord_less_real @ X4 @ Xa ) ) )
       => ~ ( finite_finite_real @ X7 ) ) ) ).

% infinite_growing
thf(fact_1515_infinite__growing,axiom,
    ! [X7: set_rat] :
      ( ( X7 != bot_bot_set_rat )
     => ( ! [X4: rat] :
            ( ( member_rat @ X4 @ X7 )
           => ? [Xa: rat] :
                ( ( member_rat @ Xa @ X7 )
                & ( ord_less_rat @ X4 @ Xa ) ) )
       => ~ ( finite_finite_rat @ X7 ) ) ) ).

% infinite_growing
thf(fact_1516_infinite__growing,axiom,
    ! [X7: set_num] :
      ( ( X7 != bot_bot_set_num )
     => ( ! [X4: num] :
            ( ( member_num @ X4 @ X7 )
           => ? [Xa: num] :
                ( ( member_num @ Xa @ X7 )
                & ( ord_less_num @ X4 @ Xa ) ) )
       => ~ ( finite_finite_num @ X7 ) ) ) ).

% infinite_growing
thf(fact_1517_infinite__growing,axiom,
    ! [X7: set_nat] :
      ( ( X7 != bot_bot_set_nat )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ X7 )
           => ? [Xa: nat] :
                ( ( member_nat @ Xa @ X7 )
                & ( ord_less_nat @ X4 @ Xa ) ) )
       => ~ ( finite_finite_nat @ X7 ) ) ) ).

% infinite_growing
thf(fact_1518_infinite__growing,axiom,
    ! [X7: set_int] :
      ( ( X7 != bot_bot_set_int )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ X7 )
           => ? [Xa: int] :
                ( ( member_int @ Xa @ X7 )
                & ( ord_less_int @ X4 @ Xa ) ) )
       => ~ ( finite_finite_int @ X7 ) ) ) ).

% infinite_growing
thf(fact_1519_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_1520_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ F @ ( some_P7363390416028606310at_nat @ A ) @ ( some_P7363390416028606310at_nat @ B ) )
      = ( some_P7363390416028606310at_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1521_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: num > num > num,A: num,B: num] :
      ( ( vEBT_V819420779217536731ft_num @ F @ ( some_num @ A ) @ ( some_num @ B ) )
      = ( some_num @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1522_VEBT__internal_Ooption__shift_Osimps_I3_J,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ F @ ( some_nat @ A ) @ ( some_nat @ B ) )
      = ( some_nat @ ( F @ A @ B ) ) ) ).

% VEBT_internal.option_shift.simps(3)
thf(fact_1523_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Uv: option4927543243414619207at_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uu @ none_P5556105721700978146at_nat @ Uv )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1524_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: num > num > num,Uv: option_num] :
      ( ( vEBT_V819420779217536731ft_num @ Uu @ none_num @ Uv )
      = none_num ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1525_VEBT__internal_Ooption__shift_Osimps_I1_J,axiom,
    ! [Uu: nat > nat > nat,Uv: option_nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uu @ none_nat @ Uv )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(1)
thf(fact_1526_old_Oprod_Oexhaust,axiom,
    ! [Y3: product_prod_int_int] :
      ~ ! [A4: int,B4: int] :
          ( Y3
         != ( product_Pair_int_int @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1527_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc8763457246119570046nteger] :
      ~ ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
          ( Y3
         != ( produc6137756002093451184nteger @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1528_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc1908205239877642774nteger] :
      ~ ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
          ( Y3
         != ( produc8603105652947943368nteger @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1529_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc2285326912895808259nt_int] :
      ~ ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( Y3
         != ( produc5700946648718959541nt_int @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1530_old_Oprod_Oexhaust,axiom,
    ! [Y3: produc7773217078559923341nt_int] :
      ~ ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( Y3
         != ( produc4305682042979456191nt_int @ A4 @ B4 ) ) ).

% old.prod.exhaust
thf(fact_1531_surj__pair,axiom,
    ! [P4: product_prod_int_int] :
    ? [X4: int,Y5: int] :
      ( P4
      = ( product_Pair_int_int @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_1532_surj__pair,axiom,
    ! [P4: produc8763457246119570046nteger] :
    ? [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
      ( P4
      = ( produc6137756002093451184nteger @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_1533_surj__pair,axiom,
    ! [P4: produc1908205239877642774nteger] :
    ? [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
      ( P4
      = ( produc8603105652947943368nteger @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_1534_surj__pair,axiom,
    ! [P4: produc2285326912895808259nt_int] :
    ? [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
      ( P4
      = ( produc5700946648718959541nt_int @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_1535_surj__pair,axiom,
    ! [P4: produc7773217078559923341nt_int] :
    ? [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
      ( P4
      = ( produc4305682042979456191nt_int @ X4 @ Y5 ) ) ).

% surj_pair
thf(fact_1536_prod__cases,axiom,
    ! [P: product_prod_int_int > $o,P4: product_prod_int_int] :
      ( ! [A4: int,B4: int] : ( P @ ( product_Pair_int_int @ A4 @ B4 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1537_prod__cases,axiom,
    ! [P: produc8763457246119570046nteger > $o,P4: produc8763457246119570046nteger] :
      ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] : ( P @ ( produc6137756002093451184nteger @ A4 @ B4 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1538_prod__cases,axiom,
    ! [P: produc1908205239877642774nteger > $o,P4: produc1908205239877642774nteger] :
      ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] : ( P @ ( produc8603105652947943368nteger @ A4 @ B4 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1539_prod__cases,axiom,
    ! [P: produc2285326912895808259nt_int > $o,P4: produc2285326912895808259nt_int] :
      ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ B4 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1540_prod__cases,axiom,
    ! [P: produc7773217078559923341nt_int > $o,P4: produc7773217078559923341nt_int] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ B4 ) )
     => ( P @ P4 ) ) ).

% prod_cases
thf(fact_1541_Pair__inject,axiom,
    ! [A: int,B: int,A6: int,B6: int] :
      ( ( ( product_Pair_int_int @ A @ B )
        = ( product_Pair_int_int @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1542_Pair__inject,axiom,
    ! [A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: code_integer > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
      ( ( ( produc6137756002093451184nteger @ A @ B )
        = ( produc6137756002093451184nteger @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1543_Pair__inject,axiom,
    ! [A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger,A6: produc6241069584506657477e_term > option6357759511663192854e_term,B6: produc8923325533196201883nteger] :
      ( ( ( produc8603105652947943368nteger @ A @ B )
        = ( produc8603105652947943368nteger @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1544_Pair__inject,axiom,
    ! [A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int,A6: produc8551481072490612790e_term > option6357759511663192854e_term,B6: product_prod_int_int] :
      ( ( ( produc5700946648718959541nt_int @ A @ B )
        = ( produc5700946648718959541nt_int @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1545_Pair__inject,axiom,
    ! [A: int > option6357759511663192854e_term,B: product_prod_int_int,A6: int > option6357759511663192854e_term,B6: product_prod_int_int] :
      ( ( ( produc4305682042979456191nt_int @ A @ B )
        = ( produc4305682042979456191nt_int @ A6 @ B6 ) )
     => ~ ( ( A = A6 )
         => ( B != B6 ) ) ) ).

% Pair_inject
thf(fact_1546_prod__cases3,axiom,
    ! [Y3: produc8763457246119570046nteger] :
      ~ ! [A4: code_integer > option6357759511663192854e_term,B4: code_integer,C2: code_integer] :
          ( Y3
         != ( produc6137756002093451184nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_1547_prod__cases3,axiom,
    ! [Y3: produc1908205239877642774nteger] :
      ~ ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: code_integer,C2: code_integer] :
          ( Y3
         != ( produc8603105652947943368nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_1548_prod__cases3,axiom,
    ! [Y3: produc2285326912895808259nt_int] :
      ~ ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C2: int] :
          ( Y3
         != ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_1549_prod__cases3,axiom,
    ! [Y3: produc7773217078559923341nt_int] :
      ~ ! [A4: int > option6357759511663192854e_term,B4: int,C2: int] :
          ( Y3
         != ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) ) ).

% prod_cases3
thf(fact_1550_prod__induct3,axiom,
    ! [P: produc8763457246119570046nteger > $o,X3: produc8763457246119570046nteger] :
      ( ! [A4: code_integer > option6357759511663192854e_term,B4: code_integer,C2: code_integer] : ( P @ ( produc6137756002093451184nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_1551_prod__induct3,axiom,
    ! [P: produc1908205239877642774nteger > $o,X3: produc1908205239877642774nteger] :
      ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: code_integer,C2: code_integer] : ( P @ ( produc8603105652947943368nteger @ A4 @ ( produc1086072967326762835nteger @ B4 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_1552_prod__induct3,axiom,
    ! [P: produc2285326912895808259nt_int > $o,X3: produc2285326912895808259nt_int] :
      ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: int,C2: int] : ( P @ ( produc5700946648718959541nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_1553_prod__induct3,axiom,
    ! [P: produc7773217078559923341nt_int > $o,X3: produc7773217078559923341nt_int] :
      ( ! [A4: int > option6357759511663192854e_term,B4: int,C2: int] : ( P @ ( produc4305682042979456191nt_int @ A4 @ ( product_Pair_int_int @ B4 @ C2 ) ) )
     => ( P @ X3 ) ) ).

% prod_induct3
thf(fact_1554_vebt__maxt_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_maxt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_maxt.simps(2)
thf(fact_1555_vebt__mint_Osimps_I2_J,axiom,
    ! [Uu: nat,Uv: list_VEBT_VEBT,Uw: vEBT_VEBT] :
      ( ( vEBT_vebt_mint @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu @ Uv @ Uw ) )
      = none_nat ) ).

% vebt_mint.simps(2)
thf(fact_1556_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,V: product_prod_nat_nat] :
      ( ( vEBT_V1502963449132264192at_nat @ Uw @ ( some_P7363390416028606310at_nat @ V ) @ none_P5556105721700978146at_nat )
      = none_P5556105721700978146at_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1557_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: num > num > num,V: num] :
      ( ( vEBT_V819420779217536731ft_num @ Uw @ ( some_num @ V ) @ none_num )
      = none_num ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1558_VEBT__internal_Ooption__shift_Osimps_I2_J,axiom,
    ! [Uw: nat > nat > nat,V: nat] :
      ( ( vEBT_V4262088993061758097ft_nat @ Uw @ ( some_nat @ V ) @ none_nat )
      = none_nat ) ).

% VEBT_internal.option_shift.simps(2)
thf(fact_1559_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat,Xa2: option4927543243414619207at_nat,Xb: option4927543243414619207at_nat,Y3: option4927543243414619207at_nat] :
      ( ( ( vEBT_V1502963449132264192at_nat @ X3 @ Xa2 @ Xb )
        = Y3 )
     => ( ( ( Xa2 = none_P5556105721700978146at_nat )
         => ( Y3 != none_P5556105721700978146at_nat ) )
       => ( ( ? [V2: product_prod_nat_nat] :
                ( Xa2
                = ( some_P7363390416028606310at_nat @ V2 ) )
           => ( ( Xb = none_P5556105721700978146at_nat )
             => ( Y3 != none_P5556105721700978146at_nat ) ) )
         => ~ ! [A4: product_prod_nat_nat] :
                ( ( Xa2
                  = ( some_P7363390416028606310at_nat @ A4 ) )
               => ! [B4: product_prod_nat_nat] :
                    ( ( Xb
                      = ( some_P7363390416028606310at_nat @ B4 ) )
                   => ( Y3
                     != ( some_P7363390416028606310at_nat @ ( X3 @ A4 @ B4 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1560_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: num > num > num,Xa2: option_num,Xb: option_num,Y3: option_num] :
      ( ( ( vEBT_V819420779217536731ft_num @ X3 @ Xa2 @ Xb )
        = Y3 )
     => ( ( ( Xa2 = none_num )
         => ( Y3 != none_num ) )
       => ( ( ? [V2: num] :
                ( Xa2
                = ( some_num @ V2 ) )
           => ( ( Xb = none_num )
             => ( Y3 != none_num ) ) )
         => ~ ! [A4: num] :
                ( ( Xa2
                  = ( some_num @ A4 ) )
               => ! [B4: num] :
                    ( ( Xb
                      = ( some_num @ B4 ) )
                   => ( Y3
                     != ( some_num @ ( X3 @ A4 @ B4 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1561_VEBT__internal_Ooption__shift_Oelims,axiom,
    ! [X3: nat > nat > nat,Xa2: option_nat,Xb: option_nat,Y3: option_nat] :
      ( ( ( vEBT_V4262088993061758097ft_nat @ X3 @ Xa2 @ Xb )
        = Y3 )
     => ( ( ( Xa2 = none_nat )
         => ( Y3 != none_nat ) )
       => ( ( ? [V2: nat] :
                ( Xa2
                = ( some_nat @ V2 ) )
           => ( ( Xb = none_nat )
             => ( Y3 != none_nat ) ) )
         => ~ ! [A4: nat] :
                ( ( Xa2
                  = ( some_nat @ A4 ) )
               => ! [B4: nat] :
                    ( ( Xb
                      = ( some_nat @ B4 ) )
                   => ( Y3
                     != ( some_nat @ ( X3 @ A4 @ B4 ) ) ) ) ) ) ) ) ).

% VEBT_internal.option_shift.elims
thf(fact_1562_vebt__buildup_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ( ( X3
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va2: nat] :
              ( X3
             != ( suc @ ( suc @ Va2 ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_1563_finite__has__minimal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1564_finite__has__minimal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( ( member_set_int @ X4 @ A2 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1565_finite__has__minimal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1566_finite__has__minimal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1567_finite__has__minimal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1568_finite__has__minimal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal
thf(fact_1569_finite__has__maximal,axiom,
    ! [A2: set_real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A2 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1570_finite__has__maximal,axiom,
    ! [A2: set_set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( A2 != bot_bot_set_set_int )
       => ? [X4: set_int] :
            ( ( member_set_int @ X4 @ A2 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1571_finite__has__maximal,axiom,
    ! [A2: set_rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( A2 != bot_bot_set_rat )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A2 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1572_finite__has__maximal,axiom,
    ! [A2: set_num] :
      ( ( finite_finite_num @ A2 )
     => ( ( A2 != bot_bot_set_num )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A2 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1573_finite__has__maximal,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1574_finite__has__maximal,axiom,
    ! [A2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A2 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal
thf(fact_1575_bot__empty__eq,axiom,
    ( bot_bot_set_nat_o
    = ( ^ [X: set_nat] : ( member_set_nat @ X @ bot_bot_set_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1576_bot__empty__eq,axiom,
    ( bot_bo482883023278783056_nat_o
    = ( ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq
thf(fact_1577_bot__empty__eq,axiom,
    ( bot_bot_real_o
    = ( ^ [X: real] : ( member_real @ X @ bot_bot_set_real ) ) ) ).

% bot_empty_eq
thf(fact_1578_bot__empty__eq,axiom,
    ( bot_bot_nat_o
    = ( ^ [X: nat] : ( member_nat @ X @ bot_bot_set_nat ) ) ) ).

% bot_empty_eq
thf(fact_1579_bot__empty__eq,axiom,
    ( bot_bot_int_o
    = ( ^ [X: int] : ( member_int @ X @ bot_bot_set_int ) ) ) ).

% bot_empty_eq
thf(fact_1580_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_1581_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_1582_Collect__empty__eq__bot,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ( ( collec3392354462482085612at_nat @ P )
        = bot_bo2099793752762293965at_nat )
      = ( P = bot_bo482883023278783056_nat_o ) ) ).

% Collect_empty_eq_bot
thf(fact_1583_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_1584_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_1585_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_1586_infinite__nat__iff__unbounded__le,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M2: nat] :
          ? [N: nat] :
            ( ( ord_less_eq_nat @ M2 @ N )
            & ( member_nat @ N @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_1587_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M2: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N5 )
         => ( ord_less_eq_nat @ X @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_1588_unbounded__k__infinite,axiom,
    ! [K: nat,S3: set_nat] :
      ( ! [M3: nat] :
          ( ( ord_less_nat @ K @ M3 )
         => ? [N6: nat] :
              ( ( ord_less_nat @ M3 @ N6 )
              & ( member_nat @ N6 @ S3 ) ) )
     => ~ ( finite_finite_nat @ S3 ) ) ).

% unbounded_k_infinite
thf(fact_1589_bounded__nat__set__is__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ N7 )
         => ( ord_less_nat @ X4 @ N2 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% bounded_nat_set_is_finite
thf(fact_1590_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_1591_finite__maxlen,axiom,
    ! [M7: set_list_o] :
      ( ( finite_finite_list_o @ M7 )
     => ? [N3: nat] :
        ! [X2: list_o] :
          ( ( member_list_o @ X2 @ M7 )
         => ( ord_less_nat @ ( size_size_list_o @ X2 ) @ N3 ) ) ) ).

% finite_maxlen
thf(fact_1592_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_1593_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_1594_subrelI,axiom,
    ! [R2: set_Pr1261947904930325089at_nat,S: set_Pr1261947904930325089at_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ R2 )
         => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X4 @ Y5 ) @ S ) )
     => ( ord_le3146513528884898305at_nat @ R2 @ S ) ) ).

% subrelI
thf(fact_1595_subrelI,axiom,
    ! [R2: set_Pr958786334691620121nt_int,S: set_Pr958786334691620121nt_int] :
      ( ! [X4: int,Y5: int] :
          ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ R2 )
         => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X4 @ Y5 ) @ S ) )
     => ( ord_le2843351958646193337nt_int @ R2 @ S ) ) ).

% subrelI
thf(fact_1596_subrelI,axiom,
    ! [R2: set_Pr8056137968301705908nteger,S: set_Pr8056137968301705908nteger] :
      ( ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
          ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y5 ) @ R2 )
         => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X4 @ Y5 ) @ S ) )
     => ( ord_le3216752416896350996nteger @ R2 @ S ) ) ).

% subrelI
thf(fact_1597_subrelI,axiom,
    ! [R2: set_Pr1281608226676607948nteger,S: set_Pr1281608226676607948nteger] :
      ( ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
          ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y5 ) @ R2 )
         => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X4 @ Y5 ) @ S ) )
     => ( ord_le653643898420964396nteger @ R2 @ S ) ) ).

% subrelI
thf(fact_1598_subrelI,axiom,
    ! [R2: set_Pr9222295170931077689nt_int,S: set_Pr9222295170931077689nt_int] :
      ( ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
          ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y5 ) @ R2 )
         => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X4 @ Y5 ) @ S ) )
     => ( ord_le8725513860283290265nt_int @ R2 @ S ) ) ).

% subrelI
thf(fact_1599_subrelI,axiom,
    ! [R2: set_Pr1872883991513573699nt_int,S: set_Pr1872883991513573699nt_int] :
      ( ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
          ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y5 ) @ R2 )
         => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X4 @ Y5 ) @ S ) )
     => ( ord_le135402666524580259nt_int @ R2 @ S ) ) ).

% subrelI
thf(fact_1600_bounded__Max__nat,axiom,
    ! [P: nat > $o,X3: nat,M7: nat] :
      ( ( P @ X3 )
     => ( ! [X4: nat] :
            ( ( P @ X4 )
           => ( ord_less_eq_nat @ X4 @ M7 ) )
       => ~ ! [M3: nat] :
              ( ( P @ M3 )
             => ~ ! [X2: nat] :
                    ( ( P @ X2 )
                   => ( ord_less_eq_nat @ X2 @ M3 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_1601_finite__has__minimal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A2 )
            & ( ord_less_eq_real @ X4 @ A )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1602_finite__has__minimal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A2 )
            & ( ord_less_eq_set_nat @ X4 @ A )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1603_finite__has__minimal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X4: set_int] :
            ( ( member_set_int @ X4 @ A2 )
            & ( ord_less_eq_set_int @ X4 @ A )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1604_finite__has__minimal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A2 )
            & ( ord_less_eq_rat @ X4 @ A )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1605_finite__has__minimal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A2 )
            & ( ord_less_eq_num @ X4 @ A )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1606_finite__has__minimal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ( ord_less_eq_nat @ X4 @ A )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1607_finite__has__minimal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A2 )
            & ( ord_less_eq_int @ X4 @ A )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ Xa @ X4 )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_minimal2
thf(fact_1608_finite__has__maximal2,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ? [X4: real] :
            ( ( member_real @ X4 @ A2 )
            & ( ord_less_eq_real @ A @ X4 )
            & ! [Xa: real] :
                ( ( member_real @ Xa @ A2 )
               => ( ( ord_less_eq_real @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1609_finite__has__maximal2,axiom,
    ! [A2: set_set_nat,A: set_nat] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( member_set_nat @ A @ A2 )
       => ? [X4: set_nat] :
            ( ( member_set_nat @ X4 @ A2 )
            & ( ord_less_eq_set_nat @ A @ X4 )
            & ! [Xa: set_nat] :
                ( ( member_set_nat @ Xa @ A2 )
               => ( ( ord_less_eq_set_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1610_finite__has__maximal2,axiom,
    ! [A2: set_set_int,A: set_int] :
      ( ( finite6197958912794628473et_int @ A2 )
     => ( ( member_set_int @ A @ A2 )
       => ? [X4: set_int] :
            ( ( member_set_int @ X4 @ A2 )
            & ( ord_less_eq_set_int @ A @ X4 )
            & ! [Xa: set_int] :
                ( ( member_set_int @ Xa @ A2 )
               => ( ( ord_less_eq_set_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1611_finite__has__maximal2,axiom,
    ! [A2: set_rat,A: rat] :
      ( ( finite_finite_rat @ A2 )
     => ( ( member_rat @ A @ A2 )
       => ? [X4: rat] :
            ( ( member_rat @ X4 @ A2 )
            & ( ord_less_eq_rat @ A @ X4 )
            & ! [Xa: rat] :
                ( ( member_rat @ Xa @ A2 )
               => ( ( ord_less_eq_rat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1612_finite__has__maximal2,axiom,
    ! [A2: set_num,A: num] :
      ( ( finite_finite_num @ A2 )
     => ( ( member_num @ A @ A2 )
       => ? [X4: num] :
            ( ( member_num @ X4 @ A2 )
            & ( ord_less_eq_num @ A @ X4 )
            & ! [Xa: num] :
                ( ( member_num @ Xa @ A2 )
               => ( ( ord_less_eq_num @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1613_finite__has__maximal2,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ? [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
            & ( ord_less_eq_nat @ A @ X4 )
            & ! [Xa: nat] :
                ( ( member_nat @ Xa @ A2 )
               => ( ( ord_less_eq_nat @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1614_finite__has__maximal2,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ? [X4: int] :
            ( ( member_int @ X4 @ A2 )
            & ( ord_less_eq_int @ A @ X4 )
            & ! [Xa: int] :
                ( ( member_int @ Xa @ A2 )
               => ( ( ord_less_eq_int @ X4 @ Xa )
                 => ( X4 = Xa ) ) ) ) ) ) ).

% finite_has_maximal2
thf(fact_1615_finite__transitivity__chain,axiom,
    ! [A2: set_set_nat,R: set_nat > set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ! [X4: set_nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: set_nat,Y5: set_nat,Z3: set_nat] :
              ( ( R @ X4 @ Y5 )
             => ( ( R @ Y5 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: set_nat] :
                ( ( member_set_nat @ X4 @ A2 )
               => ? [Y6: set_nat] :
                    ( ( member_set_nat @ Y6 @ A2 )
                    & ( R @ X4 @ Y6 ) ) )
           => ( A2 = bot_bot_set_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1616_finite__transitivity__chain,axiom,
    ! [A2: set_complex,R: complex > complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: complex,Y5: complex,Z3: complex] :
              ( ( R @ X4 @ Y5 )
             => ( ( R @ Y5 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ A2 )
               => ? [Y6: complex] :
                    ( ( member_complex @ Y6 @ A2 )
                    & ( R @ X4 @ Y6 ) ) )
           => ( A2 = bot_bot_set_complex ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1617_finite__transitivity__chain,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,R: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ! [X4: product_prod_nat_nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: product_prod_nat_nat,Y5: product_prod_nat_nat,Z3: product_prod_nat_nat] :
              ( ( R @ X4 @ Y5 )
             => ( ( R @ Y5 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X4 @ A2 )
               => ? [Y6: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ Y6 @ A2 )
                    & ( R @ X4 @ Y6 ) ) )
           => ( A2 = bot_bo2099793752762293965at_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1618_finite__transitivity__chain,axiom,
    ! [A2: set_real,R: real > real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X4: real] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: real,Y5: real,Z3: real] :
              ( ( R @ X4 @ Y5 )
             => ( ( R @ Y5 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ A2 )
               => ? [Y6: real] :
                    ( ( member_real @ Y6 @ A2 )
                    & ( R @ X4 @ Y6 ) ) )
           => ( A2 = bot_bot_set_real ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1619_finite__transitivity__chain,axiom,
    ! [A2: set_nat,R: nat > nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: nat,Y5: nat,Z3: nat] :
              ( ( R @ X4 @ Y5 )
             => ( ( R @ Y5 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ A2 )
               => ? [Y6: nat] :
                    ( ( member_nat @ Y6 @ A2 )
                    & ( R @ X4 @ Y6 ) ) )
           => ( A2 = bot_bot_set_nat ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1620_finite__transitivity__chain,axiom,
    ! [A2: set_int,R: int > int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ~ ( R @ X4 @ X4 )
       => ( ! [X4: int,Y5: int,Z3: int] :
              ( ( R @ X4 @ Y5 )
             => ( ( R @ Y5 @ Z3 )
               => ( R @ X4 @ Z3 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ A2 )
               => ? [Y6: int] :
                    ( ( member_int @ Y6 @ A2 )
                    & ( R @ X4 @ Y6 ) ) )
           => ( A2 = bot_bot_set_int ) ) ) ) ) ).

% finite_transitivity_chain
thf(fact_1621_infinite__imp__nonempty,axiom,
    ! [S3: set_complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ( S3 != bot_bot_set_complex ) ) ).

% infinite_imp_nonempty
thf(fact_1622_infinite__imp__nonempty,axiom,
    ! [S3: set_Pr1261947904930325089at_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S3 )
     => ( S3 != bot_bo2099793752762293965at_nat ) ) ).

% infinite_imp_nonempty
thf(fact_1623_infinite__imp__nonempty,axiom,
    ! [S3: set_real] :
      ( ~ ( finite_finite_real @ S3 )
     => ( S3 != bot_bot_set_real ) ) ).

% infinite_imp_nonempty
thf(fact_1624_infinite__imp__nonempty,axiom,
    ! [S3: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ( S3 != bot_bot_set_nat ) ) ).

% infinite_imp_nonempty
thf(fact_1625_infinite__imp__nonempty,axiom,
    ! [S3: set_int] :
      ( ~ ( finite_finite_int @ S3 )
     => ( S3 != bot_bot_set_int ) ) ).

% infinite_imp_nonempty
thf(fact_1626_finite_OemptyI,axiom,
    finite3207457112153483333omplex @ bot_bot_set_complex ).

% finite.emptyI
thf(fact_1627_finite_OemptyI,axiom,
    finite6177210948735845034at_nat @ bot_bo2099793752762293965at_nat ).

% finite.emptyI
thf(fact_1628_finite_OemptyI,axiom,
    finite_finite_real @ bot_bot_set_real ).

% finite.emptyI
thf(fact_1629_finite_OemptyI,axiom,
    finite_finite_nat @ bot_bot_set_nat ).

% finite.emptyI
thf(fact_1630_finite_OemptyI,axiom,
    finite_finite_int @ bot_bot_set_int ).

% finite.emptyI
thf(fact_1631_rev__finite__subset,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1632_rev__finite__subset,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1633_rev__finite__subset,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
       => ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1634_rev__finite__subset,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ A2 @ B2 )
       => ( finite_finite_int @ A2 ) ) ) ).

% rev_finite_subset
thf(fact_1635_infinite__super,axiom,
    ! [S3: set_nat,T3: set_nat] :
      ( ( ord_less_eq_set_nat @ S3 @ T3 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_1636_infinite__super,axiom,
    ! [S3: set_complex,T3: set_complex] :
      ( ( ord_le211207098394363844omplex @ S3 @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ T3 ) ) ) ).

% infinite_super
thf(fact_1637_infinite__super,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,T3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ S3 @ T3 )
     => ( ~ ( finite6177210948735845034at_nat @ S3 )
       => ~ ( finite6177210948735845034at_nat @ T3 ) ) ) ).

% infinite_super
thf(fact_1638_infinite__super,axiom,
    ! [S3: set_int,T3: set_int] :
      ( ( ord_less_eq_set_int @ S3 @ T3 )
     => ( ~ ( finite_finite_int @ S3 )
       => ~ ( finite_finite_int @ T3 ) ) ) ).

% infinite_super
thf(fact_1639_finite__subset,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( finite_finite_nat @ B2 )
       => ( finite_finite_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_1640_finite__subset,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_subset
thf(fact_1641_finite__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ( finite6177210948735845034at_nat @ B2 )
       => ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% finite_subset
thf(fact_1642_finite__subset,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( finite_finite_int @ B2 )
       => ( finite_finite_int @ A2 ) ) ) ).

% finite_subset
thf(fact_1643_finite__psubset__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [A7: set_nat] :
            ( ( finite_finite_nat @ A7 )
           => ( ! [B7: set_nat] :
                  ( ( ord_less_set_nat @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_1644_finite__psubset__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ! [A7: set_int] :
            ( ( finite_finite_int @ A7 )
           => ( ! [B7: set_int] :
                  ( ( ord_less_set_int @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_1645_finite__psubset__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [A7: set_complex] :
            ( ( finite3207457112153483333omplex @ A7 )
           => ( ! [B7: set_complex] :
                  ( ( ord_less_set_complex @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_1646_finite__psubset__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ! [A7: set_Pr1261947904930325089at_nat] :
            ( ( finite6177210948735845034at_nat @ A7 )
           => ( ! [B7: set_Pr1261947904930325089at_nat] :
                  ( ( ord_le7866589430770878221at_nat @ B7 @ A7 )
                 => ( P @ B7 ) )
             => ( P @ A7 ) ) )
       => ( P @ A2 ) ) ) ).

% finite_psubset_induct
thf(fact_1647_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N5: set_nat] :
        ? [M2: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N5 )
         => ( ord_less_nat @ X @ M2 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_1648_infinite__nat__iff__unbounded,axiom,
    ! [S3: set_nat] :
      ( ( ~ ( finite_finite_nat @ S3 ) )
      = ( ! [M2: nat] :
          ? [N: nat] :
            ( ( ord_less_nat @ M2 @ N )
            & ( member_nat @ N @ S3 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_1649_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( ( member_complex @ X2 @ S3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8794016678065449205x_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1650_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( ( member_real @ X2 @ S3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic8440615504127631091l_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1651_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ S3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic488527866317076247t_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1652_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( ( member_int @ X2 @ S3 )
              & ( ord_less_real @ ( F @ X2 ) @ ( F @ ( lattic2675449441010098035t_real @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1653_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( ( member_complex @ X2 @ S3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1654_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( ( member_real @ X2 @ S3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1655_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ~ ? [X2: nat] :
              ( ( member_nat @ X2 @ S3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1656_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_int,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ~ ? [X2: int] :
              ( ( member_int @ X2 @ S3 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1657_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_complex,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ~ ? [X2: complex] :
              ( ( member_complex @ X2 @ S3 )
              & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1922116423962787043ex_num @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1658_arg__min__if__finite_I2_J,axiom,
    ! [S3: set_real,F: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ~ ? [X2: real] :
              ( ( member_real @ X2 @ S3 )
              & ( ord_less_num @ ( F @ X2 ) @ ( F @ ( lattic1613168225601753569al_num @ F @ S3 ) ) ) ) ) ) ).

% arg_min_if_finite(2)
thf(fact_1659_arg__min__least,axiom,
    ! [S3: set_complex,Y3: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y3 @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic4729654577720512673ex_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1660_arg__min__least,axiom,
    ! [S3: set_real,Y3: real,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real @ Y3 @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic4420706379359479199al_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1661_arg__min__least,axiom,
    ! [S3: set_nat,Y3: nat,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat @ Y3 @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic6811802900495863747at_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1662_arg__min__least,axiom,
    ! [S3: set_int,Y3: int,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( member_int @ Y3 @ S3 )
         => ( ord_less_eq_rat @ ( F @ ( lattic7811156612396918303nt_rat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1663_arg__min__least,axiom,
    ! [S3: set_complex,Y3: complex,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y3 @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic1922116423962787043ex_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1664_arg__min__least,axiom,
    ! [S3: set_real,Y3: real,F: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real @ Y3 @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic1613168225601753569al_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1665_arg__min__least,axiom,
    ! [S3: set_nat,Y3: nat,F: nat > num] :
      ( ( finite_finite_nat @ S3 )
     => ( ( S3 != bot_bot_set_nat )
       => ( ( member_nat @ Y3 @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic4004264746738138117at_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1666_arg__min__least,axiom,
    ! [S3: set_int,Y3: int,F: int > num] :
      ( ( finite_finite_int @ S3 )
     => ( ( S3 != bot_bot_set_int )
       => ( ( member_int @ Y3 @ S3 )
         => ( ord_less_eq_num @ ( F @ ( lattic5003618458639192673nt_num @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1667_arg__min__least,axiom,
    ! [S3: set_complex,Y3: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( S3 != bot_bot_set_complex )
       => ( ( member_complex @ Y3 @ S3 )
         => ( ord_less_eq_nat @ ( F @ ( lattic5364784637807008409ex_nat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1668_arg__min__least,axiom,
    ! [S3: set_real,Y3: real,F: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( S3 != bot_bot_set_real )
       => ( ( member_real @ Y3 @ S3 )
         => ( ord_less_eq_nat @ ( F @ ( lattic5055836439445974935al_nat @ F @ S3 ) ) @ ( F @ Y3 ) ) ) ) ) ).

% arg_min_least
thf(fact_1669_nat__descend__induct,axiom,
    ! [N2: nat,P: nat > $o,M: nat] :
      ( ! [K2: nat] :
          ( ( ord_less_nat @ N2 @ K2 )
         => ( P @ K2 ) )
     => ( ! [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
           => ( ! [I4: nat] :
                  ( ( ord_less_nat @ K2 @ I4 )
                 => ( P @ I4 ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_1670_subset__emptyI,axiom,
    ! [A2: set_set_nat] :
      ( ! [X4: set_nat] :
          ~ ( member_set_nat @ X4 @ A2 )
     => ( ord_le6893508408891458716et_nat @ A2 @ bot_bot_set_set_nat ) ) ).

% subset_emptyI
thf(fact_1671_subset__emptyI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ! [X4: product_prod_nat_nat] :
          ~ ( member8440522571783428010at_nat @ X4 @ A2 )
     => ( ord_le3146513528884898305at_nat @ A2 @ bot_bo2099793752762293965at_nat ) ) ).

% subset_emptyI
thf(fact_1672_subset__emptyI,axiom,
    ! [A2: set_real] :
      ( ! [X4: real] :
          ~ ( member_real @ X4 @ A2 )
     => ( ord_less_eq_set_real @ A2 @ bot_bot_set_real ) ) ).

% subset_emptyI
thf(fact_1673_subset__emptyI,axiom,
    ! [A2: set_nat] :
      ( ! [X4: nat] :
          ~ ( member_nat @ X4 @ A2 )
     => ( ord_less_eq_set_nat @ A2 @ bot_bot_set_nat ) ) ).

% subset_emptyI
thf(fact_1674_subset__emptyI,axiom,
    ! [A2: set_int] :
      ( ! [X4: int] :
          ~ ( member_int @ X4 @ A2 )
     => ( ord_less_eq_set_int @ A2 @ bot_bot_set_int ) ) ).

% subset_emptyI
thf(fact_1675_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_12: nat] : ( P @ X_12 )
       => ? [N3: nat] :
            ( ~ ( P @ N3 )
            & ( P @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_1676_list__decode_Ocases,axiom,
    ! [X3: nat] :
      ( ( X3 != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X3
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_1677_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_1678_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_1679_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_1680_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1681_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_1682_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_1683_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_1684_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_1685_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_1686_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_1687_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_1688_field__lbound__gt__zero,axiom,
    ! [D1: rat,D2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ D1 )
     => ( ( ord_less_rat @ zero_zero_rat @ D2 )
       => ? [E: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ E )
            & ( ord_less_rat @ E @ D1 )
            & ( ord_less_rat @ E @ D2 ) ) ) ) ).

% field_lbound_gt_zero
thf(fact_1689_less__by__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( A2 = bot_bo2099793752762293965at_nat )
     => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).

% less_by_empty
thf(fact_1690_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1691_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1692_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1693_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1694_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1695_verit__la__disequality,axiom,
    ! [A: rat,B: rat] :
      ( ( A = B )
      | ~ ( ord_less_eq_rat @ A @ B )
      | ~ ( ord_less_eq_rat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1696_verit__la__disequality,axiom,
    ! [A: num,B: num] :
      ( ( A = B )
      | ~ ( ord_less_eq_num @ A @ B )
      | ~ ( ord_less_eq_num @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1697_verit__la__disequality,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
      | ~ ( ord_less_eq_nat @ A @ B )
      | ~ ( ord_less_eq_nat @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1698_verit__la__disequality,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
      | ~ ( ord_less_eq_int @ A @ B )
      | ~ ( ord_less_eq_int @ B @ A ) ) ).

% verit_la_disequality
thf(fact_1699_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1700_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1701_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1702_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1703_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1704_ssubst__Pair__rhs,axiom,
    ! [R2: nat,S: nat,R: set_Pr1261947904930325089at_nat,S4: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1705_ssubst__Pair__rhs,axiom,
    ! [R2: int,S: int,R: set_Pr958786334691620121nt_int,S4: int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member5262025264175285858nt_int @ ( product_Pair_int_int @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1706_ssubst__Pair__rhs,axiom,
    ! [R2: code_integer > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr8056137968301705908nteger,S4: produc8923325533196201883nteger] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1707_ssubst__Pair__rhs,axiom,
    ! [R2: produc6241069584506657477e_term > option6357759511663192854e_term,S: produc8923325533196201883nteger,R: set_Pr1281608226676607948nteger,S4: produc8923325533196201883nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1708_ssubst__Pair__rhs,axiom,
    ! [R2: produc8551481072490612790e_term > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr9222295170931077689nt_int,S4: product_prod_int_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1709_ssubst__Pair__rhs,axiom,
    ! [R2: int > option6357759511663192854e_term,S: product_prod_int_int,R: set_Pr1872883991513573699nt_int,S4: product_prod_int_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S ) @ R )
     => ( ( S4 = S )
       => ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ R2 @ S4 ) @ R ) ) ) ).

% ssubst_Pair_rhs
thf(fact_1710_prod__decode__aux_Ocases,axiom,
    ! [X3: product_prod_nat_nat] :
      ~ ! [K2: nat,M3: nat] :
          ( X3
         != ( product_Pair_nat_nat @ K2 @ M3 ) ) ).

% prod_decode_aux.cases
thf(fact_1711_verit__comp__simplify1_I3_J,axiom,
    ! [B6: real,A6: real] :
      ( ( ~ ( ord_less_eq_real @ B6 @ A6 ) )
      = ( ord_less_real @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1712_verit__comp__simplify1_I3_J,axiom,
    ! [B6: rat,A6: rat] :
      ( ( ~ ( ord_less_eq_rat @ B6 @ A6 ) )
      = ( ord_less_rat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1713_verit__comp__simplify1_I3_J,axiom,
    ! [B6: num,A6: num] :
      ( ( ~ ( ord_less_eq_num @ B6 @ A6 ) )
      = ( ord_less_num @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1714_verit__comp__simplify1_I3_J,axiom,
    ! [B6: nat,A6: nat] :
      ( ( ~ ( ord_less_eq_nat @ B6 @ A6 ) )
      = ( ord_less_nat @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1715_verit__comp__simplify1_I3_J,axiom,
    ! [B6: int,A6: int] :
      ( ( ~ ( ord_less_eq_int @ B6 @ A6 ) )
      = ( ord_less_int @ A6 @ B6 ) ) ).

% verit_comp_simplify1(3)
thf(fact_1716_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_1717_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_1718_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_1719_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1720_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_1721_minf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ~ ( ord_less_eq_real @ T @ X2 ) ) ).

% minf(8)
thf(fact_1722_minf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ~ ( ord_less_eq_rat @ T @ X2 ) ) ).

% minf(8)
thf(fact_1723_minf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ~ ( ord_less_eq_num @ T @ X2 ) ) ).

% minf(8)
thf(fact_1724_minf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ~ ( ord_less_eq_nat @ T @ X2 ) ) ).

% minf(8)
thf(fact_1725_minf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ~ ( ord_less_eq_int @ T @ X2 ) ) ).

% minf(8)
thf(fact_1726_minf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( ord_less_eq_real @ X2 @ T ) ) ).

% minf(6)
thf(fact_1727_minf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( ord_less_eq_rat @ X2 @ T ) ) ).

% minf(6)
thf(fact_1728_minf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ( ord_less_eq_num @ X2 @ T ) ) ).

% minf(6)
thf(fact_1729_minf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( ord_less_eq_nat @ X2 @ T ) ) ).

% minf(6)
thf(fact_1730_minf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( ord_less_eq_int @ X2 @ T ) ) ).

% minf(6)
thf(fact_1731_pinf_I8_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( ord_less_eq_real @ T @ X2 ) ) ).

% pinf(8)
thf(fact_1732_pinf_I8_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( ord_less_eq_rat @ T @ X2 ) ) ).

% pinf(8)
thf(fact_1733_pinf_I8_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ( ord_less_eq_num @ T @ X2 ) ) ).

% pinf(8)
thf(fact_1734_pinf_I8_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( ord_less_eq_nat @ T @ X2 ) ) ).

% pinf(8)
thf(fact_1735_pinf_I8_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( ord_less_eq_int @ T @ X2 ) ) ).

% pinf(8)
thf(fact_1736_pinf_I6_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ~ ( ord_less_eq_real @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1737_pinf_I6_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ~ ( ord_less_eq_rat @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1738_pinf_I6_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ~ ( ord_less_eq_num @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1739_pinf_I6_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ~ ( ord_less_eq_nat @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1740_pinf_I6_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ~ ( ord_less_eq_int @ X2 @ T ) ) ).

% pinf(6)
thf(fact_1741_complete__interval,axiom,
    ! [A: real,B: real,P: real > $o] :
      ( ( ord_less_real @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C2: real] :
              ( ( ord_less_eq_real @ A @ C2 )
              & ( ord_less_eq_real @ C2 @ B )
              & ! [X2: real] :
                  ( ( ( ord_less_eq_real @ A @ X2 )
                    & ( ord_less_real @ X2 @ C2 ) )
                 => ( P @ X2 ) )
              & ! [D3: real] :
                  ( ! [X4: real] :
                      ( ( ( ord_less_eq_real @ A @ X4 )
                        & ( ord_less_real @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_real @ D3 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1742_complete__interval,axiom,
    ! [A: nat,B: nat,P: nat > $o] :
      ( ( ord_less_nat @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C2: nat] :
              ( ( ord_less_eq_nat @ A @ C2 )
              & ( ord_less_eq_nat @ C2 @ B )
              & ! [X2: nat] :
                  ( ( ( ord_less_eq_nat @ A @ X2 )
                    & ( ord_less_nat @ X2 @ C2 ) )
                 => ( P @ X2 ) )
              & ! [D3: nat] :
                  ( ! [X4: nat] :
                      ( ( ( ord_less_eq_nat @ A @ X4 )
                        & ( ord_less_nat @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_nat @ D3 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1743_complete__interval,axiom,
    ! [A: int,B: int,P: int > $o] :
      ( ( ord_less_int @ A @ B )
     => ( ( P @ A )
       => ( ~ ( P @ B )
         => ? [C2: int] :
              ( ( ord_less_eq_int @ A @ C2 )
              & ( ord_less_eq_int @ C2 @ B )
              & ! [X2: int] :
                  ( ( ( ord_less_eq_int @ A @ X2 )
                    & ( ord_less_int @ X2 @ C2 ) )
                 => ( P @ X2 ) )
              & ! [D3: int] :
                  ( ! [X4: int] :
                      ( ( ( ord_less_eq_int @ A @ X4 )
                        & ( ord_less_int @ X4 @ D3 ) )
                     => ( P @ X4 ) )
                 => ( ord_less_eq_int @ D3 @ C2 ) ) ) ) ) ) ).

% complete_interval
thf(fact_1744_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1745_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_1746_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1747_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1748_find__Some__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat,X3: product_prod_nat_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
        = ( some_P7363390416028606310at_nat @ X3 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
            & ( X3
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1749_find__Some__iff,axiom,
    ! [P: num > $o,Xs: list_num,X3: num] :
      ( ( ( find_num @ P @ Xs )
        = ( some_num @ X3 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_num @ Xs ) )
            & ( P @ ( nth_num @ Xs @ I2 ) )
            & ( X3
              = ( nth_num @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1750_find__Some__iff,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P @ Xs )
        = ( some_VEBT_VEBT @ X3 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) )
            & ( X3
              = ( nth_VEBT_VEBT @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1751_find__Some__iff,axiom,
    ! [P: $o > $o,Xs: list_o,X3: $o] :
      ( ( ( find_o @ P @ Xs )
        = ( some_o @ X3 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
            & ( P @ ( nth_o @ Xs @ I2 ) )
            & ( X3
              = ( nth_o @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1752_find__Some__iff,axiom,
    ! [P: nat > $o,Xs: list_nat,X3: nat] :
      ( ( ( find_nat @ P @ Xs )
        = ( some_nat @ X3 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I2 ) )
            & ( X3
              = ( nth_nat @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1753_find__Some__iff,axiom,
    ! [P: int > $o,Xs: list_int,X3: int] :
      ( ( ( find_int @ P @ Xs )
        = ( some_int @ X3 ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
            & ( P @ ( nth_int @ Xs @ I2 ) )
            & ( X3
              = ( nth_int @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff
thf(fact_1754_find__Some__iff2,axiom,
    ! [X3: product_prod_nat_nat,P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( ( some_P7363390416028606310at_nat @ X3 )
        = ( find_P8199882355184865565at_nat @ P @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
            & ( X3
              = ( nth_Pr7617993195940197384at_nat @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1755_find__Some__iff2,axiom,
    ! [X3: num,P: num > $o,Xs: list_num] :
      ( ( ( some_num @ X3 )
        = ( find_num @ P @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_num @ Xs ) )
            & ( P @ ( nth_num @ Xs @ I2 ) )
            & ( X3
              = ( nth_num @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_num @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1756_find__Some__iff2,axiom,
    ! [X3: vEBT_VEBT,P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( ( some_VEBT_VEBT @ X3 )
        = ( find_VEBT_VEBT @ P @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( P @ ( nth_VEBT_VEBT @ Xs @ I2 ) )
            & ( X3
              = ( nth_VEBT_VEBT @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_VEBT_VEBT @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1757_find__Some__iff2,axiom,
    ! [X3: $o,P: $o > $o,Xs: list_o] :
      ( ( ( some_o @ X3 )
        = ( find_o @ P @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs ) )
            & ( P @ ( nth_o @ Xs @ I2 ) )
            & ( X3
              = ( nth_o @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_o @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1758_find__Some__iff2,axiom,
    ! [X3: nat,P: nat > $o,Xs: list_nat] :
      ( ( ( some_nat @ X3 )
        = ( find_nat @ P @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs ) )
            & ( P @ ( nth_nat @ Xs @ I2 ) )
            & ( X3
              = ( nth_nat @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_nat @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1759_find__Some__iff2,axiom,
    ! [X3: int,P: int > $o,Xs: list_int] :
      ( ( ( some_int @ X3 )
        = ( find_int @ P @ Xs ) )
      = ( ? [I2: nat] :
            ( ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs ) )
            & ( P @ ( nth_int @ Xs @ I2 ) )
            & ( X3
              = ( nth_int @ Xs @ I2 ) )
            & ! [J3: nat] :
                ( ( ord_less_nat @ J3 @ I2 )
               => ~ ( P @ ( nth_int @ Xs @ J3 ) ) ) ) ) ) ).

% find_Some_iff2
thf(fact_1760_triangle__0,axiom,
    ( ( nat_triangle @ zero_zero_nat )
    = zero_zero_nat ) ).

% triangle_0
thf(fact_1761_find__cong,axiom,
    ! [Xs: list_P6011104703257516679at_nat,Ys: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( find_P8199882355184865565at_nat @ P @ Xs )
          = ( find_P8199882355184865565at_nat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1762_find__cong,axiom,
    ! [Xs: list_real,Ys: list_real,P: real > $o,Q: real > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_real2 @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( find_real @ P @ Xs )
          = ( find_real @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1763_find__cong,axiom,
    ! [Xs: list_set_nat,Ys: list_set_nat,P: set_nat > $o,Q: set_nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: set_nat] :
            ( ( member_set_nat @ X4 @ ( set_set_nat2 @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( find_set_nat @ P @ Xs )
          = ( find_set_nat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1764_find__cong,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,P: vEBT_VEBT > $o,Q: vEBT_VEBT > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( find_VEBT_VEBT @ P @ Xs )
          = ( find_VEBT_VEBT @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1765_find__cong,axiom,
    ! [Xs: list_int,Ys: list_int,P: int > $o,Q: int > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( set_int2 @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( find_int @ P @ Xs )
          = ( find_int @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1766_find__cong,axiom,
    ! [Xs: list_nat,Ys: list_nat,P: nat > $o,Q: nat > $o] :
      ( ( Xs = Ys )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( set_nat2 @ Ys ) )
           => ( ( P @ X4 )
              = ( Q @ X4 ) ) )
       => ( ( find_nat @ P @ Xs )
          = ( find_nat @ Q @ Ys ) ) ) ) ).

% find_cong
thf(fact_1767_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_1768_pinf_I1_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1769_pinf_I1_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1770_pinf_I1_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( ( ord_less_num @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1771_pinf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1772_pinf_I1_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_1773_pinf_I2_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1774_pinf_I2_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1775_pinf_I2_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( ( ord_less_num @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1776_pinf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1777_pinf_I2_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ Z4 @ X4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z3 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_1778_pinf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1779_pinf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1780_pinf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1781_pinf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1782_pinf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(3)
thf(fact_1783_pinf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1784_pinf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1785_pinf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1786_pinf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1787_pinf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( X2 != T ) ) ).

% pinf(4)
thf(fact_1788_pinf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ~ ( ord_less_real @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1789_pinf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ~ ( ord_less_rat @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1790_pinf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ~ ( ord_less_num @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1791_pinf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ~ ( ord_less_nat @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1792_pinf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ~ ( ord_less_int @ X2 @ T ) ) ).

% pinf(5)
thf(fact_1793_pinf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( ord_less_real @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1794_pinf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( ord_less_rat @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1795_pinf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z3 @ X2 )
     => ( ord_less_num @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1796_pinf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( ord_less_nat @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1797_pinf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( ord_less_int @ T @ X2 ) ) ).

% pinf(7)
thf(fact_1798_minf_I1_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1799_minf_I1_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1800_minf_I1_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( ( ord_less_num @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1801_minf_I1_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1802_minf_I1_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                & ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_1803_minf_I2_J,axiom,
    ! [P: real > $o,P5: real > $o,Q: real > $o,Q3: real > $o] :
      ( ? [Z4: real] :
        ! [X4: real] :
          ( ( ord_less_real @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: real] :
          ! [X4: real] :
            ( ( ord_less_real @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1804_minf_I2_J,axiom,
    ! [P: rat > $o,P5: rat > $o,Q: rat > $o,Q3: rat > $o] :
      ( ? [Z4: rat] :
        ! [X4: rat] :
          ( ( ord_less_rat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: rat] :
          ! [X4: rat] :
            ( ( ord_less_rat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1805_minf_I2_J,axiom,
    ! [P: num > $o,P5: num > $o,Q: num > $o,Q3: num > $o] :
      ( ? [Z4: num] :
        ! [X4: num] :
          ( ( ord_less_num @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: num] :
          ! [X4: num] :
            ( ( ord_less_num @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: num] :
          ! [X2: num] :
            ( ( ord_less_num @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1806_minf_I2_J,axiom,
    ! [P: nat > $o,P5: nat > $o,Q: nat > $o,Q3: nat > $o] :
      ( ? [Z4: nat] :
        ! [X4: nat] :
          ( ( ord_less_nat @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: nat] :
          ! [X4: nat] :
            ( ( ord_less_nat @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1807_minf_I2_J,axiom,
    ! [P: int > $o,P5: int > $o,Q: int > $o,Q3: int > $o] :
      ( ? [Z4: int] :
        ! [X4: int] :
          ( ( ord_less_int @ X4 @ Z4 )
         => ( ( P @ X4 )
            = ( P5 @ X4 ) ) )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( Q @ X4 )
              = ( Q3 @ X4 ) ) )
       => ? [Z3: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z3 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P5 @ X2 )
                | ( Q3 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_1808_minf_I3_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1809_minf_I3_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1810_minf_I3_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1811_minf_I3_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1812_minf_I3_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(3)
thf(fact_1813_minf_I4_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1814_minf_I4_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1815_minf_I4_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1816_minf_I4_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1817_minf_I4_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( X2 != T ) ) ).

% minf(4)
thf(fact_1818_minf_I5_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( ord_less_real @ X2 @ T ) ) ).

% minf(5)
thf(fact_1819_minf_I5_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( ord_less_rat @ X2 @ T ) ) ).

% minf(5)
thf(fact_1820_minf_I5_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ( ord_less_num @ X2 @ T ) ) ).

% minf(5)
thf(fact_1821_minf_I5_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( ord_less_nat @ X2 @ T ) ) ).

% minf(5)
thf(fact_1822_minf_I5_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( ord_less_int @ X2 @ T ) ) ).

% minf(5)
thf(fact_1823_minf_I7_J,axiom,
    ! [T: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ~ ( ord_less_real @ T @ X2 ) ) ).

% minf(7)
thf(fact_1824_minf_I7_J,axiom,
    ! [T: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ~ ( ord_less_rat @ T @ X2 ) ) ).

% minf(7)
thf(fact_1825_minf_I7_J,axiom,
    ! [T: num] :
    ? [Z3: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z3 )
     => ~ ( ord_less_num @ T @ X2 ) ) ).

% minf(7)
thf(fact_1826_minf_I7_J,axiom,
    ! [T: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ~ ( ord_less_nat @ T @ X2 ) ) ).

% minf(7)
thf(fact_1827_minf_I7_J,axiom,
    ! [T: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ~ ( ord_less_int @ T @ X2 ) ) ).

% minf(7)
thf(fact_1828_is__num__normalize_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1829_is__num__normalize_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1830_is__num__normalize_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% is_num_normalize(1)
thf(fact_1831_find__None__iff2,axiom,
    ! [P: real > $o,Xs: list_real] :
      ( ( none_real
        = ( find_real @ P @ Xs ) )
      = ( ~ ? [X: real] :
              ( ( member_real @ X @ ( set_real2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_1832_find__None__iff2,axiom,
    ! [P: set_nat > $o,Xs: list_set_nat] :
      ( ( none_set_nat
        = ( find_set_nat @ P @ Xs ) )
      = ( ~ ? [X: set_nat] :
              ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_1833_find__None__iff2,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( none_VEBT_VEBT
        = ( find_VEBT_VEBT @ P @ Xs ) )
      = ( ~ ? [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_1834_find__None__iff2,axiom,
    ! [P: int > $o,Xs: list_int] :
      ( ( none_int
        = ( find_int @ P @ Xs ) )
      = ( ~ ? [X: int] :
              ( ( member_int @ X @ ( set_int2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_1835_find__None__iff2,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( none_nat
        = ( find_nat @ P @ Xs ) )
      = ( ~ ? [X: nat] :
              ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_1836_find__None__iff2,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( none_P5556105721700978146at_nat
        = ( find_P8199882355184865565at_nat @ P @ Xs ) )
      = ( ~ ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_1837_find__None__iff2,axiom,
    ! [P: num > $o,Xs: list_num] :
      ( ( none_num
        = ( find_num @ P @ Xs ) )
      = ( ~ ? [X: num] :
              ( ( member_num @ X @ ( set_num2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff2
thf(fact_1838_find__None__iff,axiom,
    ! [P: real > $o,Xs: list_real] :
      ( ( ( find_real @ P @ Xs )
        = none_real )
      = ( ~ ? [X: real] :
              ( ( member_real @ X @ ( set_real2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_1839_find__None__iff,axiom,
    ! [P: set_nat > $o,Xs: list_set_nat] :
      ( ( ( find_set_nat @ P @ Xs )
        = none_set_nat )
      = ( ~ ? [X: set_nat] :
              ( ( member_set_nat @ X @ ( set_set_nat2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_1840_find__None__iff,axiom,
    ! [P: vEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ( ( find_VEBT_VEBT @ P @ Xs )
        = none_VEBT_VEBT )
      = ( ~ ? [X: vEBT_VEBT] :
              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_1841_find__None__iff,axiom,
    ! [P: int > $o,Xs: list_int] :
      ( ( ( find_int @ P @ Xs )
        = none_int )
      = ( ~ ? [X: int] :
              ( ( member_int @ X @ ( set_int2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_1842_find__None__iff,axiom,
    ! [P: nat > $o,Xs: list_nat] :
      ( ( ( find_nat @ P @ Xs )
        = none_nat )
      = ( ~ ? [X: nat] :
              ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_1843_find__None__iff,axiom,
    ! [P: product_prod_nat_nat > $o,Xs: list_P6011104703257516679at_nat] :
      ( ( ( find_P8199882355184865565at_nat @ P @ Xs )
        = none_P5556105721700978146at_nat )
      = ( ~ ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_1844_find__None__iff,axiom,
    ! [P: num > $o,Xs: list_num] :
      ( ( ( find_num @ P @ Xs )
        = none_num )
      = ( ~ ? [X: num] :
              ( ( member_num @ X @ ( set_num2 @ Xs ) )
              & ( P @ X ) ) ) ) ).

% find_None_iff
thf(fact_1845_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr4953567300277697838T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) @ I )
          = ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1846_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Pr4606735188037164562VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) @ I )
          = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1847_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr1791586995822124652BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) @ I )
          = ( produc738532404422230701BT_nat @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1848_nth__zip,axiom,
    ! [I: nat,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr6837108013167703752BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) @ I )
          = ( produc736041933913180425BT_int @ ( nth_VEBT_VEBT @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1849_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr6777367263587873994T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) @ I )
          = ( produc2982872950893828659T_VEBT @ ( nth_o @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1850_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys: list_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Product_prod_o_o @ ( zip_o_o @ Xs @ Ys ) @ I )
          = ( product_Pair_o_o @ ( nth_o @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1851_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys: list_nat] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_nat @ Ys ) )
       => ( ( nth_Pr5826913651314560976_o_nat @ ( zip_o_nat @ Xs @ Ys ) @ I )
          = ( product_Pair_o_nat @ ( nth_o @ Xs @ I ) @ ( nth_nat @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1852_nth__zip,axiom,
    ! [I: nat,Xs: list_o,Ys: list_int] :
      ( ( ord_less_nat @ I @ ( size_size_list_o @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_int @ Ys ) )
       => ( ( nth_Pr1649062631805364268_o_int @ ( zip_o_int @ Xs @ Ys ) @ I )
          = ( product_Pair_o_int @ ( nth_o @ Xs @ I ) @ ( nth_int @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1853_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_s6755466524823107622T_VEBT @ Ys ) )
       => ( ( nth_Pr744662078594809490T_VEBT @ ( zip_nat_VEBT_VEBT @ Xs @ Ys ) @ I )
          = ( produc599794634098209291T_VEBT @ ( nth_nat @ Xs @ I ) @ ( nth_VEBT_VEBT @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1854_nth__zip,axiom,
    ! [I: nat,Xs: list_nat,Ys: list_o] :
      ( ( ord_less_nat @ I @ ( size_size_list_nat @ Xs ) )
     => ( ( ord_less_nat @ I @ ( size_size_list_o @ Ys ) )
       => ( ( nth_Pr112076138515278198_nat_o @ ( zip_nat_o @ Xs @ Ys ) @ I )
          = ( product_Pair_nat_o @ ( nth_nat @ Xs @ I ) @ ( nth_o @ Ys @ I ) ) ) ) ) ).

% nth_zip
thf(fact_1855__C5_C_I10_J,axiom,
    ( ( mi != ma )
   => ! [I4: nat] :
        ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ n )
              = I4 )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ ma @ n ) ) )
          & ! [X2: nat] :
              ( ( ( ( vEBT_VEBT_high @ X2 @ n )
                  = I4 )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ ( vEBT_VEBT_low @ X2 @ n ) ) )
             => ( ( ord_less_nat @ mi @ X2 )
                & ( ord_less_eq_nat @ X2 @ ma ) ) ) ) ) ) ).

% "5"(10)
thf(fact_1856_vebt__member_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X4: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,X4: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) @ X4 ) )
       => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT,X4: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) @ X4 ) )
         => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) @ X4 ) )
           => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,X4: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) @ X4 ) ) ) ) ) ) ).

% vebt_member.cases
thf(fact_1857_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X4: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X4 ) )
         => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
                ( X3
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) @ X4 ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X4: nat] :
                  ( X3
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ X4 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_1858_count__notin,axiom,
    ! [X3: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ( count_4203492906077236349at_nat @ Xs @ X3 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1859_count__notin,axiom,
    ! [X3: real,Xs: list_real] :
      ( ~ ( member_real @ X3 @ ( set_real2 @ Xs ) )
     => ( ( count_list_real @ Xs @ X3 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1860_count__notin,axiom,
    ! [X3: set_nat,Xs: list_set_nat] :
      ( ~ ( member_set_nat @ X3 @ ( set_set_nat2 @ Xs ) )
     => ( ( count_list_set_nat @ Xs @ X3 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1861_count__notin,axiom,
    ! [X3: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ~ ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ( count_list_VEBT_VEBT @ Xs @ X3 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1862_count__notin,axiom,
    ! [X3: int,Xs: list_int] :
      ( ~ ( member_int @ X3 @ ( set_int2 @ Xs ) )
     => ( ( count_list_int @ Xs @ X3 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1863_count__notin,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ~ ( member_nat @ X3 @ ( set_nat2 @ Xs ) )
     => ( ( count_list_nat @ Xs @ X3 )
        = zero_zero_nat ) ) ).

% count_notin
thf(fact_1864_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 ) )
      = ( ? [Y: vEBT_VEBT,N: nat] :
            ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ ( nth_VEBT_VEBT @ Xs @ N ) @ Y ) @ R2 )
            & ( ord_less_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( Ys
              = ( list_u1324408373059187874T_VEBT @ Xs @ N @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1865_listrel1__iff__update,axiom,
    ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
      = ( ? [Y: $o,N: nat] :
            ( ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ ( nth_o @ Xs @ N ) @ Y ) @ R2 )
            & ( ord_less_nat @ N @ ( size_size_list_o @ Xs ) )
            & ( Ys
              = ( list_update_o @ Xs @ N @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1866_listrel1__iff__update,axiom,
    ! [Xs: list_nat,Ys: list_nat,R2: set_Pr1261947904930325089at_nat] :
      ( ( member7340969449405702474st_nat @ ( produc2694037385005941721st_nat @ Xs @ Ys ) @ ( listrel1_nat @ R2 ) )
      = ( ? [Y: nat,N: nat] :
            ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ ( nth_nat @ Xs @ N ) @ Y ) @ R2 )
            & ( ord_less_nat @ N @ ( size_size_list_nat @ Xs ) )
            & ( Ys
              = ( list_update_nat @ Xs @ N @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1867_listrel1__iff__update,axiom,
    ! [Xs: list_int,Ys: list_int,R2: set_Pr958786334691620121nt_int] :
      ( ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs @ Ys ) @ ( listrel1_int @ R2 ) )
      = ( ? [Y: int,N: nat] :
            ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ ( nth_int @ Xs @ N ) @ Y ) @ R2 )
            & ( ord_less_nat @ N @ ( size_size_list_int @ Xs ) )
            & ( Ys
              = ( list_update_int @ Xs @ N @ Y ) ) ) ) ) ).

% listrel1_iff_update
thf(fact_1868_pair__lessI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ) ).

% pair_lessI2
thf(fact_1869_pair__less__iff1,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( product_Pair_nat_nat @ X3 @ Z2 ) ) @ fun_pair_less )
      = ( ord_less_nat @ Y3 @ Z2 ) ) ).

% pair_less_iff1
thf(fact_1870_gen__length__def,axiom,
    ( gen_length_VEBT_VEBT
    = ( ^ [N: nat,Xs2: list_VEBT_VEBT] : ( plus_plus_nat @ N @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ).

% gen_length_def
thf(fact_1871_gen__length__def,axiom,
    ( gen_length_o
    = ( ^ [N: nat,Xs2: list_o] : ( plus_plus_nat @ N @ ( size_size_list_o @ Xs2 ) ) ) ) ).

% gen_length_def
thf(fact_1872_gen__length__def,axiom,
    ( gen_length_nat
    = ( ^ [N: nat,Xs2: list_nat] : ( plus_plus_nat @ N @ ( size_size_list_nat @ Xs2 ) ) ) ) ).

% gen_length_def
thf(fact_1873_gen__length__def,axiom,
    ( gen_length_int
    = ( ^ [N: nat,Xs2: list_int] : ( plus_plus_nat @ N @ ( size_size_list_int @ Xs2 ) ) ) ) ).

% gen_length_def
thf(fact_1874_set__encode__empty,axiom,
    ( ( nat_set_encode @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% set_encode_empty
thf(fact_1875_Leaf__0__not,axiom,
    ! [A: $o,B: $o] :
      ~ ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ zero_zero_nat ) ).

% Leaf_0_not
thf(fact_1876_power__shift,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ( power_power_nat @ X3 @ Y3 )
        = Z2 )
      = ( ( vEBT_VEBT_power @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z2 ) ) ) ).

% power_shift
thf(fact_1877__C5_C_I3_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "5"(3)
thf(fact_1878__C5_C_I9_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "5"(9)
thf(fact_1879_VEBT_Oinject_I2_J,axiom,
    ! [X21: $o,X222: $o,Y21: $o,Y22: $o] :
      ( ( ( vEBT_Leaf @ X21 @ X222 )
        = ( vEBT_Leaf @ Y21 @ Y22 ) )
      = ( ( X21 = Y21 )
        & ( X222 = Y22 ) ) ) ).

% VEBT.inject(2)
thf(fact_1880_local_Opower__def,axiom,
    ( vEBT_VEBT_power
    = ( vEBT_V4262088993061758097ft_nat @ power_power_nat ) ) ).

% local.power_def
thf(fact_1881_high__bound__aux,axiom,
    ! [Ma: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ord_less_nat @ ( vEBT_VEBT_high @ Ma @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% high_bound_aux
thf(fact_1882_member__bound,axiom,
    ! [Tree: vEBT_VEBT,X3: nat,N2: nat] :
      ( ( vEBT_vebt_member @ Tree @ X3 )
     => ( ( vEBT_invar_vebt @ Tree @ N2 )
       => ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% member_bound
thf(fact_1883__C5_C_I6_J,axiom,
    ! [I4: nat] :
      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I4 ) @ X5 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I4 ) ) ) ).

% "5"(6)
thf(fact_1884_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_1885_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_1886_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_1887_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_1888_numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% numeral_le_iff
thf(fact_1889_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_1890_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_1891_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_1892_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_1893_numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% numeral_less_iff
thf(fact_1894_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( numera1916890842035813515d_enat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1895_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1896_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1897_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1898_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_1899_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W2 ) @ Z2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_1900_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W2 ) @ Z2 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_1901_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W2 ) @ Z2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_1902_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W2 ) @ Z2 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_1903_add__numeral__left,axiom,
    ! [V: num,W2: num,Z2: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W2 ) @ Z2 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) @ Z2 ) ) ).

% add_numeral_left
thf(fact_1904_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_1905_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_1906_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_1907_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_1908_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1909_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_1910_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_1911_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_1912_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_1913_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_1914_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_1915_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_1916_VEBT__internal_Oexp__split__high__low_I1_J,axiom,
    ! [X3: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(1)
thf(fact_1917_VEBT__internal_Oexp__split__high__low_I2_J,axiom,
    ! [X3: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_nat @ ( vEBT_VEBT_low @ X3 @ N2 ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% VEBT_internal.exp_split_high_low(2)
thf(fact_1918_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_1919_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_1920_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_1921_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D4: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D4 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT,Deg3: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_1922_VEBT_Odistinct_I1_J,axiom,
    ! [X11: option4927543243414619207at_nat,X12: nat,X13: list_VEBT_VEBT,X14: vEBT_VEBT,X21: $o,X222: $o] :
      ( ( vEBT_Node @ X11 @ X12 @ X13 @ X14 )
     != ( vEBT_Leaf @ X21 @ X222 ) ) ).

% VEBT.distinct(1)
thf(fact_1923_VEBT_Oexhaust,axiom,
    ! [Y3: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y3
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y3
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_1924_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1925_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_z5237406670263579293d_enat
     != ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1926_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1927_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1928_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1929_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1930_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_1931_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1932_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_1933_listrel1__eq__len,axiom,
    ! [Xs: list_o,Ys: list_o,R2: set_Product_prod_o_o] :
      ( ( member4159035015898711888list_o @ ( produc8435520187683070743list_o @ Xs @ Ys ) @ ( listrel1_o @ R2 ) )
     => ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys ) ) ) ).

% listrel1_eq_len
thf(fact_1934_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_1935_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_1936_VEBT__internal_OminNull_Osimps_I3_J,axiom,
    ! [Uu: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ Uu @ $true ) ) ).

% VEBT_internal.minNull.simps(3)
thf(fact_1937_VEBT__internal_OminNull_Osimps_I2_J,axiom,
    ! [Uv: $o] :
      ~ ( vEBT_VEBT_minNull @ ( vEBT_Leaf @ $true @ Uv ) ) ).

% VEBT_internal.minNull.simps(2)
thf(fact_1938_VEBT__internal_OminNull_Osimps_I1_J,axiom,
    vEBT_VEBT_minNull @ ( vEBT_Leaf @ $false @ $false ) ).

% VEBT_internal.minNull.simps(1)
thf(fact_1939_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_1940_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_1 )
               => ( ! [X4: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_1941_zip__same,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ A @ B ) @ ( set_Pr5518436109238095868at_nat @ ( zip_Pr4664179122662387191at_nat @ Xs @ Xs ) ) )
      = ( ( member8440522571783428010at_nat @ A @ ( set_Pr5648618587558075414at_nat @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1942_zip__same,axiom,
    ! [A: real,B: real,Xs: list_real] :
      ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ A @ B ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Xs ) ) )
      = ( ( member_real @ A @ ( set_real2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1943_zip__same,axiom,
    ! [A: set_nat,B: set_nat,Xs: list_set_nat] :
      ( ( member8277197624267554838et_nat @ ( produc4532415448927165861et_nat @ A @ B ) @ ( set_Pr9040384385603167362et_nat @ ( zip_set_nat_set_nat @ Xs @ Xs ) ) )
      = ( ( member_set_nat @ A @ ( set_set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1944_zip__same,axiom,
    ! [A: vEBT_VEBT,B: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ A @ B ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Xs ) ) )
      = ( ( member_VEBT_VEBT @ A @ ( set_VEBT_VEBT2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1945_zip__same,axiom,
    ! [A: nat,B: nat,Xs: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ A @ B ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Xs ) ) )
      = ( ( member_nat @ A @ ( set_nat2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1946_zip__same,axiom,
    ! [A: int,B: int,Xs: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ A @ B ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Xs ) ) )
      = ( ( member_int @ A @ ( set_int2 @ Xs ) )
        & ( A = B ) ) ) ).

% zip_same
thf(fact_1947_in__set__zipE,axiom,
    ! [X3: real,Y3: real,Xs: list_real,Ys: list_real] :
      ( ( member7849222048561428706l_real @ ( produc4511245868158468465l_real @ X3 @ Y3 ) @ ( set_Pr5999470521830281550l_real @ ( zip_real_real @ Xs @ Ys ) ) )
     => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ~ ( member_real @ Y3 @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1948_in__set__zipE,axiom,
    ! [X3: real,Y3: vEBT_VEBT,Xs: list_real,Ys: list_VEBT_VEBT] :
      ( ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X3 @ Y3 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) )
     => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ~ ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1949_in__set__zipE,axiom,
    ! [X3: real,Y3: int,Xs: list_real,Ys: list_int] :
      ( ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X3 @ Y3 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) )
     => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ~ ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1950_in__set__zipE,axiom,
    ! [X3: real,Y3: nat,Xs: list_real,Ys: list_nat] :
      ( ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y3 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
         => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1951_in__set__zipE,axiom,
    ! [X3: vEBT_VEBT,Y3: real,Xs: list_VEBT_VEBT,Ys: list_real] :
      ( ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X3 @ Y3 ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) )
     => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ~ ( member_real @ Y3 @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1952_in__set__zipE,axiom,
    ! [X3: vEBT_VEBT,Y3: vEBT_VEBT,Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) )
     => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ~ ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1953_in__set__zipE,axiom,
    ! [X3: vEBT_VEBT,Y3: int,Xs: list_VEBT_VEBT,Ys: list_int] :
      ( ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X3 @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) )
     => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ~ ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1954_in__set__zipE,axiom,
    ! [X3: vEBT_VEBT,Y3: nat,Xs: list_VEBT_VEBT,Ys: list_nat] :
      ( ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) )
     => ~ ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ~ ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1955_in__set__zipE,axiom,
    ! [X3: int,Y3: real,Xs: list_int,Ys: list_real] :
      ( ( member2744130022092475746t_real @ ( produc801115645435158769t_real @ X3 @ Y3 ) @ ( set_Pr112895574167722958t_real @ ( zip_int_real @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ~ ( member_real @ Y3 @ ( set_real2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1956_in__set__zipE,axiom,
    ! [X3: int,Y3: vEBT_VEBT,Xs: list_int,Ys: list_VEBT_VEBT] :
      ( ( member2056185340421749780T_VEBT @ ( produc3329399203697025711T_VEBT @ X3 @ Y3 ) @ ( set_Pr8714266321650254504T_VEBT @ ( zip_int_VEBT_VEBT @ Xs @ Ys ) ) )
     => ~ ( ( member_int @ X3 @ ( set_int2 @ Xs ) )
         => ~ ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) ) ) ) ).

% in_set_zipE
thf(fact_1957_set__zip__leftD,axiom,
    ! [X3: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1958_set__zip__leftD,axiom,
    ! [X3: int,Y3: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ( member_int @ X3 @ ( set_int2 @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1959_set__zip__leftD,axiom,
    ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
     => ( member1535805642427569193e_term @ X3 @ ( set_Co8062243466402858685e_term @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1960_set__zip__leftD,axiom,
    ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
     => ( member4242434998011752849e_term @ X3 @ ( set_Pr8342322266483756581e_term @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1961_set__zip__leftD,axiom,
    ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
     => ( member3222579708246209666e_term @ X3 @ ( set_Pr16608062948090134e_term @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1962_set__zip__leftD,axiom,
    ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
     => ( member8845023287901829240e_term @ X3 @ ( set_in5217446777445088012e_term @ Xs ) ) ) ).

% set_zip_leftD
thf(fact_1963_set__zip__rightD,axiom,
    ! [X3: nat,Y3: nat,Xs: list_nat,Ys: list_nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X3 @ Y3 ) @ ( set_Pr5648618587558075414at_nat @ ( zip_nat_nat @ Xs @ Ys ) ) )
     => ( member_nat @ Y3 @ ( set_nat2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1964_set__zip__rightD,axiom,
    ! [X3: int,Y3: int,Xs: list_int,Ys: list_int] :
      ( ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X3 @ Y3 ) @ ( set_Pr2470121279949933262nt_int @ ( zip_int_int @ Xs @ Ys ) ) )
     => ( member_int @ Y3 @ ( set_int2 @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1965_set__zip__rightD,axiom,
    ! [X3: code_integer > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_C878401137130745250e_term,Ys: list_P5578671422887162913nteger] :
      ( ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X3 @ Y3 ) @ ( set_Pr2999063419360598313nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) ) )
     => ( member157494554546826820nteger @ Y3 @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1966_set__zip__rightD,axiom,
    ! [X3: produc6241069584506657477e_term > option6357759511663192854e_term,Y3: produc8923325533196201883nteger,Xs: list_P1316552470764441098e_term,Ys: list_P5578671422887162913nteger] :
      ( ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X3 @ Y3 ) @ ( set_Pr2135590979564877377nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) ) )
     => ( member157494554546826820nteger @ Y3 @ ( set_Pr920681315882439344nteger @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1967_set__zip__rightD,axiom,
    ! [X3: produc8551481072490612790e_term > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_P1743416141875011707e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) @ ( set_Pr4943052134776177454nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) ) )
     => ( member5262025264175285858nt_int @ Y3 @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1968_set__zip__rightD,axiom,
    ! [X3: int > option6357759511663192854e_term,Y3: product_prod_int_int,Xs: list_i8448526496819171953e_term,Ys: list_P5707943133018811711nt_int] :
      ( ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) @ ( set_Pr1633835384712236856nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) ) )
     => ( member5262025264175285858nt_int @ Y3 @ ( set_Pr2470121279949933262nt_int @ Ys ) ) ) ).

% set_zip_rightD
thf(fact_1969_zip__update,axiom,
    ! [Xs: list_int,I: nat,X3: int,Ys: list_int,Y3: int] :
      ( ( zip_int_int @ ( list_update_int @ Xs @ I @ X3 ) @ ( list_update_int @ Ys @ I @ Y3 ) )
      = ( list_u3002344382305578791nt_int @ ( zip_int_int @ Xs @ Ys ) @ I @ ( product_Pair_int_int @ X3 @ Y3 ) ) ) ).

% zip_update
thf(fact_1970_zip__update,axiom,
    ! [Xs: list_C878401137130745250e_term,I: nat,X3: code_integer > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y3: produc8923325533196201883nteger] :
      ( ( zip_Co8729459035503499408nteger @ ( list_u4743598893156345252e_term @ Xs @ I @ X3 ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y3 ) )
      = ( list_u1133519416628930960nteger @ ( zip_Co8729459035503499408nteger @ Xs @ Ys ) @ I @ ( produc6137756002093451184nteger @ X3 @ Y3 ) ) ) ).

% zip_update
thf(fact_1971_zip__update,axiom,
    ! [Xs: list_P1316552470764441098e_term,I: nat,X3: produc6241069584506657477e_term > option6357759511663192854e_term,Ys: list_P5578671422887162913nteger,Y3: produc8923325533196201883nteger] :
      ( ( zip_Pr8292346330294042792nteger @ ( list_u877304756163299468e_term @ Xs @ I @ X3 ) @ ( list_u2254550707601501961nteger @ Ys @ I @ Y3 ) )
      = ( list_u234853988314817064nteger @ ( zip_Pr8292346330294042792nteger @ Xs @ Ys ) @ I @ ( produc8603105652947943368nteger @ X3 @ Y3 ) ) ) ).

% zip_update
thf(fact_1972_zip__update,axiom,
    ! [Xs: list_P1743416141875011707e_term,I: nat,X3: produc8551481072490612790e_term > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y3: product_prod_int_int] :
      ( ( zip_Pr4168994715204986005nt_int @ ( list_u3533491785856317309e_term @ Xs @ I @ X3 ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y3 ) )
      = ( list_u7736365598306452245nt_int @ ( zip_Pr4168994715204986005nt_int @ Xs @ Ys ) @ I @ ( produc5700946648718959541nt_int @ X3 @ Y3 ) ) ) ).

% zip_update
thf(fact_1973_zip__update,axiom,
    ! [Xs: list_i8448526496819171953e_term,I: nat,X3: int > option6357759511663192854e_term,Ys: list_P5707943133018811711nt_int,Y3: product_prod_int_int] :
      ( ( zip_in8766932505889695135nt_int @ ( list_u8946639151299769843e_term @ Xs @ I @ X3 ) @ ( list_u3002344382305578791nt_int @ Ys @ I @ Y3 ) )
      = ( list_u4780935413889332127nt_int @ ( zip_in8766932505889695135nt_int @ Xs @ Ys ) @ I @ ( produc4305682042979456191nt_int @ X3 @ Y3 ) ) ) ).

% zip_update
thf(fact_1974_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1975_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_le_numeral
thf(fact_1976_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1977_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1978_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_1979_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_1980_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_1981_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_1982_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_1983_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_1984_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_less_zero
thf(fact_1985_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_1986_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_1987_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_1988_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_1989_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1990_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_less_numeral
thf(fact_1991_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1992_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1993_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_1994_set__encode__eq,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( ( nat_set_encode @ A2 )
            = ( nat_set_encode @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% set_encode_eq
thf(fact_1995_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_1996_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_real,Y3: real] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real @ Y3 @ ( set_real2 @ Ys ) )
       => ~ ! [X4: vEBT_VEBT] :
              ~ ( member8675245146396747942T_real @ ( produc8117437818029410057T_real @ X4 @ Y3 ) @ ( set_Pr1087130671499945274T_real @ ( zip_VEBT_VEBT_real @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1997_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) )
       => ~ ! [X4: vEBT_VEBT] :
              ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X4 @ Y3 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1998_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,Y3: $o] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o @ Y3 @ ( set_o2 @ Ys ) )
       => ~ ! [X4: vEBT_VEBT] :
              ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X4 @ Y3 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_1999_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,Y3: nat] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
       => ~ ! [X4: vEBT_VEBT] :
              ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X4 @ Y3 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2000_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,Y3: int] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int @ Y3 @ ( set_int2 @ Ys ) )
       => ~ ! [X4: vEBT_VEBT] :
              ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X4 @ Y3 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2001_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_real,Y3: real] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_real @ Ys ) )
     => ( ( member_real @ Y3 @ ( set_real2 @ Ys ) )
       => ~ ! [X4: $o] :
              ~ ( member7400031367953476362o_real @ ( product_Pair_o_real @ X4 @ Y3 ) @ ( set_Pr2600826154070092190o_real @ ( zip_o_real @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2002_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT,Y3: vEBT_VEBT] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Ys ) )
       => ~ ! [X4: $o] :
              ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X4 @ Y3 ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2003_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_o,Y3: $o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o @ Y3 @ ( set_o2 @ Ys ) )
       => ~ ! [X4: $o] :
              ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X4 @ Y3 ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2004_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_nat,Y3: nat] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_nat @ Y3 @ ( set_nat2 @ Ys ) )
       => ~ ! [X4: $o] :
              ~ ( member2802428098988154798_o_nat @ ( product_Pair_o_nat @ X4 @ Y3 ) @ ( set_Pr7006799604034136130_o_nat @ ( zip_o_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2005_in__set__impl__in__set__zip2,axiom,
    ! [Xs: list_o,Ys: list_int,Y3: int] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_int @ Y3 @ ( set_int2 @ Ys ) )
       => ~ ! [X4: $o] :
              ~ ( member7847949116333733898_o_int @ ( product_Pair_o_int @ X4 @ Y3 ) @ ( set_Pr2828948584524939422_o_int @ ( zip_o_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip2
thf(fact_2006_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_VEBT_VEBT,X3: real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
       => ~ ! [Y5: vEBT_VEBT] :
              ~ ( member7262085504369356948T_VEBT @ ( produc6931449550656315951T_VEBT @ X3 @ Y5 ) @ ( set_Pr8897343066327330088T_VEBT @ ( zip_real_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2007_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_o,X3: real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
       => ~ ! [Y5: $o] :
              ~ ( member772602641336174712real_o @ ( product_Pair_real_o @ X3 @ Y5 ) @ ( set_Pr5196769464307566348real_o @ ( zip_real_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2008_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_nat,X3: real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
       => ~ ! [Y5: nat] :
              ~ ( member5805532792777349510al_nat @ ( produc3181502643871035669al_nat @ X3 @ Y5 ) @ ( set_Pr3174298344852596722al_nat @ ( zip_real_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2009_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_real,Ys: list_int,X3: real] :
      ( ( ( size_size_list_real @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_real @ X3 @ ( set_real2 @ Xs ) )
       => ~ ! [Y5: int] :
              ~ ( member1627681773268152802al_int @ ( produc3179012173361985393al_int @ X3 @ Y5 ) @ ( set_Pr8219819362198175822al_int @ ( zip_real_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2010_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT,X3: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y5: vEBT_VEBT] :
              ~ ( member568628332442017744T_VEBT @ ( produc537772716801021591T_VEBT @ X3 @ Y5 ) @ ( set_Pr9182192707038809660T_VEBT @ ( zip_VE537291747668921783T_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2011_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o,X3: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y5: $o] :
              ~ ( member3307348790968139188VEBT_o @ ( produc8721562602347293563VEBT_o @ X3 @ Y5 ) @ ( set_Pr7708085864119495200VEBT_o @ ( zip_VEBT_VEBT_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2012_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_nat,X3: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y5: nat] :
              ~ ( member373505688050248522BT_nat @ ( produc738532404422230701BT_nat @ X3 @ Y5 ) @ ( set_Pr7031586669278753246BT_nat @ ( zip_VEBT_VEBT_nat @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2013_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_int,X3: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ( member_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) )
       => ~ ! [Y5: int] :
              ~ ( member5419026705395827622BT_int @ ( produc736041933913180425BT_int @ X3 @ Y5 ) @ ( set_Pr2853735649769556538BT_int @ ( zip_VEBT_VEBT_int @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2014_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT,X3: $o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
       => ~ ! [Y5: vEBT_VEBT] :
              ~ ( member5477980866518848620T_VEBT @ ( produc2982872950893828659T_VEBT @ X3 @ Y5 ) @ ( set_Pr655345902815428824T_VEBT @ ( zip_o_VEBT_VEBT @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2015_in__set__impl__in__set__zip1,axiom,
    ! [Xs: list_o,Ys: list_o,X3: $o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ( member_o @ X3 @ ( set_o2 @ Xs ) )
       => ~ ! [Y5: $o] :
              ~ ( member7466972457876170832od_o_o @ ( product_Pair_o_o @ X3 @ Y5 ) @ ( set_Product_prod_o_o2 @ ( zip_o_o @ Xs @ Ys ) ) ) ) ) ).

% in_set_impl_in_set_zip1
thf(fact_2016_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X3: produc9072475918466114483BT_nat] :
      ( ! [A4: $o,B4: $o,X4: nat] :
          ( X3
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A4 @ B4 ) @ X4 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X3
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S2: vEBT_VEBT,X4: nat] :
              ( X3
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ X4 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_2017_invar__vebt_Ointros_I1_J,axiom,
    ! [A: $o,B: $o] : ( vEBT_invar_vebt @ ( vEBT_Leaf @ A @ B ) @ ( suc @ zero_zero_nat ) ) ).

% invar_vebt.intros(1)
thf(fact_2018_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_2019_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A3: $o,B3: $o] :
                ( A1
                = ( vEBT_Leaf @ A3 @ B3 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X5 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X5 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary3: vEBT_VEBT,Mi3: nat,Ma3: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ A22 @ TreeList @ Summary3 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary3 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ! [I2: nat] :
                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ X5 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I2 ) ) )
              & ( ( Mi3 = Ma3 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
              & ( ord_less_eq_nat @ Mi3 @ Ma3 )
              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi3 != Ma3 )
               => ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N )
                          = I2 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ Ma3 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I2 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I2 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_2020_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A4: $o,B4: $o] :
              ( A12
              = ( vEBT_Leaf @ A4 @ B4 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( A23 = 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_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( A23 = 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_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X_12 )
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary2: vEBT_VEBT,M3: nat,Deg2: nat,Mi2: nat,Ma2: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                 => ( ( A23 = 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 ) )
                             => ( ! [I4: nat] :
                                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X5 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                               => ( ( ( Mi2 = Ma2 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                   => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi2 != Ma2 )
                                         => ! [I4: nat] :
                                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                    = I4 )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                & ! [X2: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
                                                        = I4 )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( 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] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Deg2 @ TreeList3 @ Summary2 ) )
                   => ( ( A23 = 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 ) )
                               => ( ! [I4: nat] :
                                      ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                     => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ X5 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary2 @ I4 ) ) )
                                 => ( ( ( Mi2 = Ma2 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                         => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_12 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                                     => ( ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi2 != Ma2 )
                                           => ! [I4: nat] :
                                                ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M3 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N3 )
                                                      = I4 )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ Ma2 @ N3 ) ) )
                                                  & ! [X2: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
                                                          = I4 )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I4 ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
                                                     => ( ( ord_less_nat @ Mi2 @ X2 )
                                                        & ( ord_less_eq_nat @ X2 @ Ma2 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_2021_VEBT__internal_OminNull_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( X3
       != ( vEBT_Leaf @ $false @ $false ) )
     => ( ! [Uv2: $o] :
            ( X3
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X3
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X3
               != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X3
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.cases
thf(fact_2022_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_2023_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X4: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_2024_VEBT__internal_OminNull_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X3 )
     => ( ! [Uv2: $o] :
            ( X3
           != ( vEBT_Leaf @ $true @ Uv2 ) )
       => ( ! [Uu2: $o] :
              ( X3
             != ( vEBT_Leaf @ Uu2 @ $true ) )
         => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                ( X3
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ).

% VEBT_internal.minNull.elims(3)
thf(fact_2025_VEBT__internal_OminNull_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X3 )
     => ( ( X3
         != ( vEBT_Leaf @ $false @ $false ) )
       => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ).

% VEBT_internal.minNull.elims(2)
thf(fact_2026_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X4: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X4 @ X_1 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I3 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X4: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X4 @ N2 )
                                        = I3 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ ( vEBT_VEBT_low @ X4 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X4 )
                                      & ( ord_less_eq_nat @ X4 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_2027_set__encode__inf,axiom,
    ! [A2: set_nat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( nat_set_encode @ A2 )
        = zero_zero_nat ) ) ).

% set_encode_inf
thf(fact_2028_pair__lessI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_less ) ) ).

% pair_lessI1
thf(fact_2029_count__le__length,axiom,
    ! [Xs: list_VEBT_VEBT,X3: vEBT_VEBT] : ( ord_less_eq_nat @ ( count_list_VEBT_VEBT @ Xs @ X3 ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ).

% count_le_length
thf(fact_2030_count__le__length,axiom,
    ! [Xs: list_o,X3: $o] : ( ord_less_eq_nat @ ( count_list_o @ Xs @ X3 ) @ ( size_size_list_o @ Xs ) ) ).

% count_le_length
thf(fact_2031_count__le__length,axiom,
    ! [Xs: list_nat,X3: nat] : ( ord_less_eq_nat @ ( count_list_nat @ Xs @ X3 ) @ ( size_size_list_nat @ Xs ) ) ).

% count_le_length
thf(fact_2032_count__le__length,axiom,
    ! [Xs: list_int,X3: int] : ( ord_less_eq_nat @ ( count_list_int @ Xs @ X3 ) @ ( size_size_list_int @ Xs ) ) ).

% count_le_length
thf(fact_2033_vebt__mint_Ocases,axiom,
    ! [X3: vEBT_VEBT] :
      ( ! [A4: $o,B4: $o] :
          ( X3
         != ( vEBT_Leaf @ A4 @ B4 ) )
     => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
            ( X3
           != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
       => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ).

% vebt_mint.cases
thf(fact_2034_VEBT__internal_OminNull_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y3: $o] :
      ( ( ( vEBT_VEBT_minNull @ X3 )
        = Y3 )
     => ( ( ( X3
            = ( vEBT_Leaf @ $false @ $false ) )
         => ~ Y3 )
       => ( ( ? [Uv2: $o] :
                ( X3
                = ( vEBT_Leaf @ $true @ Uv2 ) )
           => Y3 )
         => ( ( ? [Uu2: $o] :
                  ( X3
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
             => Y3 )
           => ( ( ? [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ Y3 )
             => ~ ( ? [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => Y3 ) ) ) ) ) ) ).

% VEBT_internal.minNull.elims(1)
thf(fact_2035_length__code,axiom,
    ( size_s6755466524823107622T_VEBT
    = ( gen_length_VEBT_VEBT @ zero_zero_nat ) ) ).

% length_code
thf(fact_2036_length__code,axiom,
    ( size_size_list_o
    = ( gen_length_o @ zero_zero_nat ) ) ).

% length_code
thf(fact_2037_length__code,axiom,
    ( size_size_list_nat
    = ( gen_length_nat @ zero_zero_nat ) ) ).

% length_code
thf(fact_2038_length__code,axiom,
    ( size_size_list_int
    = ( gen_length_int @ zero_zero_nat ) ) ).

% length_code
thf(fact_2039_sum__power2__eq__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2040_sum__power2__eq__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2041_sum__power2__eq__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_2042_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_2043_zero__less__power2,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_power2
thf(fact_2044_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_2045_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_2046_power2__less__eq__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% power2_less_eq_zero_iff
thf(fact_2047_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_2048_power2__eq__iff__nonneg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2049_power2__eq__iff__nonneg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2050_power2__eq__iff__nonneg,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2051_power2__eq__iff__nonneg,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X3 = Y3 ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_2052_zero__eq__power2,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_power2
thf(fact_2053_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_2054_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_2055_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_2056_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_2057_power__mono__iff,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
            = ( ord_less_eq_real @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2058_power__mono__iff,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
            = ( ord_less_eq_rat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2059_power__mono__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
            = ( ord_less_eq_nat @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2060_power__mono__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
            = ( ord_less_eq_int @ A @ B ) ) ) ) ) ).

% power_mono_iff
thf(fact_2061_power__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( power_power_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% power_eq_0_iff
thf(fact_2062_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_2063_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_2064_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_2065_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_2066_member__inv,axiom,
    ! [Mi: nat,Ma: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ X3 )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg )
        & ( ( X3 = Mi )
          | ( X3 = Ma )
          | ( ( ord_less_nat @ X3 @ Ma )
            & ( ord_less_nat @ Mi @ X3 )
            & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
            & ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% member_inv
thf(fact_2067_both__member__options__ding,axiom,
    ! [Info: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,N2: nat,X3: nat] :
      ( ( vEBT_invar_vebt @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ N2 )
     => ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ Info @ Deg @ TreeList2 @ Summary ) @ X3 ) ) ) ) ).

% both_member_options_ding
thf(fact_2068_nat__zero__less__power__iff,axiom,
    ! [X3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X3 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_2069_pow__sum,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ).

% pow_sum
thf(fact_2070_high__def,axiom,
    ( vEBT_VEBT_high
    = ( ^ [X: nat,N: nat] : ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% high_def
thf(fact_2071_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_2072_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_2073_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_2074_divide__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2075_divide__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2076_divide__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_right
thf(fact_2077_divide__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ C @ A )
        = ( divide1717551699836669952omplex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2078_divide__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( divide_divide_real @ C @ A )
        = ( divide_divide_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2079_divide__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( divide_divide_rat @ C @ A )
        = ( divide_divide_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% divide_cancel_left
thf(fact_2080_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_2081_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_2082_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_2083_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_2084_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_2085_div__by__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% div_by_0
thf(fact_2086_divide__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divide_eq_0_iff
thf(fact_2087_divide__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_eq_0_iff
thf(fact_2088_divide__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_eq_0_iff
thf(fact_2089_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_2090_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_2091_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_2092_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_2093_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_2094_div__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% div_0
thf(fact_2095_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_2096_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_2097_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_2098_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_2099_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_2100_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_2101_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_2102_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_2103_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_2104_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_2105_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2106_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2107_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2108_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_2109_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_2110_nat__power__eq__Suc__0__iff,axiom,
    ! [X3: nat,M: nat] :
      ( ( ( power_power_nat @ X3 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X3
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_2111_add__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_2112_add__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_2113_divide__right__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( divide_divide_real @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2114_divide__right__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( divide_divide_rat @ A @ C ) ) ) ) ).

% divide_right_mono_neg
thf(fact_2115_divide__nonpos__nonpos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2116_divide__nonpos__nonpos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_2117_divide__nonpos__nonneg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2118_divide__nonpos__nonneg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_2119_divide__nonneg__nonpos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2120_divide__nonneg__nonpos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_2121_divide__nonneg__nonneg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2122_divide__nonneg__nonneg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_2123_zero__le__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2124_zero__le__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_divide_iff
thf(fact_2125_divide__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2126_divide__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_right_mono
thf(fact_2127_divide__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2128_divide__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_le_0_iff
thf(fact_2129_divide__neg__neg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_neg_neg
thf(fact_2130_divide__neg__neg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_neg_neg
thf(fact_2131_divide__neg__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_2132_divide__neg__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_2133_divide__pos__neg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_2134_divide__pos__neg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_2135_divide__pos__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_pos_pos
thf(fact_2136_divide__pos__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_pos_pos
thf(fact_2137_divide__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2138_divide__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% divide_less_0_iff
thf(fact_2139_divide__less__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) )
        & ( C != zero_zero_real ) ) ) ).

% divide_less_cancel
thf(fact_2140_divide__less__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) )
        & ( C != zero_zero_rat ) ) ) ).

% divide_less_cancel
thf(fact_2141_zero__less__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2142_zero__less__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_divide_iff
thf(fact_2143_divide__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2144_divide__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono
thf(fact_2145_divide__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2146_divide__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ) ).

% divide_strict_right_mono_neg
thf(fact_2147_frac__le,axiom,
    ! [Y3: real,X3: real,W2: real,Z2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z2 )
           => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_2148_frac__le,axiom,
    ! [Y3: rat,X3: rat,W2: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X3 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z2 )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_le
thf(fact_2149_frac__less,axiom,
    ! [X3: real,Y3: real,W2: real,Z2: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_eq_real @ W2 @ Z2 )
           => ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_2150_frac__less,axiom,
    ! [X3: rat,Y3: rat,W2: rat,Z2: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ X3 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_eq_rat @ W2 @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less
thf(fact_2151_frac__less2,axiom,
    ! [X3: real,Y3: real,W2: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ( ord_less_real @ zero_zero_real @ W2 )
         => ( ( ord_less_real @ W2 @ Z2 )
           => ( ord_less_real @ ( divide_divide_real @ X3 @ Z2 ) @ ( divide_divide_real @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_2152_frac__less2,axiom,
    ! [X3: rat,Y3: rat,W2: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ Y3 )
       => ( ( ord_less_rat @ zero_zero_rat @ W2 )
         => ( ( ord_less_rat @ W2 @ Z2 )
           => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ ( divide_divide_rat @ Y3 @ W2 ) ) ) ) ) ) ).

% frac_less2
thf(fact_2153_divide__le__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2154_divide__le__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% divide_le_cancel
thf(fact_2155_divide__nonneg__neg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_2156_divide__nonneg__neg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_2157_divide__nonneg__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2158_divide__nonneg__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_nonneg_pos
thf(fact_2159_divide__nonpos__neg,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ Y3 @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2160_divide__nonpos__neg,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y3 @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% divide_nonpos_neg
thf(fact_2161_divide__nonpos__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_2162_divide__nonpos__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_2163_field__sum__of__halves,axiom,
    ! [X3: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X3 ) ).

% field_sum_of_halves
thf(fact_2164_field__sum__of__halves,axiom,
    ! [X3: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X3 ) ).

% field_sum_of_halves
thf(fact_2165_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_2166_half__gt__zero,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% half_gt_zero
thf(fact_2167_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_2168_half__gt__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% half_gt_zero_iff
thf(fact_2169_field__less__half__sum,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_real @ X3 @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_2170_field__less__half__sum,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ X3 @ Y3 )
     => ( ord_less_rat @ X3 @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_2171_power__not__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( A != zero_zero_rat )
     => ( ( power_power_rat @ A @ N2 )
       != zero_zero_rat ) ) ).

% power_not_zero
thf(fact_2172_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_2173_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_2174_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_2175_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_2176_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_2177_zero__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_power
thf(fact_2178_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_2179_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_2180_power__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2181_power__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2182_power__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2183_power__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono
thf(fact_2184_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_2185_zero__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_less_power
thf(fact_2186_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_2187_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_2188_nat__power__less__imp__less,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I )
     => ( ( ord_less_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_power_less_imp_less
thf(fact_2189_power__less__imp__less__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_real @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2190_power__less__imp__less__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2191_power__less__imp__less__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2192_power__less__imp__less__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_int @ A @ B ) ) ) ).

% power_less_imp_less_base
thf(fact_2193_power__le__imp__le__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( suc @ N2 ) ) @ ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2194_power__le__imp__le__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2195_power__le__imp__le__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ A @ ( suc @ N2 ) ) @ ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2196_power__le__imp__le__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( suc @ N2 ) ) @ ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% power_le_imp_le_base
thf(fact_2197_power__inject__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ ( suc @ N2 ) )
        = ( power_power_real @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2198_power__inject__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ ( suc @ N2 ) )
        = ( power_power_rat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2199_power__inject__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ ( suc @ N2 ) )
        = ( power_power_nat @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2200_power__inject__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ ( suc @ N2 ) )
        = ( power_power_int @ B @ ( suc @ N2 ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( A = B ) ) ) ) ).

% power_inject_base
thf(fact_2201_zero__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_rat @ zero_zero_rat @ N2 )
        = zero_zero_rat ) ) ).

% zero_power
thf(fact_2202_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_2203_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_2204_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_2205_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_2206_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_2207_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_2208_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ( power_power_real @ A @ N2 )
              = ( power_power_real @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2209_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ( power_power_rat @ A @ N2 )
              = ( power_power_rat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2210_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ( power_power_nat @ A @ N2 )
              = ( power_power_nat @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2211_power__eq__iff__eq__base,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ( power_power_int @ A @ N2 )
              = ( power_power_int @ B @ N2 ) )
            = ( A = B ) ) ) ) ) ).

% power_eq_iff_eq_base
thf(fact_2212_power__eq__imp__eq__base,axiom,
    ! [A: real,N2: nat,B: real] :
      ( ( ( power_power_real @ A @ N2 )
        = ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2213_power__eq__imp__eq__base,axiom,
    ! [A: rat,N2: nat,B: rat] :
      ( ( ( power_power_rat @ A @ N2 )
        = ( power_power_rat @ B @ N2 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2214_power__eq__imp__eq__base,axiom,
    ! [A: nat,N2: nat,B: nat] :
      ( ( ( power_power_nat @ A @ N2 )
        = ( power_power_nat @ B @ N2 ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2215_power__eq__imp__eq__base,axiom,
    ! [A: int,N2: nat,B: int] :
      ( ( ( power_power_int @ A @ N2 )
        = ( power_power_int @ B @ N2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_nat @ zero_zero_nat @ N2 )
           => ( A = B ) ) ) ) ) ).

% power_eq_imp_eq_base
thf(fact_2216_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_2217_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_2218_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_2219_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_2220_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_2221_less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% less_exp
thf(fact_2222_power2__nat__le__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_imp_le
thf(fact_2223_power2__nat__le__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% power2_nat_le_eq_le
thf(fact_2224_self__le__ge2__pow,axiom,
    ! [K: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ M @ ( power_power_nat @ K @ M ) ) ) ).

% self_le_ge2_pow
thf(fact_2225_power2__le__imp__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_2226_power2__le__imp__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_2227_power2__le__imp__le,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_2228_power2__le__imp__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).

% power2_le_imp_le
thf(fact_2229_power2__eq__imp__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( X3 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2230_power2__eq__imp__eq,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
         => ( X3 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2231_power2__eq__imp__eq,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
         => ( X3 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2232_power2__eq__imp__eq,axiom,
    ! [X3: int,Y3: int] :
      ( ( ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( X3 = Y3 ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_2233_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_2234_zero__le__power2,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% zero_le_power2
thf(fact_2235_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_2236_power__strict__mono,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( 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 @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2237_power__strict__mono,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2238_power__strict__mono,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( 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 @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2239_power__strict__mono,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( ord_less_int @ A @ B )
     => ( ( 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 @ B @ N2 ) ) ) ) ) ).

% power_strict_mono
thf(fact_2240_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_2241_power2__less__0,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ zero_zero_rat ) ).

% power2_less_0
thf(fact_2242_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_2243_power2__less__imp__less,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ X3 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_2244_power2__less__imp__less,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ord_less_rat @ X3 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_2245_power2__less__imp__less,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y3 )
       => ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_2246_power2__less__imp__less,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ord_less_int @ X3 @ Y3 ) ) ) ).

% power2_less_imp_less
thf(fact_2247_sum__power2__le__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2248_sum__power2__le__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2249_sum__power2__le__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_2250_sum__power2__ge__zero,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2251_sum__power2__ge__zero,axiom,
    ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2252_sum__power2__ge__zero,axiom,
    ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_2253_not__sum__power2__lt__zero,axiom,
    ! [X3: real,Y3: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_2254_not__sum__power2__lt__zero,axiom,
    ! [X3: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_2255_not__sum__power2__lt__zero,axiom,
    ! [X3: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_2256_sum__power2__gt__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X3 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2257_sum__power2__gt__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X3 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2258_sum__power2__gt__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X3 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_2259_both__member__options__from__chilf__to__complete__tree,axiom,
    ! [X3: nat,Deg: nat,TreeList2: list_VEBT_VEBT,Mi: nat,Ma: nat,Summary: vEBT_VEBT] :
      ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( 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 @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( 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 ) @ X3 ) ) ) ) ).

% both_member_options_from_chilf_to_complete_tree
thf(fact_2260_add__self__div__2,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = M ) ).

% add_self_div_2
thf(fact_2261_div2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( suc @ ( divide_divide_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% div2_Suc_Suc
thf(fact_2262_both__member__options__from__complete__tree__to__child,axiom,
    ! [Deg: nat,Mi: nat,Ma: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: 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 ) @ X3 )
       => ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          | ( X3 = Mi )
          | ( X3 = Ma ) ) ) ) ).

% both_member_options_from_complete_tree_to_child
thf(fact_2263_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_2264_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_2265_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_2266_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
      ( ! [X4: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X4 @ N2 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_2267_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_2268_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_2269_deg1Leaf,axiom,
    ! [T: vEBT_VEBT] :
      ( ( vEBT_invar_vebt @ T @ one_one_nat )
      = ( ? [A3: $o,B3: $o] :
            ( T
            = ( vEBT_Leaf @ A3 @ B3 ) ) ) ) ).

% deg1Leaf
thf(fact_2270_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_2271_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_2272_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_2273_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_2274_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_2275_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_2276_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_2277_div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% div_by_1
thf(fact_2278_semiring__norm_I78_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(78)
thf(fact_2279_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_2280_divide__eq__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = one_one_complex )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2281_divide__eq__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = one_one_real )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2282_divide__eq__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = one_one_rat )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_1_iff
thf(fact_2283_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_2284_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_2285_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_2286_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_2287_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_2288_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ A @ A )
        = one_one_Code_integer ) ) ).

% div_self
thf(fact_2289_one__eq__divide__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( one_one_complex
        = ( divide1717551699836669952omplex @ A @ B ) )
      = ( ( B != zero_zero_complex )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2290_one__eq__divide__iff,axiom,
    ! [A: real,B: real] :
      ( ( one_one_real
        = ( divide_divide_real @ A @ B ) )
      = ( ( B != zero_zero_real )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2291_one__eq__divide__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ A @ B ) )
      = ( ( B != zero_zero_rat )
        & ( A = B ) ) ) ).

% one_eq_divide_iff
thf(fact_2292_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_2293_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_2294_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_2295_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_2296_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_2297_divide__self__if,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( divide_divide_rat @ A @ A )
          = one_one_rat ) ) ) ).

% divide_self_if
thf(fact_2298_divide__eq__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ( divide_divide_real @ B @ A )
        = one_one_real )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_2299_divide__eq__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ A )
        = one_one_rat )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% divide_eq_eq_1
thf(fact_2300_eq__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( one_one_real
        = ( divide_divide_real @ B @ A ) )
      = ( ( A != zero_zero_real )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_2301_eq__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( one_one_rat
        = ( divide_divide_rat @ B @ A ) )
      = ( ( A != zero_zero_rat )
        & ( A = B ) ) ) ).

% eq_divide_eq_1
thf(fact_2302_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_2303_one__divide__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( divide_divide_rat @ one_one_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% one_divide_eq_0_iff
thf(fact_2304_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_2305_zero__eq__1__divide__iff,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( divide_divide_rat @ one_one_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% zero_eq_1_divide_iff
thf(fact_2306_power__inject__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( power_power_real @ A @ M )
          = ( power_power_real @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2307_power__inject__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ( power_power_rat @ A @ M )
          = ( power_power_rat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2308_power__inject__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ( power_power_nat @ A @ M )
          = ( power_power_nat @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2309_power__inject__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ( power_power_int @ A @ M )
          = ( power_power_int @ A @ N2 ) )
        = ( M = N2 ) ) ) ).

% power_inject_exp
thf(fact_2310_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_2311_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_2312_divide__le__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% divide_le_0_1_iff
thf(fact_2313_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_2314_zero__le__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% zero_le_divide_1_iff
thf(fact_2315_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_2316_zero__less__divide__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ one_one_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% zero_less_divide_1_iff
thf(fact_2317_less__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2318_less__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% less_divide_eq_1_pos
thf(fact_2319_less__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2320_less__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% less_divide_eq_1_neg
thf(fact_2321_divide__less__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2322_divide__less__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ B @ A ) ) ) ).

% divide_less_eq_1_pos
thf(fact_2323_divide__less__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_real @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2324_divide__less__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_rat @ A @ B ) ) ) ).

% divide_less_eq_1_neg
thf(fact_2325_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_2326_divide__less__0__1__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% divide_less_0_1_iff
thf(fact_2327_power__strict__increasing__iff,axiom,
    ! [B: real,X3: nat,Y3: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y3 ) )
        = ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_2328_power__strict__increasing__iff,axiom,
    ! [B: rat,X3: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X3 ) @ ( power_power_rat @ B @ Y3 ) )
        = ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_2329_power__strict__increasing__iff,axiom,
    ! [B: nat,X3: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y3 ) )
        = ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_2330_power__strict__increasing__iff,axiom,
    ! [B: int,X3: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X3 ) @ ( power_power_int @ B @ Y3 ) )
        = ( ord_less_nat @ X3 @ Y3 ) ) ) ).

% power_strict_increasing_iff
thf(fact_2331_le__divide__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2332_le__divide__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% le_divide_eq_1_pos
thf(fact_2333_le__divide__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2334_le__divide__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% le_divide_eq_1_neg
thf(fact_2335_divide__le__eq__1__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2336_divide__le__eq__1__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% divide_le_eq_1_pos
thf(fact_2337_divide__le__eq__1__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2338_divide__le__eq__1__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% divide_le_eq_1_neg
thf(fact_2339_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2340_one__add__one,axiom,
    ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2341_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2342_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2343_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2344_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_2345_power__strict__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2346_power__strict__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2347_power__strict__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2348_power__strict__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_nat @ N2 @ M ) ) ) ) ).

% power_strict_decreasing_iff
thf(fact_2349_power__increasing__iff,axiom,
    ! [B: real,X3: nat,Y3: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X3 ) @ ( power_power_real @ B @ Y3 ) )
        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_2350_power__increasing__iff,axiom,
    ! [B: rat,X3: nat,Y3: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X3 ) @ ( power_power_rat @ B @ Y3 ) )
        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_2351_power__increasing__iff,axiom,
    ! [B: nat,X3: nat,Y3: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X3 ) @ ( power_power_nat @ B @ Y3 ) )
        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_2352_power__increasing__iff,axiom,
    ! [B: int,X3: nat,Y3: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X3 ) @ ( power_power_int @ B @ Y3 ) )
        = ( ord_less_eq_nat @ X3 @ Y3 ) ) ) ).

% power_increasing_iff
thf(fact_2353_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_2354_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_2355_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_2356_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_2357_one__plus__numeral,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% one_plus_numeral
thf(fact_2358_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_2359_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_2360_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_2361_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_2362_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_2363_numeral__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( numeral_numeral_rat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% numeral_plus_one
thf(fact_2364_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_2365_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_2366_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_2367_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_2368_numeral__le__one__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat )
      = ( ord_less_eq_num @ N2 @ one ) ) ).

% numeral_le_one_iff
thf(fact_2369_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_2370_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_2371_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_2372_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_2373_one__less__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ord_less_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_num @ one @ N2 ) ) ).

% one_less_numeral_iff
thf(fact_2374_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_2375_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_2376_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_2377_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_2378_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_2379_power__decreasing__iff,axiom,
    ! [B: real,M: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( ord_less_real @ B @ one_one_real )
       => ( ( ord_less_eq_real @ ( power_power_real @ B @ M ) @ ( power_power_real @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2380_power__decreasing__iff,axiom,
    ! [B: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ B )
     => ( ( ord_less_rat @ B @ one_one_rat )
       => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ M ) @ ( power_power_rat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2381_power__decreasing__iff,axiom,
    ! [B: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ B @ one_one_nat )
       => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ M ) @ ( power_power_nat @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2382_power__decreasing__iff,axiom,
    ! [B: int,M: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ B @ one_one_int )
       => ( ( ord_less_eq_int @ ( power_power_int @ B @ M ) @ ( power_power_int @ B @ N2 ) )
          = ( ord_less_eq_nat @ N2 @ M ) ) ) ) ).

% power_decreasing_iff
thf(fact_2383_one__reorient,axiom,
    ! [X3: complex] :
      ( ( one_one_complex = X3 )
      = ( X3 = one_one_complex ) ) ).

% one_reorient
thf(fact_2384_one__reorient,axiom,
    ! [X3: real] :
      ( ( one_one_real = X3 )
      = ( X3 = one_one_real ) ) ).

% one_reorient
thf(fact_2385_one__reorient,axiom,
    ! [X3: rat] :
      ( ( one_one_rat = X3 )
      = ( X3 = one_one_rat ) ) ).

% one_reorient
thf(fact_2386_one__reorient,axiom,
    ! [X3: nat] :
      ( ( one_one_nat = X3 )
      = ( X3 = one_one_nat ) ) ).

% one_reorient
thf(fact_2387_one__reorient,axiom,
    ! [X3: int] :
      ( ( one_one_int = X3 )
      = ( X3 = one_one_int ) ) ).

% one_reorient
thf(fact_2388_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_2389_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_2390_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_2391_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_2392_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_2393_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_2394_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_2395_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_2396_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_2397_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_2398_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_2399_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_2400_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_2401_div__add__self2,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self2
thf(fact_2402_div__add__self2,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self2
thf(fact_2403_div__add__self2,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self2
thf(fact_2404_div__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ B @ A ) @ B )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% div_add_self1
thf(fact_2405_div__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ B @ A ) @ B )
        = ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% div_add_self1
thf(fact_2406_div__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% div_add_self1
thf(fact_2407_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_2408_zero__less__one__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one_class.zero_le_one
thf(fact_2409_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_2410_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_2411_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_2412_linordered__nonzero__semiring__class_Ozero__le__one,axiom,
    ord_less_eq_rat @ zero_zero_rat @ one_one_rat ).

% linordered_nonzero_semiring_class.zero_le_one
thf(fact_2413_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_2414_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_2415_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_2416_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_2417_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_2418_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_2419_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_2420_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_2421_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_2422_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_2423_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_2424_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_2425_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_2426_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_2427_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_2428_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_2429_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_2430_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_2431_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% one_le_numeral
thf(fact_2432_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).

% one_le_numeral
thf(fact_2433_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_2434_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_2435_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_2436_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ).

% not_numeral_less_one
thf(fact_2437_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_2438_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_2439_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_2440_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_2441_add__mono1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( plus_plus_real @ A @ one_one_real ) @ ( plus_plus_real @ B @ one_one_real ) ) ) ).

% add_mono1
thf(fact_2442_add__mono1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( plus_plus_rat @ B @ one_one_rat ) ) ) ).

% add_mono1
thf(fact_2443_add__mono1,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ord_less_nat @ ( plus_plus_nat @ A @ one_one_nat ) @ ( plus_plus_nat @ B @ one_one_nat ) ) ) ).

% add_mono1
thf(fact_2444_add__mono1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( plus_plus_int @ A @ one_one_int ) @ ( plus_plus_int @ B @ one_one_int ) ) ) ).

% add_mono1
thf(fact_2445_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_2446_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_2447_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_2448_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_2449_right__inverse__eq,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = one_one_complex )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2450_right__inverse__eq,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( ( divide_divide_real @ A @ B )
          = one_one_real )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2451_right__inverse__eq,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( divide_divide_rat @ A @ B )
          = one_one_rat )
        = ( A = B ) ) ) ).

% right_inverse_eq
thf(fact_2452_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X3 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_2453_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X3 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ one_on7984719198319812577d_enat ) ) ).

% one_plus_numeral_commute
thf(fact_2454_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_2455_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_2456_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_2457_one__plus__numeral__commute,axiom,
    ! [X3: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_2458_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_2459_one__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ord_less_eq_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% one_le_power
thf(fact_2460_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_2461_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_2462_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_2463_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_2464_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_2465_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_2466_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_2467_div__eq__dividend__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ( divide_divide_nat @ M @ N2 )
          = M )
        = ( N2 = one_one_nat ) ) ) ).

% div_eq_dividend_iff
thf(fact_2468_div__less__dividend,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ) ) ).

% div_less_dividend
thf(fact_2469_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_2470_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_2471_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_2472_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_2473_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_2474_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_2475_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_2476_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_2477_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_2478_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_2479_power__le__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ one_one_rat ) ) ) ).

% power_le_one
thf(fact_2480_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_2481_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_2482_less__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2483_less__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% less_divide_eq_1
thf(fact_2484_divide__less__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_less_eq_1
thf(fact_2485_divide__less__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_less_eq_1
thf(fact_2486_less__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ A @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) ) ) ).

% less_half_sum
thf(fact_2487_less__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ A @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) ) ) ).

% less_half_sum
thf(fact_2488_gt__half__sum,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ one_one_real @ one_one_real ) ) @ B ) ) ).

% gt_half_sum
thf(fact_2489_gt__half__sum,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ) @ B ) ) ).

% gt_half_sum
thf(fact_2490_power__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( power_power_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% power_0_left
thf(fact_2491_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_2492_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_2493_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_2494_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_2495_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_2496_power__gt1,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) ) ) ).

% power_gt1
thf(fact_2497_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_2498_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_2499_power__less__imp__less__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2500_power__less__imp__less__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2501_power__less__imp__less__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2502_power__less__imp__less__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% power_less_imp_less_exp
thf(fact_2503_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_2504_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_2505_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_2506_power__strict__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_strict_increasing
thf(fact_2507_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_2508_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_2509_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_2510_power__increasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N7 ) ) ) ) ).

% power_increasing
thf(fact_2511_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_2512_vebt__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X3: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Leaf @ A @ B ) @ X3 )
      = ( ( ( X3 = zero_zero_nat )
         => A )
        & ( ( X3 != zero_zero_nat )
         => ( ( ( X3 = one_one_nat )
             => B )
            & ( X3 = one_one_nat ) ) ) ) ) ).

% vebt_member.simps(1)
thf(fact_2513_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X3: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X3 )
      = ( ( ( X3 = zero_zero_nat )
         => A )
        & ( ( X3 != zero_zero_nat )
         => ( ( ( X3 = one_one_nat )
             => B )
            & ( X3 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_2514_le__divide__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( divide_divide_real @ B @ A ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ A @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2515_le__divide__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( divide_divide_rat @ B @ A ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% le_divide_eq_1
thf(fact_2516_divide__le__eq__1,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ A ) @ one_one_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ A ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ A @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% divide_le_eq_1
thf(fact_2517_divide__le__eq__1,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ A ) @ one_one_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ A ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ A @ B ) )
        | ( A = zero_zero_rat ) ) ) ).

% divide_le_eq_1
thf(fact_2518_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_2519_power__Suc__le__self,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ A ) ) ) ).

% power_Suc_le_self
thf(fact_2520_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_2521_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_2522_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_2523_power__Suc__less__one,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ N2 ) ) @ one_one_rat ) ) ) ).

% power_Suc_less_one
thf(fact_2524_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_2525_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_2526_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2527_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2528_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2529_power__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N7 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_2530_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N7 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2531_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N7 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2532_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N7 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2533_power__decreasing,axiom,
    ! [N2: nat,N7: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N7 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N7 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_2534_power__le__imp__le__exp,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_eq_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2535_power__le__imp__le__exp,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2536_power__le__imp__le__exp,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ A )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2537_power__le__imp__le__exp,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( ord_less_int @ one_one_int @ A )
     => ( ( ord_less_eq_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_le_imp_le_exp
thf(fact_2538_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_2539_self__le__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% self_le_power
thf(fact_2540_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_2541_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_2542_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_2543_one__less__power,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% one_less_power
thf(fact_2544_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_2545_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_2546_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_2547_vebt__maxt_Osimps_I1_J,axiom,
    ! [B: $o,A: $o] :
      ( ( B
       => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ one_one_nat ) ) )
      & ( ~ B
       => ( ( A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ zero_zero_nat ) ) )
          & ( ~ A
           => ( ( vEBT_vebt_maxt @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_maxt.simps(1)
thf(fact_2548_vebt__mint_Osimps_I1_J,axiom,
    ! [A: $o,B: $o] :
      ( ( A
       => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
          = ( some_nat @ zero_zero_nat ) ) )
      & ( ~ A
       => ( ( B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = ( some_nat @ one_one_nat ) ) )
          & ( ~ B
           => ( ( vEBT_vebt_mint @ ( vEBT_Leaf @ A @ B ) )
              = none_nat ) ) ) ) ) ).

% vebt_mint.simps(1)
thf(fact_2549_div__le__mono,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ M @ K ) @ ( divide_divide_nat @ N2 @ K ) ) ) ).

% div_le_mono
thf(fact_2550_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_2551_ex__power__ivl2,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
       => ? [N3: nat] :
            ( ( ord_less_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl2
thf(fact_2552_ex__power__ivl1,axiom,
    ! [B: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_eq_nat @ one_one_nat @ K )
       => ? [N3: nat] :
            ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N3 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N3 @ one_one_nat ) ) ) ) ) ) ).

% ex_power_ivl1
thf(fact_2553_Euclidean__Division_Odiv__eq__0__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( ord_less_nat @ M @ N2 )
        | ( N2 = zero_zero_nat ) ) ) ).

% Euclidean_Division.div_eq_0_iff
thf(fact_2554_Suc__div__le__mono,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ ( divide_divide_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_div_le_mono
thf(fact_2555_vebt__maxt_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X3 )
        = Y3 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( B4
                 => ( Y3
                    = ( some_nat @ one_one_nat ) ) )
                & ( ~ B4
                 => ( ( A4
                     => ( Y3
                        = ( some_nat @ zero_zero_nat ) ) )
                    & ( ~ A4
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat,Ma2: nat] :
                ( ? [Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y3
                 != ( some_nat @ Ma2 ) ) ) ) ) ) ).

% vebt_maxt.elims
thf(fact_2556_vebt__mint_Oelims,axiom,
    ! [X3: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_mint @ X3 )
        = Y3 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ~ ( ( A4
                 => ( Y3
                    = ( some_nat @ zero_zero_nat ) ) )
                & ( ~ A4
                 => ( ( B4
                     => ( Y3
                        = ( some_nat @ one_one_nat ) ) )
                    & ( ~ B4
                     => ( Y3 = none_nat ) ) ) ) ) )
       => ( ( ? [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => ( Y3 != none_nat ) )
         => ~ ! [Mi2: nat] :
                ( ? [Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
               => ( Y3
                 != ( some_nat @ Mi2 ) ) ) ) ) ) ).

% vebt_mint.elims
thf(fact_2557_div__greater__zero__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ N2 @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% div_greater_zero_iff
thf(fact_2558_div__le__mono2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ord_less_eq_nat @ ( divide_divide_nat @ K @ N2 ) @ ( divide_divide_nat @ K @ M ) ) ) ) ).

% div_le_mono2
thf(fact_2559_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_2560_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_2561_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_2562_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_2563_div__exp__eq,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2564_div__exp__eq,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide_divide_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2565_div__exp__eq,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% div_exp_eq
thf(fact_2566_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2567_exp__add__not__zero__imp__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_right
thf(fact_2568_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_nat )
     => ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
       != zero_zero_nat ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2569_exp__add__not__zero__imp__left,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) )
       != zero_zero_int )
     => ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
       != zero_zero_int ) ) ).

% exp_add_not_zero_imp_left
thf(fact_2570_bit__concat__def,axiom,
    ( vEBT_VEBT_bit_concat
    = ( ^ [H: nat,L2: nat,D5: nat] : ( plus_plus_nat @ ( times_times_nat @ H @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ D5 ) ) @ L2 ) ) ) ).

% bit_concat_def
thf(fact_2571_low__inv,axiom,
    ! [X3: nat,N2: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_low @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X3 ) @ N2 )
        = X3 ) ) ).

% low_inv
thf(fact_2572_high__inv,axiom,
    ! [X3: nat,N2: nat,Y3: nat] :
      ( ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( vEBT_VEBT_high @ ( plus_plus_nat @ ( times_times_nat @ Y3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ X3 ) @ N2 )
        = Y3 ) ) ).

% high_inv
thf(fact_2573_enat__ord__number_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(1)
thf(fact_2574_enat__ord__number_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% enat_ord_number(2)
thf(fact_2575_mul__shift,axiom,
    ! [X3: nat,Y3: nat,Z2: nat] :
      ( ( ( times_times_nat @ X3 @ Y3 )
        = Z2 )
      = ( ( vEBT_VEBT_mul @ ( some_nat @ X3 ) @ ( some_nat @ Y3 ) )
        = ( some_nat @ Z2 ) ) ) ).

% mul_shift
thf(fact_2576_mul__def,axiom,
    ( vEBT_VEBT_mul
    = ( vEBT_V4262088993061758097ft_nat @ times_times_nat ) ) ).

% mul_def
thf(fact_2577_i0__less,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = ( N2 != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_2578_mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( times_times_complex @ A @ C )
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2579_mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( times_times_real @ A @ C )
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2580_mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( times_times_rat @ A @ C )
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2581_mult__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( times_times_nat @ A @ C )
        = ( times_times_nat @ B @ C ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2582_mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( times_times_int @ A @ C )
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_right
thf(fact_2583_mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( times_times_complex @ C @ A )
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2584_mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( times_times_real @ C @ A )
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2585_mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( times_times_rat @ C @ A )
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2586_mult__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( times_times_nat @ C @ A )
        = ( times_times_nat @ C @ B ) )
      = ( ( C = zero_zero_nat )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2587_mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( times_times_int @ C @ A )
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( A = B ) ) ) ).

% mult_cancel_left
thf(fact_2588_mult__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
      = ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% mult_eq_0_iff
thf(fact_2589_mult__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
      = ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% mult_eq_0_iff
thf(fact_2590_mult__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
      = ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% mult_eq_0_iff
thf(fact_2591_mult__eq__0__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% mult_eq_0_iff
thf(fact_2592_mult__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
      = ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% mult_eq_0_iff
thf(fact_2593_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_2594_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_2595_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_2596_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_2597_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_2598_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_2599_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_2600_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_2601_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_2602_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_2603_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_2604_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_2605_bits__div__by__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% bits_div_by_0
thf(fact_2606_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_2607_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_2608_bits__div__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_div_0
thf(fact_2609_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_2610_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_2611_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_2612_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_2613_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_2614_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_2615_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_2616_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_2617_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_2618_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_2619_half__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% half_negative_int_iff
thf(fact_2620_times__divide__eq__left,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ B @ C ) @ A )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2621_times__divide__eq__left,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( divide_divide_real @ ( times_times_real @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2622_times__divide__eq__left,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( divide_divide_rat @ ( times_times_rat @ B @ A ) @ C ) ) ).

% times_divide_eq_left
thf(fact_2623_divide__divide__eq__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2624_divide__divide__eq__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2625_divide__divide__eq__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% divide_divide_eq_left
thf(fact_2626_divide__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_2627_divide__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_2628_divide__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ B ) ) ).

% divide_divide_eq_right
thf(fact_2629_times__divide__eq__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( divide1717551699836669952omplex @ B @ C ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2630_times__divide__eq__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( divide_divide_real @ ( times_times_real @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2631_times__divide__eq__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ C ) ) ).

% times_divide_eq_right
thf(fact_2632_mult__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ K )
        = ( times_times_nat @ N2 @ K ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel2
thf(fact_2633_mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( M = N2 )
        | ( K = zero_zero_nat ) ) ) ).

% mult_cancel1
thf(fact_2634_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2635_mult__is__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ( M = zero_zero_nat )
        | ( N2 = zero_zero_nat ) ) ) ).

% mult_is_0
thf(fact_2636_nat__1__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( one_one_nat
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_1_eq_mult_iff
thf(fact_2637_nat__mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = one_one_nat )
      = ( ( M = one_one_nat )
        & ( N2 = one_one_nat ) ) ) ).

% nat_mult_eq_1_iff
thf(fact_2638_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_2639_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_2640_mult__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ( times_times_rat @ A @ C )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_right2
thf(fact_2641_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_2642_mult__cancel__right1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_right1
thf(fact_2643_mult__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_right1
thf(fact_2644_mult__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_right1
thf(fact_2645_mult__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_right1
thf(fact_2646_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_2647_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_2648_mult__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ( times_times_rat @ C @ A )
        = C )
      = ( ( C = zero_zero_rat )
        | ( A = one_one_rat ) ) ) ).

% mult_cancel_left2
thf(fact_2649_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_2650_mult__cancel__left1,axiom,
    ! [C: complex,B: complex] :
      ( ( C
        = ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( B = one_one_complex ) ) ) ).

% mult_cancel_left1
thf(fact_2651_mult__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( C
        = ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( B = one_one_real ) ) ) ).

% mult_cancel_left1
thf(fact_2652_mult__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( C
        = ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( B = one_one_rat ) ) ) ).

% mult_cancel_left1
thf(fact_2653_mult__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( C
        = ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( B = one_one_int ) ) ) ).

% mult_cancel_left1
thf(fact_2654_sum__squares__eq__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
        = zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2655_sum__squares__eq__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
        = zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2656_sum__squares__eq__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
        = zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2657_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2658_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2659_nonzero__mult__divide__mult__cancel__right2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right2
thf(fact_2660_nonzero__mult__div__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2661_nonzero__mult__div__cancel__right,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2662_nonzero__mult__div__cancel__right,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2663_nonzero__mult__div__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2664_nonzero__mult__div__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2665_nonzero__mult__div__cancel__right,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
        = A ) ) ).

% nonzero_mult_div_cancel_right
thf(fact_2666_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2667_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2668_nonzero__mult__divide__mult__cancel__right,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_right
thf(fact_2669_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ B @ C ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2670_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ B @ C ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2671_nonzero__mult__divide__mult__cancel__left2,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ B @ C ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left2
thf(fact_2672_nonzero__mult__div__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2673_nonzero__mult__div__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2674_nonzero__mult__div__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2675_nonzero__mult__div__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2676_nonzero__mult__div__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2677_nonzero__mult__div__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ A )
        = B ) ) ).

% nonzero_mult_div_cancel_left
thf(fact_2678_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2679_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2680_nonzero__mult__divide__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_mult_divide_mult_cancel_left
thf(fact_2681_mult__divide__mult__cancel__left__if,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( ( C = zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = zero_zero_complex ) )
      & ( ( C != zero_zero_complex )
       => ( ( divide1717551699836669952omplex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
          = ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2682_mult__divide__mult__cancel__left__if,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ( C = zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = zero_zero_real ) )
      & ( ( C != zero_zero_real )
       => ( ( divide_divide_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
          = ( divide_divide_real @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2683_mult__divide__mult__cancel__left__if,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ( C = zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = zero_zero_rat ) )
      & ( ( C != zero_zero_rat )
       => ( ( divide_divide_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
          = ( divide_divide_rat @ A @ B ) ) ) ) ).

% mult_divide_mult_cancel_left_if
thf(fact_2684_div__mult__mult1__if,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ( C = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = zero_zero_nat ) )
      & ( ( C != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2685_div__mult__mult1__if,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ( C = zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = zero_zero_int ) )
      & ( ( C != zero_zero_int )
       => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2686_div__mult__mult1__if,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ( C = zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( C != zero_z3403309356797280102nteger )
       => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_mult1_if
thf(fact_2687_div__mult__mult2,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2688_div__mult__mult2,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2689_div__mult__mult2,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult2
thf(fact_2690_div__mult__mult1,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( divide_divide_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
        = ( divide_divide_nat @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2691_div__mult__mult1,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( divide_divide_int @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2692_div__mult__mult1,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% div_mult_mult1
thf(fact_2693_distrib__right__numeral,axiom,
    ! [A: complex,B: complex,V: num] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2694_distrib__right__numeral,axiom,
    ! [A: extended_enat,B: extended_enat,V: num] :
      ( ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ A @ B ) @ ( numera1916890842035813515d_enat @ V ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ V ) ) @ ( times_7803423173614009249d_enat @ B @ ( numera1916890842035813515d_enat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2695_distrib__right__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( plus_plus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2696_distrib__right__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2697_distrib__right__numeral,axiom,
    ! [A: nat,B: nat,V: num] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ ( numeral_numeral_nat @ V ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( numeral_numeral_nat @ V ) ) @ ( times_times_nat @ B @ ( numeral_numeral_nat @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2698_distrib__right__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( plus_plus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% distrib_right_numeral
thf(fact_2699_distrib__left__numeral,axiom,
    ! [V: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2700_distrib__left__numeral,axiom,
    ! [V: num,B: extended_enat,C: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ B @ C ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ B ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2701_distrib__left__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2702_distrib__left__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2703_distrib__left__numeral,axiom,
    ! [V: num,B: nat,C: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ B ) @ ( times_times_nat @ ( numeral_numeral_nat @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2704_distrib__left__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% distrib_left_numeral
thf(fact_2705_mult__eq__1__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( times_times_nat @ M @ N2 )
        = ( suc @ zero_zero_nat ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% mult_eq_1_iff
thf(fact_2706_one__eq__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( suc @ zero_zero_nat )
        = ( times_times_nat @ M @ N2 ) )
      = ( ( M
          = ( suc @ zero_zero_nat ) )
        & ( N2
          = ( suc @ zero_zero_nat ) ) ) ) ).

% one_eq_mult_iff
thf(fact_2707_mult__less__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% mult_less_cancel2
thf(fact_2708_nat__0__less__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ M )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% nat_0_less_mult_iff
thf(fact_2709_nat__mult__less__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
        & ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel_disj
thf(fact_2710_mult__Suc__right,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ M @ ( suc @ N2 ) )
      = ( plus_plus_nat @ M @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc_right
thf(fact_2711_nat__mult__div__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( K = zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = zero_zero_nat ) )
      & ( ( K != zero_zero_nat )
       => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
          = ( divide_divide_nat @ M @ N2 ) ) ) ) ).

% nat_mult_div_cancel_disj
thf(fact_2712_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2713_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2714_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2715_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2716_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2717_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2718_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2719_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W2 )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W2 ) )
            = B ) )
        & ( ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2720_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ( numeral_numeral_real @ W2 )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W2 )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2721_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ( numeral_numeral_rat @ W2 )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2722_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2723_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2724_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W2 ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2725_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W2 ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W2 ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2726_nonzero__divide__mult__cancel__left,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2727_nonzero__divide__mult__cancel__left,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2728_nonzero__divide__mult__cancel__left,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ B ) ) ) ).

% nonzero_divide_mult_cancel_left
thf(fact_2729_nonzero__divide__mult__cancel__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ B @ ( times_times_complex @ A @ B ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2730_nonzero__divide__mult__cancel__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ B @ ( times_times_real @ A @ B ) )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2731_nonzero__divide__mult__cancel__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ B @ ( times_times_rat @ A @ B ) )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_divide_mult_cancel_right
thf(fact_2732_div__mult__self1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2733_div__mult__self1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2734_div__mult__self1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self1
thf(fact_2735_div__mult__self2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2736_div__mult__self2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2737_div__mult__self2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self2
thf(fact_2738_div__mult__self3,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2739_div__mult__self3,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2740_div__mult__self3,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self3
thf(fact_2741_div__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
        = ( plus_plus_nat @ C @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2742_div__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
        = ( plus_plus_int @ C @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2743_div__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
        = ( plus_p5714425477246183910nteger @ C @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_mult_self4
thf(fact_2744_one__le__mult__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) )
      = ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ M )
        & ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) ) ).

% one_le_mult_iff
thf(fact_2745_mult__le__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% mult_le_cancel2
thf(fact_2746_nat__mult__le__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel_disj
thf(fact_2747_div__mult__self__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ M @ N2 ) @ N2 )
        = M ) ) ).

% div_mult_self_is_m
thf(fact_2748_div__mult__self1__is__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( divide_divide_nat @ ( times_times_nat @ N2 @ M ) @ N2 )
        = M ) ) ).

% div_mult_self1_is_m
thf(fact_2749_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2750_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2751_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2752_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2753_ab__semigroup__mult__class_Omult__ac_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% ab_semigroup_mult_class.mult_ac(1)
thf(fact_2754_mult_Oassoc,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( times_times_complex @ A @ B ) @ C )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2755_mult_Oassoc,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( times_times_real @ A @ B ) @ C )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2756_mult_Oassoc,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( times_times_rat @ A @ B ) @ C )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2757_mult_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2758_mult_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( times_times_int @ A @ B ) @ C )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.assoc
thf(fact_2759_mult_Ocommute,axiom,
    ( times_times_complex
    = ( ^ [A3: complex,B3: complex] : ( times_times_complex @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2760_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A3: real,B3: real] : ( times_times_real @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2761_mult_Ocommute,axiom,
    ( times_times_rat
    = ( ^ [A3: rat,B3: rat] : ( times_times_rat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2762_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B3: nat] : ( times_times_nat @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2763_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B3: int] : ( times_times_int @ B3 @ A3 ) ) ) ).

% mult.commute
thf(fact_2764_mult_Oleft__commute,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( times_times_complex @ B @ ( times_times_complex @ A @ C ) )
      = ( times_times_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2765_mult_Oleft__commute,axiom,
    ! [B: real,A: real,C: real] :
      ( ( times_times_real @ B @ ( times_times_real @ A @ C ) )
      = ( times_times_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2766_mult_Oleft__commute,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( times_times_rat @ B @ ( times_times_rat @ A @ C ) )
      = ( times_times_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2767_mult_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( times_times_nat @ B @ ( times_times_nat @ A @ C ) )
      = ( times_times_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2768_mult_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( times_times_int @ B @ ( times_times_int @ A @ C ) )
      = ( times_times_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% mult.left_commute
thf(fact_2769_not__iless0,axiom,
    ! [N2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_2770_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N2: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M6: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M6 @ N3 )
             => ( P @ M6 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% enat_less_induct
thf(fact_2771_mult__right__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = ( times_times_complex @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2772_mult__right__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = ( times_times_real @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2773_mult__right__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = ( times_times_rat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2774_mult__right__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ A @ C )
          = ( times_times_nat @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2775_mult__right__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ A @ C )
          = ( times_times_int @ B @ C ) )
        = ( A = B ) ) ) ).

% mult_right_cancel
thf(fact_2776_mult__left__cancel,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ C @ A )
          = ( times_times_complex @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2777_mult__left__cancel,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ C @ A )
          = ( times_times_real @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2778_mult__left__cancel,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ C @ A )
          = ( times_times_rat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2779_mult__left__cancel,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( C != zero_zero_nat )
     => ( ( ( times_times_nat @ C @ A )
          = ( times_times_nat @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2780_mult__left__cancel,axiom,
    ! [C: int,A: int,B: int] :
      ( ( C != zero_zero_int )
     => ( ( ( times_times_int @ C @ A )
          = ( times_times_int @ C @ B ) )
        = ( A = B ) ) ) ).

% mult_left_cancel
thf(fact_2781_no__zero__divisors,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( times_times_complex @ A @ B )
         != zero_zero_complex ) ) ) ).

% no_zero_divisors
thf(fact_2782_no__zero__divisors,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( times_times_real @ A @ B )
         != zero_zero_real ) ) ) ).

% no_zero_divisors
thf(fact_2783_no__zero__divisors,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( times_times_rat @ A @ B )
         != zero_zero_rat ) ) ) ).

% no_zero_divisors
thf(fact_2784_no__zero__divisors,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( B != zero_zero_nat )
       => ( ( times_times_nat @ A @ B )
         != zero_zero_nat ) ) ) ).

% no_zero_divisors
thf(fact_2785_no__zero__divisors,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( B != zero_zero_int )
       => ( ( times_times_int @ A @ B )
         != zero_zero_int ) ) ) ).

% no_zero_divisors
thf(fact_2786_divisors__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = zero_zero_complex )
     => ( ( A = zero_zero_complex )
        | ( B = zero_zero_complex ) ) ) ).

% divisors_zero
thf(fact_2787_divisors__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = zero_zero_real )
     => ( ( A = zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divisors_zero
thf(fact_2788_divisors__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = zero_zero_rat )
     => ( ( A = zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divisors_zero
thf(fact_2789_divisors__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
        = zero_zero_nat )
     => ( ( A = zero_zero_nat )
        | ( B = zero_zero_nat ) ) ) ).

% divisors_zero
thf(fact_2790_divisors__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
        = zero_zero_int )
     => ( ( A = zero_zero_int )
        | ( B = zero_zero_int ) ) ) ).

% divisors_zero
thf(fact_2791_mult__not__zero,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
       != zero_zero_complex )
     => ( ( A != zero_zero_complex )
        & ( B != zero_zero_complex ) ) ) ).

% mult_not_zero
thf(fact_2792_mult__not__zero,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
       != zero_zero_real )
     => ( ( A != zero_zero_real )
        & ( B != zero_zero_real ) ) ) ).

% mult_not_zero
thf(fact_2793_mult__not__zero,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
       != zero_zero_rat )
     => ( ( A != zero_zero_rat )
        & ( B != zero_zero_rat ) ) ) ).

% mult_not_zero
thf(fact_2794_mult__not__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( ( times_times_nat @ A @ B )
       != zero_zero_nat )
     => ( ( A != zero_zero_nat )
        & ( B != zero_zero_nat ) ) ) ).

% mult_not_zero
thf(fact_2795_mult__not__zero,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ B )
       != zero_zero_int )
     => ( ( A != zero_zero_int )
        & ( B != zero_zero_int ) ) ) ).

% mult_not_zero
thf(fact_2796_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_2797_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_2798_comm__monoid__mult__class_Omult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% comm_monoid_mult_class.mult_1
thf(fact_2799_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_2800_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_2801_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_2802_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_2803_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_2804_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_2805_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_2806_combine__common__factor,axiom,
    ! [A: complex,E2: complex,B: complex,C: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2807_combine__common__factor,axiom,
    ! [A: real,E2: real,B: real,C: real] :
      ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ C ) )
      = ( plus_plus_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2808_combine__common__factor,axiom,
    ! [A: rat,E2: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2809_combine__common__factor,axiom,
    ! [A: nat,E2: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ A @ E2 ) @ ( plus_plus_nat @ ( times_times_nat @ B @ E2 ) @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2810_combine__common__factor,axiom,
    ! [A: int,E2: int,B: int,C: int] :
      ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ C ) )
      = ( plus_plus_int @ ( times_times_int @ ( plus_plus_int @ A @ B ) @ E2 ) @ C ) ) ).

% combine_common_factor
thf(fact_2811_distrib__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% distrib_right
thf(fact_2812_distrib__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% distrib_right
thf(fact_2813_distrib__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2814_distrib__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% distrib_right
thf(fact_2815_distrib__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% distrib_right
thf(fact_2816_distrib__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% distrib_left
thf(fact_2817_distrib__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% distrib_left
thf(fact_2818_distrib__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2819_distrib__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( plus_plus_nat @ B @ C ) )
      = ( plus_plus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% distrib_left
thf(fact_2820_distrib__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% distrib_left
thf(fact_2821_comm__semiring__class_Odistrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2822_comm__semiring__class_Odistrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2823_comm__semiring__class_Odistrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2824_comm__semiring__class_Odistrib,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2825_comm__semiring__class_Odistrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% comm_semiring_class.distrib
thf(fact_2826_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( plus_plus_complex @ B @ C ) )
      = ( plus_plus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2827_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( plus_plus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2828_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( plus_plus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2829_ring__class_Oring__distribs_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( plus_plus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% ring_class.ring_distribs(1)
thf(fact_2830_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2831_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( plus_plus_real @ A @ B ) @ C )
      = ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2832_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ C )
      = ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2833_ring__class_Oring__distribs_I2_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% ring_class.ring_distribs(2)
thf(fact_2834_crossproduct__noteq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) )
       != ( plus_plus_complex @ ( times_times_complex @ A @ D ) @ ( times_times_complex @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2835_crossproduct__noteq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) )
       != ( plus_plus_real @ ( times_times_real @ A @ D ) @ ( times_times_real @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2836_crossproduct__noteq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) )
       != ( plus_plus_rat @ ( times_times_rat @ A @ D ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2837_crossproduct__noteq,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) )
       != ( plus_plus_nat @ ( times_times_nat @ A @ D ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2838_crossproduct__noteq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( A != B )
        & ( C != D ) )
      = ( ( plus_plus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) )
       != ( plus_plus_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ C ) ) ) ) ).

% crossproduct_noteq
thf(fact_2839_crossproduct__eq,axiom,
    ! [W2: complex,Y3: complex,X3: complex,Z2: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ W2 @ Y3 ) @ ( times_times_complex @ X3 @ Z2 ) )
        = ( plus_plus_complex @ ( times_times_complex @ W2 @ Z2 ) @ ( times_times_complex @ X3 @ Y3 ) ) )
      = ( ( W2 = X3 )
        | ( Y3 = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_2840_crossproduct__eq,axiom,
    ! [W2: real,Y3: real,X3: real,Z2: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W2 @ Y3 ) @ ( times_times_real @ X3 @ Z2 ) )
        = ( plus_plus_real @ ( times_times_real @ W2 @ Z2 ) @ ( times_times_real @ X3 @ Y3 ) ) )
      = ( ( W2 = X3 )
        | ( Y3 = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_2841_crossproduct__eq,axiom,
    ! [W2: rat,Y3: rat,X3: rat,Z2: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W2 @ Y3 ) @ ( times_times_rat @ X3 @ Z2 ) )
        = ( plus_plus_rat @ ( times_times_rat @ W2 @ Z2 ) @ ( times_times_rat @ X3 @ Y3 ) ) )
      = ( ( W2 = X3 )
        | ( Y3 = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_2842_crossproduct__eq,axiom,
    ! [W2: nat,Y3: nat,X3: nat,Z2: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W2 @ Y3 ) @ ( times_times_nat @ X3 @ Z2 ) )
        = ( plus_plus_nat @ ( times_times_nat @ W2 @ Z2 ) @ ( times_times_nat @ X3 @ Y3 ) ) )
      = ( ( W2 = X3 )
        | ( Y3 = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_2843_crossproduct__eq,axiom,
    ! [W2: int,Y3: int,X3: int,Z2: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W2 @ Y3 ) @ ( times_times_int @ X3 @ Z2 ) )
        = ( plus_plus_int @ ( times_times_int @ W2 @ Z2 ) @ ( times_times_int @ X3 @ Y3 ) ) )
      = ( ( W2 = X3 )
        | ( Y3 = Z2 ) ) ) ).

% crossproduct_eq
thf(fact_2844_times__divide__times__eq,axiom,
    ! [X3: complex,Y3: complex,Z2: complex,W2: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ Y3 @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_2845_times__divide__times__eq,axiom,
    ! [X3: real,Y3: real,Z2: real,W2: real] :
      ( ( times_times_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z2 @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y3 @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_2846_times__divide__times__eq,axiom,
    ! [X3: rat,Y3: rat,Z2: rat,W2: rat] :
      ( ( times_times_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y3 @ W2 ) ) ) ).

% times_divide_times_eq
thf(fact_2847_divide__divide__times__eq,axiom,
    ! [X3: complex,Y3: complex,Z2: complex,W2: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ Z2 @ W2 ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X3 @ W2 ) @ ( times_times_complex @ Y3 @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_2848_divide__divide__times__eq,axiom,
    ! [X3: real,Y3: real,Z2: real,W2: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ Z2 @ W2 ) )
      = ( divide_divide_real @ ( times_times_real @ X3 @ W2 ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_2849_divide__divide__times__eq,axiom,
    ! [X3: rat,Y3: rat,Z2: rat,W2: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ Z2 @ W2 ) )
      = ( divide_divide_rat @ ( times_times_rat @ X3 @ W2 ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ).

% divide_divide_times_eq
thf(fact_2850_divide__divide__eq__left_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ A @ B ) @ C )
      = ( divide1717551699836669952omplex @ A @ ( times_times_complex @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_2851_divide__divide__eq__left_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ A @ B ) @ C )
      = ( divide_divide_real @ A @ ( times_times_real @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_2852_divide__divide__eq__left_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( divide_divide_rat @ A @ B ) @ C )
      = ( divide_divide_rat @ A @ ( times_times_rat @ C @ B ) ) ) ).

% divide_divide_eq_left'
thf(fact_2853_Suc__mult__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ ( suc @ K ) @ M )
        = ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( M = N2 ) ) ).

% Suc_mult_cancel1
thf(fact_2854_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2855_nat__mult__eq__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ( times_times_nat @ K @ M )
        = ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( M = N2 ) ) ) ).

% nat_mult_eq_cancel_disj
thf(fact_2856_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_2857_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_2858_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_2859_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_2860_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_2861_add__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% add_mult_distrib2
thf(fact_2862_add__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( plus_plus_nat @ M @ N2 ) @ K )
      = ( plus_plus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% add_mult_distrib
thf(fact_2863_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_2864_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_2865_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_2866_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2867_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2868_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2869_ordered__comm__semiring__class_Ocomm__mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% ordered_comm_semiring_class.comm_mult_left_mono
thf(fact_2870_zero__le__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2871_zero__le__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2872_zero__le__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) ) ) ).

% zero_le_mult_iff
thf(fact_2873_mult__nonneg__nonpos2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2874_mult__nonneg__nonpos2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2875_mult__nonneg__nonpos2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2876_mult__nonneg__nonpos2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos2
thf(fact_2877_mult__nonpos__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2878_mult__nonpos__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2879_mult__nonpos__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2880_mult__nonpos__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonpos_nonneg
thf(fact_2881_mult__nonneg__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2882_mult__nonneg__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2883_mult__nonneg__nonpos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ B @ zero_zero_nat )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2884_mult__nonneg__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_nonneg_nonpos
thf(fact_2885_mult__nonneg__nonneg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2886_mult__nonneg__nonneg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2887_mult__nonneg__nonneg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ord_less_eq_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2888_mult__nonneg__nonneg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonneg_nonneg
thf(fact_2889_split__mult__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) )
     => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ).

% split_mult_neg_le
thf(fact_2890_split__mult__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) )
     => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ).

% split_mult_neg_le
thf(fact_2891_split__mult__neg__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ( ord_less_eq_nat @ zero_zero_nat @ A )
          & ( ord_less_eq_nat @ B @ zero_zero_nat ) )
        | ( ( ord_less_eq_nat @ A @ zero_zero_nat )
          & ( ord_less_eq_nat @ zero_zero_nat @ B ) ) )
     => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ).

% split_mult_neg_le
thf(fact_2892_split__mult__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) )
     => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ).

% split_mult_neg_le
thf(fact_2893_mult__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2894_mult__le__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2895_mult__le__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ B @ zero_zero_int ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ B ) ) ) ) ).

% mult_le_0_iff
thf(fact_2896_mult__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2897_mult__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2898_mult__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2899_mult__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono
thf(fact_2900_mult__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2901_mult__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2902_mult__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_right_mono_neg
thf(fact_2903_mult__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2904_mult__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2905_mult__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2906_mult__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono
thf(fact_2907_mult__nonpos__nonpos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2908_mult__nonpos__nonpos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2909_mult__nonpos__nonpos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_eq_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_nonpos_nonpos
thf(fact_2910_mult__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2911_mult__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2912_mult__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( ord_less_eq_int @ C @ zero_zero_int )
       => ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_left_mono_neg
thf(fact_2913_split__mult__pos__le,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2914_split__mult__pos__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          & ( ord_less_eq_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_eq_rat @ A @ zero_zero_rat )
          & ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2915_split__mult__pos__le,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          & ( ord_less_eq_int @ zero_zero_int @ B ) )
        | ( ( ord_less_eq_int @ A @ zero_zero_int )
          & ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% split_mult_pos_le
thf(fact_2916_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_2917_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_2918_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_2919_mult__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2920_mult__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2921_mult__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2922_mult__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono'
thf(fact_2923_mult__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2924_mult__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2925_mult__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2926_mult__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_mono
thf(fact_2927_mult__neg__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2928_mult__neg__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2929_mult__neg__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_neg_neg
thf(fact_2930_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_2931_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_2932_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_2933_mult__less__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ zero_zero_real @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2934_mult__less__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ B @ zero_zero_rat ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ zero_zero_rat @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2935_mult__less__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ B @ zero_zero_int ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% mult_less_0_iff
thf(fact_2936_mult__neg__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_neg_pos
thf(fact_2937_mult__neg__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_neg_pos
thf(fact_2938_mult__neg__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ zero_zero_nat )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_neg_pos
thf(fact_2939_mult__neg__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_neg_pos
thf(fact_2940_mult__pos__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ B ) @ zero_zero_real ) ) ) ).

% mult_pos_neg
thf(fact_2941_mult__pos__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg
thf(fact_2942_mult__pos__neg,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ A @ B ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg
thf(fact_2943_mult__pos__neg,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ B ) @ zero_zero_int ) ) ) ).

% mult_pos_neg
thf(fact_2944_mult__pos__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2945_mult__pos__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2946_mult__pos__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2947_mult__pos__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ) ).

% mult_pos_pos
thf(fact_2948_mult__pos__neg2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ B @ A ) @ zero_zero_real ) ) ) ).

% mult_pos_neg2
thf(fact_2949_mult__pos__neg2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ B @ A ) @ zero_zero_rat ) ) ) ).

% mult_pos_neg2
thf(fact_2950_mult__pos__neg2,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ B @ zero_zero_nat )
       => ( ord_less_nat @ ( times_times_nat @ B @ A ) @ zero_zero_nat ) ) ) ).

% mult_pos_neg2
thf(fact_2951_mult__pos__neg2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ B @ A ) @ zero_zero_int ) ) ) ).

% mult_pos_neg2
thf(fact_2952_zero__less__mult__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_real @ B @ zero_zero_real ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2953_zero__less__mult__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ A )
          & ( ord_less_rat @ zero_zero_rat @ B ) )
        | ( ( ord_less_rat @ A @ zero_zero_rat )
          & ( ord_less_rat @ B @ zero_zero_rat ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2954_zero__less__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ A )
          & ( ord_less_int @ zero_zero_int @ B ) )
        | ( ( ord_less_int @ A @ zero_zero_int )
          & ( ord_less_int @ B @ zero_zero_int ) ) ) ) ).

% zero_less_mult_iff
thf(fact_2955_zero__less__mult__pos,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2956_zero__less__mult__pos,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2957_zero__less__mult__pos,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ A @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2958_zero__less__mult__pos,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos
thf(fact_2959_zero__less__mult__pos2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ B @ A ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ zero_zero_real @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2960_zero__less__mult__pos2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ B @ A ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ zero_zero_rat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2961_zero__less__mult__pos2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( times_times_nat @ B @ A ) )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ord_less_nat @ zero_zero_nat @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2962_zero__less__mult__pos2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( times_times_int @ B @ A ) )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ord_less_int @ zero_zero_int @ B ) ) ) ).

% zero_less_mult_pos2
thf(fact_2963_mult__less__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2964_mult__less__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2965_mult__less__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% mult_less_cancel_left_neg
thf(fact_2966_mult__less__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2967_mult__less__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_rat @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2968_mult__less__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_int @ A @ B ) ) ) ).

% mult_less_cancel_left_pos
thf(fact_2969_mult__strict__left__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2970_mult__strict__left__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2971_mult__strict__left__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono_neg
thf(fact_2972_mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2973_mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2974_mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2975_mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% mult_strict_left_mono
thf(fact_2976_mult__less__cancel__left__disj,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2977_mult__less__cancel__left__disj,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2978_mult__less__cancel__left__disj,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left_disj
thf(fact_2979_mult__strict__right__mono__neg,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2980_mult__strict__right__mono__neg,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2981_mult__strict__right__mono__neg,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( ord_less_int @ C @ zero_zero_int )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono_neg
thf(fact_2982_mult__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2983_mult__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2984_mult__strict__right__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2985_mult__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ) ).

% mult_strict_right_mono
thf(fact_2986_mult__less__cancel__right__disj,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
          & ( ord_less_real @ A @ B ) )
        | ( ( ord_less_real @ C @ zero_zero_real )
          & ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2987_mult__less__cancel__right__disj,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
          & ( ord_less_rat @ A @ B ) )
        | ( ( ord_less_rat @ C @ zero_zero_rat )
          & ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2988_mult__less__cancel__right__disj,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
          & ( ord_less_int @ A @ B ) )
        | ( ( ord_less_int @ C @ zero_zero_int )
          & ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right_disj
thf(fact_2989_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2990_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2991_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2992_linordered__comm__semiring__strict__class_Ocomm__mult__strict__left__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ) ).

% linordered_comm_semiring_strict_class.comm_mult_strict_left_mono
thf(fact_2993_add__scale__eq__noteq,axiom,
    ! [R2: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R2 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R2 @ C ) )
         != ( plus_plus_complex @ B @ ( times_times_complex @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2994_add__scale__eq__noteq,axiom,
    ! [R2: real,A: real,B: real,C: real,D: real] :
      ( ( R2 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R2 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2995_add__scale__eq__noteq,axiom,
    ! [R2: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R2 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R2 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2996_add__scale__eq__noteq,axiom,
    ! [R2: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R2 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R2 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2997_add__scale__eq__noteq,axiom,
    ! [R2: int,A: int,B: int,C: int,D: int] :
      ( ( R2 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R2 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R2 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_2998_less__1__mult,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_real @ one_one_real @ M )
     => ( ( ord_less_real @ one_one_real @ N2 )
       => ( ord_less_real @ one_one_real @ ( times_times_real @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_2999_less__1__mult,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_rat @ one_one_rat @ M )
     => ( ( ord_less_rat @ one_one_rat @ N2 )
       => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_3000_less__1__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ M )
     => ( ( ord_less_nat @ one_one_nat @ N2 )
       => ( ord_less_nat @ one_one_nat @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_3001_less__1__mult,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ one_one_int @ M )
     => ( ( ord_less_int @ one_one_int @ N2 )
       => ( ord_less_int @ one_one_int @ ( times_times_int @ M @ N2 ) ) ) ) ).

% less_1_mult
thf(fact_3002_frac__eq__eq,axiom,
    ! [Y3: complex,Z2: complex,X3: complex,W2: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X3 @ Y3 )
            = ( divide1717551699836669952omplex @ W2 @ Z2 ) )
          = ( ( times_times_complex @ X3 @ Z2 )
            = ( times_times_complex @ W2 @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3003_frac__eq__eq,axiom,
    ! [Y3: real,Z2: real,X3: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ( divide_divide_real @ X3 @ Y3 )
            = ( divide_divide_real @ W2 @ Z2 ) )
          = ( ( times_times_real @ X3 @ Z2 )
            = ( times_times_real @ W2 @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3004_frac__eq__eq,axiom,
    ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X3 @ Y3 )
            = ( divide_divide_rat @ W2 @ Z2 ) )
          = ( ( times_times_rat @ X3 @ Z2 )
            = ( times_times_rat @ W2 @ Y3 ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3005_divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq
thf(fact_3006_divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( divide_divide_real @ B @ C )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq
thf(fact_3007_divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( divide_divide_rat @ B @ C )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq
thf(fact_3008_eq__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq
thf(fact_3009_eq__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq
thf(fact_3010_eq__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq
thf(fact_3011_divide__eq__imp,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( B
          = ( times_times_complex @ A @ C ) )
       => ( ( divide1717551699836669952omplex @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_3012_divide__eq__imp,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( B
          = ( times_times_real @ A @ C ) )
       => ( ( divide_divide_real @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_3013_divide__eq__imp,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( B
          = ( times_times_rat @ A @ C ) )
       => ( ( divide_divide_rat @ B @ C )
          = A ) ) ) ).

% divide_eq_imp
thf(fact_3014_eq__divide__imp,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( times_times_complex @ A @ C )
          = B )
       => ( A
          = ( divide1717551699836669952omplex @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_3015_eq__divide__imp,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( ( times_times_real @ A @ C )
          = B )
       => ( A
          = ( divide_divide_real @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_3016_eq__divide__imp,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( times_times_rat @ A @ C )
          = B )
       => ( A
          = ( divide_divide_rat @ B @ C ) ) ) ) ).

% eq_divide_imp
thf(fact_3017_nonzero__divide__eq__eq,axiom,
    ! [C: complex,B: complex,A: complex] :
      ( ( C != zero_zero_complex )
     => ( ( ( divide1717551699836669952omplex @ B @ C )
          = A )
        = ( B
          = ( times_times_complex @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_3018_nonzero__divide__eq__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( C != zero_zero_real )
     => ( ( ( divide_divide_real @ B @ C )
          = A )
        = ( B
          = ( times_times_real @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_3019_nonzero__divide__eq__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( C != zero_zero_rat )
     => ( ( ( divide_divide_rat @ B @ C )
          = A )
        = ( B
          = ( times_times_rat @ A @ C ) ) ) ) ).

% nonzero_divide_eq_eq
thf(fact_3020_nonzero__eq__divide__eq,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( C != zero_zero_complex )
     => ( ( A
          = ( divide1717551699836669952omplex @ B @ C ) )
        = ( ( times_times_complex @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_3021_nonzero__eq__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( C != zero_zero_real )
     => ( ( A
          = ( divide_divide_real @ B @ C ) )
        = ( ( times_times_real @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_3022_nonzero__eq__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( C != zero_zero_rat )
     => ( ( A
          = ( divide_divide_rat @ B @ C ) )
        = ( ( times_times_rat @ A @ C )
          = B ) ) ) ).

% nonzero_eq_divide_eq
thf(fact_3023_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_3024_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_3025_power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_Suc
thf(fact_3026_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_3027_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_3028_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_3029_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_3030_power__Suc2,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ N2 ) @ A ) ) ).

% power_Suc2
thf(fact_3031_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_3032_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_3033_Suc__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% Suc_mult_less_cancel1
thf(fact_3034_power__add,axiom,
    ! [A: complex,M: nat,N2: nat] :
      ( ( power_power_complex @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_add
thf(fact_3035_power__add,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( power_power_real @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% power_add
thf(fact_3036_power__add,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( power_power_rat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_add
thf(fact_3037_power__add,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_add
thf(fact_3038_power__add,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( power_power_int @ A @ ( plus_plus_nat @ M @ N2 ) )
      = ( times_times_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_add
thf(fact_3039_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_3040_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_3041_nat__mult__eq__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ( times_times_nat @ K @ M )
          = ( times_times_nat @ K @ N2 ) )
        = ( M = N2 ) ) ) ).

% nat_mult_eq_cancel1
thf(fact_3042_nat__mult__less__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_mult_less_cancel1
thf(fact_3043_Suc__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ ( suc @ K ) @ M ) @ ( times_times_nat @ ( suc @ K ) @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_mult_le_cancel1
thf(fact_3044_mult__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( suc @ M ) @ N2 )
      = ( plus_plus_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ).

% mult_Suc
thf(fact_3045_mult__eq__self__implies__10,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( times_times_nat @ M @ N2 ) )
     => ( ( N2 = one_one_nat )
        | ( M = zero_zero_nat ) ) ) ).

% mult_eq_self_implies_10
thf(fact_3046_less__mult__imp__div__less,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I @ N2 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I ) ) ).

% less_mult_imp_div_less
thf(fact_3047_times__div__less__eq__dividend,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) @ M ) ).

% times_div_less_eq_dividend
thf(fact_3048_div__times__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) @ M ) ).

% div_times_less_eq_dividend
thf(fact_3049_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_3050_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_3051_power__odd__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( times_times_rat @ A @ ( power_power_rat @ ( power_power_rat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% power_odd_eq
thf(fact_3052_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_3053_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_3054_Suc__double__not__eq__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
     != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% Suc_double_not_eq_double
thf(fact_3055_double__not__eq__Suc__double,axiom,
    ! [M: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
     != ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% double_not_eq_Suc_double
thf(fact_3056_mult__le__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3057_mult__le__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3058_mult__le__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_left
thf(fact_3059_mult__le__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ A @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3060_mult__le__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3061_mult__le__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ A @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ A ) ) ) ) ).

% mult_le_cancel_right
thf(fact_3062_mult__left__less__imp__less,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3063_mult__left__less__imp__less,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3064_mult__left__less__imp__less,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3065_mult__left__less__imp__less,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_left_less_imp_less
thf(fact_3066_mult__strict__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3067_mult__strict__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ B )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3068_mult__strict__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ B )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3069_mult__strict__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono
thf(fact_3070_mult__less__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3071_mult__less__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3072_mult__less__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_left
thf(fact_3073_mult__right__less__imp__less,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ord_less_real @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3074_mult__right__less__imp__less,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ord_less_rat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3075_mult__right__less__imp__less,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
       => ( ord_less_nat @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3076_mult__right__less__imp__less,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ C )
       => ( ord_less_int @ A @ B ) ) ) ).

% mult_right_less_imp_less
thf(fact_3077_mult__strict__mono_H,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3078_mult__strict__mono_H,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3079_mult__strict__mono_H,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3080_mult__strict__mono_H,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_strict_mono'
thf(fact_3081_mult__less__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ A @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3082_mult__less__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3083_mult__less__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ A @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ A ) ) ) ) ).

% mult_less_cancel_right
thf(fact_3084_mult__le__cancel__left__neg,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3085_mult__le__cancel__left__neg,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3086_mult__le__cancel__left__neg,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ C @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% mult_le_cancel_left_neg
thf(fact_3087_mult__le__cancel__left__pos,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3088_mult__le__cancel__left__pos,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
        = ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3089_mult__le__cancel__left__pos,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ C )
     => ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_le_cancel_left_pos
thf(fact_3090_mult__left__le__imp__le,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3091_mult__left__le__imp__le,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3092_mult__left__le__imp__le,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3093_mult__left__le__imp__le,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_left_le_imp_le
thf(fact_3094_mult__right__le__imp__le,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ord_less_eq_real @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3095_mult__right__le__imp__le,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ord_less_eq_rat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3096_mult__right__le__imp__le,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
     => ( ( ord_less_nat @ zero_zero_nat @ C )
       => ( ord_less_eq_nat @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3097_mult__right__le__imp__le,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
     => ( ( ord_less_int @ zero_zero_int @ C )
       => ( ord_less_eq_int @ A @ B ) ) ) ).

% mult_right_le_imp_le
thf(fact_3098_mult__le__less__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ C @ D )
       => ( ( ord_less_real @ zero_zero_real @ A )
         => ( ( ord_less_eq_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3099_mult__le__less__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ C @ D )
       => ( ( ord_less_rat @ zero_zero_rat @ A )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3100_mult__le__less__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_nat @ C @ D )
       => ( ( ord_less_nat @ zero_zero_nat @ A )
         => ( ( ord_less_eq_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3101_mult__le__less__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_int @ C @ D )
       => ( ( ord_less_int @ zero_zero_int @ A )
         => ( ( ord_less_eq_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_le_less_imp_less
thf(fact_3102_mult__less__le__imp__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ D )
       => ( ( ord_less_eq_real @ zero_zero_real @ A )
         => ( ( ord_less_real @ zero_zero_real @ C )
           => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3103_mult__less__le__imp__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ D )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
         => ( ( ord_less_rat @ zero_zero_rat @ C )
           => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3104_mult__less__le__imp__less,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ D )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
         => ( ( ord_less_nat @ zero_zero_nat @ C )
           => ( ord_less_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3105_mult__less__le__imp__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_eq_int @ C @ D )
       => ( ( ord_less_eq_int @ zero_zero_int @ A )
         => ( ( ord_less_int @ zero_zero_int @ C )
           => ( ord_less_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% mult_less_le_imp_less
thf(fact_3106_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_3107_mult__left__le,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ C @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ A ) ) ) ).

% mult_left_le
thf(fact_3108_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_3109_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_3110_mult__le__one,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ one_one_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ B @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ one_one_real ) ) ) ) ).

% mult_le_one
thf(fact_3111_mult__le__one,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ one_one_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ B @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ one_one_rat ) ) ) ) ).

% mult_le_one
thf(fact_3112_mult__le__one,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ one_one_nat )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ one_one_nat )
         => ( ord_less_eq_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ) ).

% mult_le_one
thf(fact_3113_mult__le__one,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ one_one_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ) ).

% mult_le_one
thf(fact_3114_mult__right__le__one__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X3 @ Y3 ) @ X3 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3115_mult__right__le__one__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Y3 ) @ X3 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3116_mult__right__le__one__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X3 @ Y3 ) @ X3 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3117_mult__left__le__one__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y3 @ X3 ) @ X3 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3118_mult__left__le__one__le,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
       => ( ( ord_less_eq_rat @ Y3 @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y3 @ X3 ) @ X3 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3119_mult__left__le__one__le,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( ord_less_eq_int @ Y3 @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y3 @ X3 ) @ X3 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3120_sum__squares__le__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real )
      = ( ( X3 = zero_zero_real )
        & ( Y3 = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3121_sum__squares__le__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat )
      = ( ( X3 = zero_zero_rat )
        & ( Y3 = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3122_sum__squares__le__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int )
      = ( ( X3 = zero_zero_int )
        & ( Y3 = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3123_sum__squares__ge__zero,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_3124_sum__squares__ge__zero,axiom,
    ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_3125_sum__squares__ge__zero,axiom,
    ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) ) ).

% sum_squares_ge_zero
thf(fact_3126_sum__squares__gt__zero__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) )
      = ( ( X3 != zero_zero_real )
        | ( Y3 != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3127_sum__squares__gt__zero__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) )
      = ( ( X3 != zero_zero_rat )
        | ( Y3 != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3128_sum__squares__gt__zero__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) )
      = ( ( X3 != zero_zero_int )
        | ( Y3 != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3129_not__sum__squares__lt__zero,axiom,
    ! [X3: real,Y3: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_3130_not__sum__squares__lt__zero,axiom,
    ! [X3: rat,Y3: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_3131_not__sum__squares__lt__zero,axiom,
    ! [X3: int,Y3: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_3132_divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3133_divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_less_eq
thf(fact_3134_less__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3135_less__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq
thf(fact_3136_neg__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_3137_neg__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_less_eq
thf(fact_3138_neg__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3139_neg__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_divide_eq
thf(fact_3140_pos__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3141_pos__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_less_eq
thf(fact_3142_pos__less__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_3143_pos__less__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_less_divide_eq
thf(fact_3144_mult__imp__div__pos__less,axiom,
    ! [Y3: real,X3: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) )
       => ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z2 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3145_mult__imp__div__pos__less,axiom,
    ! [Y3: rat,X3: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z2 ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3146_mult__imp__less__div__pos,axiom,
    ! [Y3: real,Z2: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ ( times_times_real @ Z2 @ Y3 ) @ X3 )
       => ( ord_less_real @ Z2 @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3147_mult__imp__less__div__pos,axiom,
    ! [Y3: rat,Z2: rat,X3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_rat @ ( times_times_rat @ Z2 @ Y3 ) @ X3 )
       => ( ord_less_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3148_divide__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( ord_less_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3149_divide__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono
thf(fact_3150_divide__strict__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3151_divide__strict__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_strict_left_mono_neg
thf(fact_3152_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( numera6690914467698888265omplex @ W2 ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3153_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W2 ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3154_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W2 ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3155_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W2 )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W2 )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3156_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W2 )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W2 )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3157_eq__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W2 )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W2 )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3158_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3159_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3160_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3161_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3162_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3163_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3164_add__frac__eq,axiom,
    ! [Y3: complex,Z2: complex,X3: complex,W2: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3165_add__frac__eq,axiom,
    ! [Y3: real,Z2: real,X3: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3166_add__frac__eq,axiom,
    ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ) ) ).

% add_frac_eq
thf(fact_3167_add__frac__num,axiom,
    ! [Y3: complex,X3: complex,Z2: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ Z2 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_3168_add__frac__num,axiom,
    ! [Y3: real,X3: real,Z2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z2 )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_3169_add__frac__num,axiom,
    ! [Y3: rat,X3: rat,Z2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z2 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).

% add_frac_num
thf(fact_3170_add__num__frac,axiom,
    ! [Y3: complex,Z2: complex,X3: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ X3 @ Y3 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_3171_add__num__frac,axiom,
    ! [Y3: real,Z2: real,X3: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( plus_plus_real @ Z2 @ ( divide_divide_real @ X3 @ Y3 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_3172_add__num__frac,axiom,
    ! [Y3: rat,Z2: rat,X3: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y3 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) ) @ Y3 ) ) ) ).

% add_num_frac
thf(fact_3173_add__divide__eq__iff,axiom,
    ! [Z2: complex,X3: complex,Y3: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z2 ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_3174_add__divide__eq__iff,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ X3 @ ( divide_divide_real @ Y3 @ Z2 ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_3175_add__divide__eq__iff,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z2 ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).

% add_divide_eq_iff
thf(fact_3176_divide__add__eq__iff,axiom,
    ! [Z2: complex,X3: complex,Y3: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X3 @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_3177_divide__add__eq__iff,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X3 @ Z2 ) @ Y3 )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_3178_divide__add__eq__iff,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ X3 @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% divide_add_eq_iff
thf(fact_3179_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_3180_power__gt1__lemma,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ one_one_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_gt1_lemma
thf(fact_3181_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_3182_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_3183_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_3184_power__less__power__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ one_one_rat @ A )
     => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_less_power_Suc
thf(fact_3185_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_3186_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_3187_one__less__mult,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% one_less_mult
thf(fact_3188_n__less__m__mult__n,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ M @ N2 ) ) ) ) ).

% n_less_m_mult_n
thf(fact_3189_n__less__n__mult__m,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
       => ( ord_less_nat @ N2 @ ( times_times_nat @ N2 @ M ) ) ) ) ).

% n_less_n_mult_m
thf(fact_3190_nat__mult__le__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( ord_less_eq_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% nat_mult_le_cancel1
thf(fact_3191_nat__mult__div__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( divide_divide_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( divide_divide_nat @ M @ N2 ) ) ) ).

% nat_mult_div_cancel1
thf(fact_3192_div__less__iff__less__mult,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q4 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q4 ) @ N2 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q4 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_3193_mult__le__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3194_mult__le__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3195_mult__le__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_left1
thf(fact_3196_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_3197_mult__le__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_left2
thf(fact_3198_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_3199_mult__le__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_eq_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ one_one_real @ B ) )
        & ( ( ord_less_real @ C @ zero_zero_real )
         => ( ord_less_eq_real @ B @ one_one_real ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3200_mult__le__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_eq_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ one_one_rat @ B ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ B @ one_one_rat ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3201_mult__le__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_eq_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_int @ zero_zero_int @ C )
         => ( ord_less_eq_int @ one_one_int @ B ) )
        & ( ( ord_less_int @ C @ zero_zero_int )
         => ( ord_less_eq_int @ B @ one_one_int ) ) ) ) ).

% mult_le_cancel_right1
thf(fact_3202_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_3203_mult__le__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ A @ one_one_rat ) )
        & ( ( ord_less_rat @ C @ zero_zero_rat )
         => ( ord_less_eq_rat @ one_one_rat @ A ) ) ) ) ).

% mult_le_cancel_right2
thf(fact_3204_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_3205_mult__less__cancel__left1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ C @ B ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3206_mult__less__cancel__left1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ C @ B ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3207_mult__less__cancel__left1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ C @ B ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_left1
thf(fact_3208_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_3209_mult__less__cancel__left2,axiom,
    ! [C: rat,A: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ C @ A ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_left2
thf(fact_3210_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_3211_mult__less__cancel__right1,axiom,
    ! [C: real,B: real] :
      ( ( ord_less_real @ C @ ( times_times_real @ B @ C ) )
      = ( ( ( ord_less_eq_real @ zero_zero_real @ C )
         => ( ord_less_real @ one_one_real @ B ) )
        & ( ( ord_less_eq_real @ C @ zero_zero_real )
         => ( ord_less_real @ B @ one_one_real ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3212_mult__less__cancel__right1,axiom,
    ! [C: rat,B: rat] :
      ( ( ord_less_rat @ C @ ( times_times_rat @ B @ C ) )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ one_one_rat @ B ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ B @ one_one_rat ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3213_mult__less__cancel__right1,axiom,
    ! [C: int,B: int] :
      ( ( ord_less_int @ C @ ( times_times_int @ B @ C ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ C )
         => ( ord_less_int @ one_one_int @ B ) )
        & ( ( ord_less_eq_int @ C @ zero_zero_int )
         => ( ord_less_int @ B @ one_one_int ) ) ) ) ).

% mult_less_cancel_right1
thf(fact_3214_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_3215_mult__less__cancel__right2,axiom,
    ! [A: rat,C: rat] :
      ( ( ord_less_rat @ ( times_times_rat @ A @ C ) @ C )
      = ( ( ( ord_less_eq_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ A @ one_one_rat ) )
        & ( ( ord_less_eq_rat @ C @ zero_zero_rat )
         => ( ord_less_rat @ one_one_rat @ A ) ) ) ) ).

% mult_less_cancel_right2
thf(fact_3216_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_3217_field__le__mult__one__interval,axiom,
    ! [X3: real,Y3: real] :
      ( ! [Z3: real] :
          ( ( ord_less_real @ zero_zero_real @ Z3 )
         => ( ( ord_less_real @ Z3 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z3 @ X3 ) @ Y3 ) ) )
     => ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% field_le_mult_one_interval
thf(fact_3218_field__le__mult__one__interval,axiom,
    ! [X3: rat,Y3: rat] :
      ( ! [Z3: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z3 )
         => ( ( ord_less_rat @ Z3 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z3 @ X3 ) @ Y3 ) ) )
     => ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% field_le_mult_one_interval
thf(fact_3219_divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3220_divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% divide_le_eq
thf(fact_3221_le__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3222_le__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq
thf(fact_3223_divide__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3224_divide__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ C )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono
thf(fact_3225_neg__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3226_neg__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% neg_divide_le_eq
thf(fact_3227_neg__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3228_neg__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_divide_eq
thf(fact_3229_pos__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ A )
        = ( ord_less_eq_real @ B @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3230_pos__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ A )
        = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_divide_le_eq
thf(fact_3231_pos__le__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ C ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3232_pos__le__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ C ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ B ) ) ) ).

% pos_le_divide_eq
thf(fact_3233_mult__imp__div__pos__le,axiom,
    ! [Y3: real,X3: real,Z2: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ X3 @ ( times_times_real @ Z2 @ Y3 ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ Z2 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3234_mult__imp__div__pos__le,axiom,
    ! [Y3: rat,X3: rat,Z2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ X3 @ ( times_times_rat @ Z2 @ Y3 ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ Z2 ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3235_mult__imp__le__div__pos,axiom,
    ! [Y3: real,Z2: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ Y3 ) @ X3 )
       => ( ord_less_eq_real @ Z2 @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3236_mult__imp__le__div__pos,axiom,
    ! [Y3: rat,Z2: rat,X3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ Y3 ) @ X3 )
       => ( ord_less_eq_rat @ Z2 @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3237_divide__left__mono__neg,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ C @ zero_zero_real )
       => ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ ( divide_divide_real @ C @ A ) @ ( divide_divide_real @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3238_divide__left__mono__neg,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ C @ zero_zero_rat )
       => ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ ( divide_divide_rat @ C @ A ) @ ( divide_divide_rat @ C @ B ) ) ) ) ) ).

% divide_left_mono_neg
thf(fact_3239_convex__bound__le,axiom,
    ! [X3: real,A: real,Y3: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X3 @ A )
     => ( ( ord_less_eq_real @ Y3 @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3240_convex__bound__le,axiom,
    ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X3 @ A )
     => ( ( ord_less_eq_rat @ Y3 @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3241_convex__bound__le,axiom,
    ! [X3: int,A: int,Y3: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X3 @ A )
     => ( ( ord_less_eq_int @ Y3 @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3242_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3243_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3244_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3245_less__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3246_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_3247_power__Suc__less,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ ( times_times_rat @ A @ ( power_power_rat @ A @ N2 ) ) @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_Suc_less
thf(fact_3248_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_3249_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_3250_mult__2,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_complex @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3251_mult__2,axiom,
    ! [Z2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3252_mult__2,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_real @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3253_mult__2,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3254_mult__2,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3255_mult__2,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z2 )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2
thf(fact_3256_mult__2__right,axiom,
    ! [Z2: complex] :
      ( ( times_times_complex @ Z2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3257_mult__2__right,axiom,
    ! [Z2: extended_enat] :
      ( ( times_7803423173614009249d_enat @ Z2 @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3258_mult__2__right,axiom,
    ! [Z2: real] :
      ( ( times_times_real @ Z2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3259_mult__2__right,axiom,
    ! [Z2: rat] :
      ( ( times_times_rat @ Z2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3260_mult__2__right,axiom,
    ! [Z2: nat] :
      ( ( times_times_nat @ Z2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3261_mult__2__right,axiom,
    ! [Z2: int] :
      ( ( times_times_int @ Z2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z2 @ Z2 ) ) ).

% mult_2_right
thf(fact_3262_left__add__twice,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3263_left__add__twice,axiom,
    ! [A: extended_enat,B: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ A @ ( plus_p3455044024723400733d_enat @ A @ B ) )
      = ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3264_left__add__twice,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3265_left__add__twice,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3266_left__add__twice,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3267_left__add__twice,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ B ) ) ).

% left_add_twice
thf(fact_3268_div__nat__eqI,axiom,
    ! [N2: nat,Q4: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q4 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q4 ) ) )
       => ( ( divide_divide_nat @ M @ N2 )
          = Q4 ) ) ) ).

% div_nat_eqI
thf(fact_3269_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q4: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q4 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q4 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q4 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_3270_split__div,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_nat ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) )
               => ( P @ I2 ) ) ) ) ) ) ).

% split_div
thf(fact_3271_dividend__less__div__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ ( divide_divide_nat @ M @ N2 ) @ N2 ) ) ) ) ).

% dividend_less_div_times
thf(fact_3272_dividend__less__times__div,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ M @ ( plus_plus_nat @ N2 @ ( times_times_nat @ N2 @ ( divide_divide_nat @ M @ N2 ) ) ) ) ) ).

% dividend_less_times_div
thf(fact_3273_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_3274_convex__bound__lt,axiom,
    ! [X3: real,A: real,Y3: real,U: real,V: real] :
      ( ( ord_less_real @ X3 @ A )
     => ( ( ord_less_real @ Y3 @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X3 ) @ ( times_times_real @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3275_convex__bound__lt,axiom,
    ! [X3: rat,A: rat,Y3: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X3 @ A )
     => ( ( ord_less_rat @ Y3 @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X3 ) @ ( times_times_rat @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3276_convex__bound__lt,axiom,
    ! [X3: int,A: int,Y3: int,U: int,V: int] :
      ( ( ord_less_int @ X3 @ A )
     => ( ( ord_less_int @ Y3 @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X3 ) @ ( times_times_int @ V @ Y3 ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3277_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W2 ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3278_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W2 ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3279_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W2 ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3280_le__divide__eq__numeral_I1_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W2 ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W2 ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3281_split__div_H,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( divide_divide_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
          & ( P @ zero_zero_nat ) )
        | ? [Q5: nat] :
            ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q5 ) @ M )
            & ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q5 ) ) )
            & ( P @ Q5 ) ) ) ) ).

% split_div'
thf(fact_3282_power2__sum,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_3283_power2__sum,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_3284_power2__sum,axiom,
    ! [X3: real,Y3: real] :
      ( ( power_power_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_3285_power2__sum,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_3286_power2__sum,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_3287_power2__sum,axiom,
    ! [X3: int,Y3: int] :
      ( ( power_power_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_sum
thf(fact_3288_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_3289_zero__le__even__power_H,axiom,
    ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% zero_le_even_power'
thf(fact_3290_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_3291_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_3292_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_3293_odd__0__le__power__imp__0__le,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% odd_0_le_power_imp_0_le
thf(fact_3294_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_3295_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_3296_odd__power__less__zero,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( power_power_rat @ A @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ zero_zero_rat ) ) ).

% odd_power_less_zero
thf(fact_3297_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_3298_arith__geo__mean,axiom,
    ! [U: real,X3: real,Y3: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X3 @ Y3 ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_3299_arith__geo__mean,axiom,
    ! [U: rat,X3: rat,Y3: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X3 @ Y3 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_3300_sum__squares__bound,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_3301_sum__squares__bound,axiom,
    ! [X3: rat,Y3: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_3302_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_3303_set__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% set_bit_0
thf(fact_3304_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_3305_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_3306_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_3307_unset__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ zero_zero_nat @ A )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% unset_bit_0
thf(fact_3308_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_3309_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3310_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3311_unique__euclidean__semiring__numeral__class_Odiv__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_mult2_eq
thf(fact_3312_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_3313_discrete,axiom,
    ( ord_less_int
    = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_3314_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3315_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3316_unique__euclidean__semiring__numeral__class_Odiv__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( divide_divide_int @ A @ B )
          = zero_zero_int ) ) ) ).

% unique_euclidean_semiring_numeral_class.div_less
thf(fact_3317_div__positive,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le3102999989581377725nteger @ B @ A )
       => ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3318_div__positive,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( ord_less_nat @ zero_zero_nat @ ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3319_div__positive,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% div_positive
thf(fact_3320_zle__add1__eq__le,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% zle_add1_eq_le
thf(fact_3321_set__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% set_bit_negative_int_iff
thf(fact_3322_unset__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% unset_bit_negative_int_iff
thf(fact_3323_div__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_pos_pos_trivial
thf(fact_3324_div__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = zero_zero_int ) ) ) ).

% div_neg_neg_trivial
thf(fact_3325_nonneg1__imp__zdiv__pos__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ( ord_less_eq_int @ B @ A )
          & ( ord_less_int @ zero_zero_int @ B ) ) ) ) ).

% nonneg1_imp_zdiv_pos_iff
thf(fact_3326_pos__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% pos_imp_zdiv_nonneg_iff
thf(fact_3327_neg__imp__zdiv__nonneg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ A @ B ) )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% neg_imp_zdiv_nonneg_iff
thf(fact_3328_int__one__le__iff__zero__less,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z2 )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% int_one_le_iff_zero_less
thf(fact_3329_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I @ K ) )
        = ( ord_less_eq_int @ K @ I ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_3330_div__nonpos__pos__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonpos_pos_le0
thf(fact_3331_div__nonneg__neg__le0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_nonneg_neg_le0
thf(fact_3332_div__positive__int,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ L @ K )
     => ( ( ord_less_int @ zero_zero_int @ L )
       => ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) ) ) ) ).

% div_positive_int
thf(fact_3333_div__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ L ) )
      = ( ( K = zero_zero_int )
        | ( L = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ K )
          & ( ord_less_eq_int @ zero_zero_int @ L ) )
        | ( ( ord_less_int @ K @ zero_zero_int )
          & ( ord_less_int @ L @ zero_zero_int ) ) ) ) ).

% div_int_pos_iff
thf(fact_3334_zless__imp__add1__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ W2 @ Z2 )
     => ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 ) ) ).

% zless_imp_add1_zle
thf(fact_3335_zdiv__mono2__neg,axiom,
    ! [A: int,B6: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B6 )
       => ( ( ord_less_eq_int @ B6 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B6 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_3336_zdiv__mono1__neg,axiom,
    ! [A: int,A6: int,B: int] :
      ( ( ord_less_eq_int @ A @ A6 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A6 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_3337_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q4 ) ) ) ) ).

% int_div_pos_eq
thf(fact_3338_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( divide_divide_int @ A @ B )
            = Q4 ) ) ) ) ).

% int_div_neg_eq
thf(fact_3339_zdiv__eq__0__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( divide_divide_int @ I @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_3340_zdiv__mono2,axiom,
    ! [A: int,B6: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B6 )
       => ( ( ord_less_eq_int @ B6 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B6 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_3341_zdiv__mono1,axiom,
    ! [A: int,A6: int,B: int] :
      ( ( ord_less_eq_int @ A @ A6 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A6 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_3342_split__zdiv,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( divide_divide_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ zero_zero_int ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 ) ) ) ) ) ).

% split_zdiv
thf(fact_3343_le__imp__0__less,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ).

% le_imp_0_less
thf(fact_3344_add1__zle__eq,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W2 @ one_one_int ) @ Z2 )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% add1_zle_eq
thf(fact_3345_odd__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 ) @ zero_zero_int )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_3346_pos__zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ( times_times_int @ M @ N2 )
          = one_one_int )
        = ( ( M = one_one_int )
          & ( N2 = one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff
thf(fact_3347_zless__add1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ W2 @ ( plus_plus_int @ Z2 @ one_one_int ) )
      = ( ( ord_less_int @ W2 @ Z2 )
        | ( W2 = Z2 ) ) ) ).

% zless_add1_eq
thf(fact_3348_int__gr__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_int @ K @ I )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_gr_induct
thf(fact_3349_pos__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% pos_imp_zdiv_neg_iff
thf(fact_3350_neg__imp__zdiv__neg__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% neg_imp_zdiv_neg_iff
thf(fact_3351_int__div__less__self,axiom,
    ! [X3: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X3 )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X3 @ K ) @ X3 ) ) ) ).

% int_div_less_self
thf(fact_3352_div__neg__pos__less0,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_int @ ( divide_divide_int @ A @ B ) @ zero_zero_int ) ) ) ).

% div_neg_pos_less0
thf(fact_3353_odd__nonzero,axiom,
    ! [Z2: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z2 ) @ Z2 )
     != zero_zero_int ) ).

% odd_nonzero
thf(fact_3354_times__int__code_I2_J,axiom,
    ! [L: int] :
      ( ( times_times_int @ zero_zero_int @ L )
      = zero_zero_int ) ).

% times_int_code(2)
thf(fact_3355_times__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( times_times_int @ K @ zero_zero_int )
      = zero_zero_int ) ).

% times_int_code(1)
thf(fact_3356_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_3357_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int] :
            ( ( P @ X4 )
           => ( P @ ( plus_plus_int @ X4 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X2: int] :
              ( ( P @ X2 )
             => ( P @ ( plus_plus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% incr_mult_lemma
thf(fact_3358_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_3359_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_3360_q__pos__lemma,axiom,
    ! [B6: int,Q6: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
     => ( ( ord_less_int @ R3 @ B6 )
       => ( ( ord_less_int @ zero_zero_int @ B6 )
         => ( ord_less_eq_int @ zero_zero_int @ Q6 ) ) ) ) ).

% q_pos_lemma
thf(fact_3361_zdiv__mono2__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B6: int,Q6: int,R3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
       => ( ( ord_less_int @ R3 @ B6 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
           => ( ( ord_less_int @ zero_zero_int @ B6 )
             => ( ( ord_less_eq_int @ B6 @ B )
               => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_3362_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q4: int,R2: int,B6: int,Q6: int,R3: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 )
        = ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B6 @ Q6 ) @ R3 ) @ zero_zero_int )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B6 )
             => ( ( ord_less_eq_int @ B6 @ B )
               => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_3363_unique__quotient__lemma,axiom,
    ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( ord_less_int @ R2 @ B )
           => ( ord_less_eq_int @ Q6 @ Q4 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_3364_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q6: int,R3: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q6 ) @ R3 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( ord_less_int @ B @ R3 )
           => ( ord_less_eq_int @ Q4 @ Q6 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_3365_enat__0__less__mult__iff,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( times_7803423173614009249d_enat @ M @ N2 ) )
      = ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ M )
        & ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 ) ) ) ).

% enat_0_less_mult_iff
thf(fact_3366_realpow__pos__nth2,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R4: real] :
          ( ( ord_less_real @ zero_zero_real @ R4 )
          & ( ( power_power_real @ R4 @ ( suc @ N2 ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_3367_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X4: real] :
            ( ( ord_less_real @ zero_zero_real @ X4 )
            & ( ( power_power_real @ X4 @ N2 )
              = A )
            & ! [Y6: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y6 )
                  & ( ( power_power_real @ Y6 @ N2 )
                    = A ) )
               => ( Y6 = X4 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_3368_realpow__pos__nth,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ( ( power_power_real @ R4 @ N2 )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_3369_mult__le__cancel__iff2,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z2 @ X3 ) @ ( times_times_real @ Z2 @ Y3 ) )
        = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3370_mult__le__cancel__iff2,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X3 ) @ ( times_times_rat @ Z2 @ Y3 ) )
        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3371_mult__le__cancel__iff2,axiom,
    ! [Z2: int,X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z2 @ X3 ) @ ( times_times_int @ Z2 @ Y3 ) )
        = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3372_mult__le__cancel__iff1,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_eq_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
        = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3373_mult__le__cancel__iff1,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y3 @ Z2 ) )
        = ( ord_less_eq_rat @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3374_mult__le__cancel__iff1,axiom,
    ! [Z2: int,X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ ( times_times_int @ X3 @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
        = ( ord_less_eq_int @ X3 @ Y3 ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3375_divides__aux__eq,axiom,
    ! [Q4: nat,R2: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
      = ( R2 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_3376_divides__aux__eq,axiom,
    ! [Q4: int,R2: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q4 @ R2 ) )
      = ( R2 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_3377_low__def,axiom,
    ( vEBT_VEBT_low
    = ( ^ [X: nat,N: nat] : ( modulo_modulo_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% low_def
thf(fact_3378_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_3379_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_3380_even__succ__div__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% even_succ_div_exp
thf(fact_3381_set__decode__Suc,axiom,
    ! [N2: nat,X3: nat] :
      ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X3 ) )
      = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_3382_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_3383_length__product,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_o] :
      ( ( size_s9168528473962070013VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3384_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_3385_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_3386_length__product,axiom,
    ! [Xs: list_o,Ys: list_VEBT_VEBT] :
      ( ( size_s4313452262239582901T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) ) ).

% length_product
thf(fact_3387_length__product,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( size_s1515746228057227161od_o_o @ ( product_o_o @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3388_length__product,axiom,
    ! [Xs: list_o,Ys: list_nat] :
      ( ( size_s5443766701097040955_o_nat @ ( product_o_nat @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) ) ).

% length_product
thf(fact_3389_length__product,axiom,
    ! [Xs: list_o,Ys: list_int] :
      ( ( size_s2953683556165314199_o_int @ ( product_o_int @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) ) ).

% length_product
thf(fact_3390_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_3391_length__product,axiom,
    ! [Xs: list_nat,Ys: list_o] :
      ( ( size_s6491369823275344609_nat_o @ ( product_nat_o @ Xs @ Ys ) )
      = ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) ) ).

% length_product
thf(fact_3392_mult__less__iff1,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( ( ord_less_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ Y3 @ Z2 ) )
        = ( ord_less_real @ X3 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_3393_mult__less__iff1,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z2 )
     => ( ( ord_less_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ Y3 @ Z2 ) )
        = ( ord_less_rat @ X3 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_3394_mult__less__iff1,axiom,
    ! [Z2: int,X3: int,Y3: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_int @ ( times_times_int @ X3 @ Z2 ) @ ( times_times_int @ Y3 @ Z2 ) )
        = ( ord_less_int @ X3 @ Y3 ) ) ) ).

% mult_less_iff1
thf(fact_3395_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_3396_nat__dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ one_one_nat )
      = ( M = one_one_nat ) ) ).

% nat_dvd_1_iff_1
thf(fact_3397_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_3398_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_3399_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_3400_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_3401_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_3402_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_3403_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_3404_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_3405_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_3406_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_3407_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_3408_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_3409_dvd__add__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ A ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3410_dvd__add__triv__right__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ A ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3411_dvd__add__triv__right__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ A ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3412_dvd__add__triv__right__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ A ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3413_dvd__add__triv__right__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ A ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_right_iff
thf(fact_3414_dvd__add__triv__left__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3415_dvd__add__triv__left__iff,axiom,
    ! [A: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ A @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3416_dvd__add__triv__left__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ A @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3417_dvd__add__triv__left__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3418_dvd__add__triv__left__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ A @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_triv_left_iff
thf(fact_3419_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_3420_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_3421_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_3422_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_3423_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_3424_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_3425_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_3426_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_3427_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_3428_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_3429_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_3430_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_3431_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_3432_dvd__1__iff__1,axiom,
    ! [M: nat] :
      ( ( dvd_dvd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( M
        = ( suc @ zero_zero_nat ) ) ) ).

% dvd_1_iff_1
thf(fact_3433_div__dvd__div,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ B @ A ) @ ( divide_divide_nat @ C @ A ) )
          = ( dvd_dvd_nat @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3434_div__dvd__div,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ B @ A ) @ ( divide_divide_int @ C @ A ) )
          = ( dvd_dvd_int @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3435_div__dvd__div,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ B @ A ) @ ( divide6298287555418463151nteger @ C @ A ) )
          = ( dvd_dvd_Code_integer @ B @ C ) ) ) ) ).

% div_dvd_div
thf(fact_3436_not__real__square__gt__zero,axiom,
    ! [X3: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X3 @ X3 ) ) )
      = ( X3 = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_3437_mod__add__self2,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self2
thf(fact_3438_mod__add__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self2
thf(fact_3439_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_3440_mod__add__self1,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ B @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_add_self1
thf(fact_3441_mod__add__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_add_self1
thf(fact_3442_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_3443_nat__mult__dvd__cancel__disj,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
      = ( ( K = zero_zero_nat )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel_disj
thf(fact_3444_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_3445_dvd__mult__cancel__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3446_dvd__mult__cancel__left,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ C @ A ) @ ( times_times_complex @ C @ B ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3447_dvd__mult__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ C @ A ) @ ( times_times_real @ C @ B ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3448_dvd__mult__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ C @ A ) @ ( times_times_rat @ C @ B ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3449_dvd__mult__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_left
thf(fact_3450_dvd__mult__cancel__right,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( ( C = zero_z3403309356797280102nteger )
        | ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3451_dvd__mult__cancel__right,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) )
      = ( ( C = zero_zero_complex )
        | ( dvd_dvd_complex @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3452_dvd__mult__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) )
      = ( ( C = zero_zero_real )
        | ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3453_dvd__mult__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) )
      = ( ( C = zero_zero_rat )
        | ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3454_dvd__mult__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( ( C = zero_zero_int )
        | ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_mult_cancel_right
thf(fact_3455_dvd__times__left__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ ( times_3573771949741848930nteger @ A @ C ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3456_dvd__times__left__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3457_dvd__times__left__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_left_cancel_iff
thf(fact_3458_dvd__times__right__cancel__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ A ) @ ( times_3573771949741848930nteger @ C @ A ) )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3459_dvd__times__right__cancel__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3460_dvd__times__right__cancel__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% dvd_times_right_cancel_iff
thf(fact_3461_unit__prod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_prod
thf(fact_3462_unit__prod,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_prod
thf(fact_3463_unit__prod,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_prod
thf(fact_3464_dvd__add__times__triv__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ ( times_3573771949741848930nteger @ C @ A ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3465_dvd__add__times__triv__right__iff,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ B @ ( times_times_complex @ C @ A ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3466_dvd__add__times__triv__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ ( times_times_real @ C @ A ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3467_dvd__add__times__triv__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ ( times_times_rat @ C @ A ) ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3468_dvd__add__times__triv__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ ( times_times_nat @ C @ A ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3469_dvd__add__times__triv__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ ( times_times_int @ C @ A ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_right_iff
thf(fact_3470_dvd__add__times__triv__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ A ) @ B ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3471_dvd__add__times__triv__left__iff,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ A @ ( plus_plus_complex @ ( times_times_complex @ C @ A ) @ B ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3472_dvd__add__times__triv__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ ( plus_plus_real @ ( times_times_real @ C @ A ) @ B ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3473_dvd__add__times__triv__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ ( times_times_rat @ C @ A ) @ B ) )
      = ( dvd_dvd_rat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3474_dvd__add__times__triv__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ ( times_times_nat @ C @ A ) @ B ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3475_dvd__add__times__triv__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( plus_plus_int @ ( times_times_int @ C @ A ) @ B ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% dvd_add_times_triv_left_iff
thf(fact_3476_mod__mult__self1__is__0,axiom,
    ! [B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ B @ A ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self1_is_0
thf(fact_3477_mod__mult__self1__is__0,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ B @ A ) @ B )
      = zero_zero_int ) ).

% mod_mult_self1_is_0
thf(fact_3478_mod__mult__self1__is__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ B @ A ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self1_is_0
thf(fact_3479_mod__mult__self2__is__0,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_mult_self2_is_0
thf(fact_3480_mod__mult__self2__is__0,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_mult_self2_is_0
thf(fact_3481_mod__mult__self2__is__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_mult_self2_is_0
thf(fact_3482_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_3483_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_3484_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_3485_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_3486_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_3487_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_3488_dvd__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3489_dvd__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3490_dvd__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% dvd_div_mult_self
thf(fact_3491_dvd__mult__div__cancel,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3492_dvd__mult__div__cancel,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3493_dvd__mult__div__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ A ) )
        = B ) ) ).

% dvd_mult_div_cancel
thf(fact_3494_unit__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ one_one_nat ) ) ) ).

% unit_div
thf(fact_3495_unit__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ).

% unit_div
thf(fact_3496_unit__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ one_one_Code_integer ) ) ) ).

% unit_div
thf(fact_3497_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_3498_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_3499_unit__div__1__unit,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) @ one_one_Code_integer ) ) ).

% unit_div_1_unit
thf(fact_3500_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_3501_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_3502_unit__div__1__div__1,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = A ) ) ).

% unit_div_1_div_1
thf(fact_3503_div__add,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3504_div__add,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3505_div__add,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_add
thf(fact_3506_bits__mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% bits_mod_div_trivial
thf(fact_3507_bits__mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% bits_mod_div_trivial
thf(fact_3508_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_3509_mod__div__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( divide_divide_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = zero_zero_nat ) ).

% mod_div_trivial
thf(fact_3510_mod__div__trivial,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = zero_zero_int ) ).

% mod_div_trivial
thf(fact_3511_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_3512_mod__mult__self4,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3513_mod__mult__self4,axiom,
    ! [B: int,C: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ B @ C ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3514_mod__mult__self4,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ C ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self4
thf(fact_3515_mod__mult__self3,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( times_times_nat @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3516_mod__mult__self3,axiom,
    ! [C: int,B: int,A: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( times_times_int @ C @ B ) @ A ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3517_mod__mult__self3,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ C @ B ) @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self3
thf(fact_3518_mod__mult__self2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ B @ C ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3519_mod__mult__self2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ B @ C ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3520_mod__mult__self2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self2
thf(fact_3521_mod__mult__self1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ C @ B ) ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3522_mod__mult__self1,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( times_times_int @ C @ B ) ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3523_mod__mult__self1,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mult_self1
thf(fact_3524_dvd__imp__mod__0,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( modulo_modulo_nat @ B @ A )
        = zero_zero_nat ) ) ).

% dvd_imp_mod_0
thf(fact_3525_dvd__imp__mod__0,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( modulo_modulo_int @ B @ A )
        = zero_zero_int ) ) ).

% dvd_imp_mod_0
thf(fact_3526_dvd__imp__mod__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( modulo364778990260209775nteger @ B @ A )
        = zero_z3403309356797280102nteger ) ) ).

% dvd_imp_mod_0
thf(fact_3527_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_3528_set__decode__zero,axiom,
    ( ( nat_set_decode @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% set_decode_zero
thf(fact_3529_set__encode__inverse,axiom,
    ! [A2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( nat_set_decode @ ( nat_set_encode @ A2 ) )
        = A2 ) ) ).

% set_encode_inverse
thf(fact_3530_unit__div__mult__self,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3531_unit__div__mult__self,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3532_unit__div__mult__self,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ A ) @ A )
        = B ) ) ).

% unit_div_mult_self
thf(fact_3533_unit__mult__div__div,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( times_times_nat @ B @ ( divide_divide_nat @ one_one_nat @ A ) )
        = ( divide_divide_nat @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3534_unit__mult__div__div,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( times_times_int @ B @ ( divide_divide_int @ one_one_int @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3535_unit__mult__div__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ one_one_Code_integer @ A ) )
        = ( divide6298287555418463151nteger @ B @ A ) ) ) ).

% unit_mult_div_div
thf(fact_3536_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_3537_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_3538_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_3539_pow__divides__pow__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% pow_divides_pow_iff
thf(fact_3540_Suc__mod__mult__self4,axiom,
    ! [N2: nat,K: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ N2 @ K ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self4
thf(fact_3541_Suc__mod__mult__self3,axiom,
    ! [K: nat,N2: nat,M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ ( times_times_nat @ K @ N2 ) @ M ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self3
thf(fact_3542_Suc__mod__mult__self2,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ N2 @ K ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self2
thf(fact_3543_Suc__mod__mult__self1,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( plus_plus_nat @ M @ ( times_times_nat @ K @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_mod_mult_self1
thf(fact_3544_even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3545_even__add,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3546_even__add,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_add
thf(fact_3547_odd__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3548_odd__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3549_odd__add,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) )
      = ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
       != ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ) ).

% odd_add
thf(fact_3550_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_3551_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_3552_mod2__Suc__Suc,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% mod2_Suc_Suc
thf(fact_3553_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_3554_set__decode__0,axiom,
    ! [X3: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X3 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X3 ) ) ) ).

% set_decode_0
thf(fact_3555_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_3556_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_3557_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_3558_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_3559_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_3560_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_3561_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_3562_power__less__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq
thf(fact_3563_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_3564_even__plus__one__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_plus_one_iff
thf(fact_3565_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_3566_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_3567_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_3568_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_3569_not__mod__2__eq__1__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != one_one_Code_integer )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% not_mod_2_eq_1_eq_0
thf(fact_3570_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_3571_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_3572_not__mod__2__eq__0__eq__1,axiom,
    ! [A: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
       != zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% not_mod_2_eq_0_eq_1
thf(fact_3573_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_3574_add__self__mod__2,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ M @ M ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% add_self_mod_2
thf(fact_3575_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_3576_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) )
      = ( ( ( numeral_numeral_nat @ W2 )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_3577_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_3578_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_3579_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_3580_even__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_two
thf(fact_3581_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_3582_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_3583_odd__succ__div__two,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% odd_succ_div_two
thf(fact_3584_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_3585_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_3586_even__succ__div__2,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% even_succ_div_2
thf(fact_3587_even__power,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% even_power
thf(fact_3588_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_3589_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_3590_mod2__gr__0,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% mod2_gr_0
thf(fact_3591_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_3592_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_3593_odd__two__times__div__two__succ,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ one_one_Code_integer )
        = A ) ) ).

% odd_two_times_div_two_succ
thf(fact_3594_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_3595_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W2 ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W2 ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W2 ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_3596_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_3597_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_3598_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_3599_even__succ__mod__exp,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
          = ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ) ).

% even_succ_mod_exp
thf(fact_3600_real__arch__pow,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ? [N3: nat] : ( ord_less_real @ Y3 @ ( power_power_real @ X3 @ N3 ) ) ) ).

% real_arch_pow
thf(fact_3601_real__arch__pow__inv,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X3 @ N3 ) @ Y3 ) ) ) ).

% real_arch_pow_inv
thf(fact_3602_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_3603_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_3604_zdvd__period,axiom,
    ! [A: int,D: int,X3: int,T: int,C: int] :
      ( ( dvd_dvd_int @ A @ D )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ X3 @ T ) )
        = ( dvd_dvd_int @ A @ ( plus_plus_int @ ( plus_plus_int @ X3 @ ( times_times_int @ C @ D ) ) @ T ) ) ) ) ).

% zdvd_period
thf(fact_3605_zdvd__reduce,axiom,
    ! [K: int,N2: int,M: int] :
      ( ( dvd_dvd_int @ K @ ( plus_plus_int @ N2 @ ( times_times_int @ K @ M ) ) )
      = ( dvd_dvd_int @ K @ N2 ) ) ).

% zdvd_reduce
thf(fact_3606_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y: real] :
          ( ( ord_less_real @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_real_def
thf(fact_3607_mod__eq__0__iff__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
      = ( dvd_dvd_nat @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_3608_mod__eq__0__iff__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
      = ( dvd_dvd_int @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_3609_mod__eq__0__iff__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_eq_0_iff_dvd
thf(fact_3610_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_3611_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_3612_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A3: code_integer,B3: code_integer] :
          ( ( modulo364778990260209775nteger @ B3 @ A3 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3613_mod__0__imp__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_3614_mod__0__imp__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_3615_mod__0__imp__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% mod_0_imp_dvd
thf(fact_3616_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_3617_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_3618_dvd__refl,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).

% dvd_refl
thf(fact_3619_dvd__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_trans
thf(fact_3620_dvd__trans,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ C )
       => ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_trans
thf(fact_3621_dvd__trans,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ C )
       => ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_trans
thf(fact_3622_dvd__mod__iff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
        = ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_3623_dvd__mod__iff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
        = ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_3624_dvd__mod__iff,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
        = ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_iff
thf(fact_3625_dvd__mod__imp__dvd,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( dvd_dvd_nat @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_3626_dvd__mod__imp__dvd,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ ( modulo_modulo_int @ A @ B ) )
     => ( ( dvd_dvd_int @ C @ B )
       => ( dvd_dvd_int @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_3627_dvd__mod__imp__dvd,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ ( modulo364778990260209775nteger @ A @ B ) )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( dvd_dvd_Code_integer @ C @ A ) ) ) ).

% dvd_mod_imp_dvd
thf(fact_3628_int__ge__induct,axiom,
    ! [K: int,I: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ K @ I3 )
             => ( ( P @ I3 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_ge_induct
thf(fact_3629_unit__imp__mod__eq__0,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( modulo_modulo_nat @ A @ B )
        = zero_zero_nat ) ) ).

% unit_imp_mod_eq_0
thf(fact_3630_unit__imp__mod__eq__0,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( modulo_modulo_int @ A @ B )
        = zero_zero_int ) ) ).

% unit_imp_mod_eq_0
thf(fact_3631_unit__imp__mod__eq__0,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( modulo364778990260209775nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% unit_imp_mod_eq_0
thf(fact_3632_mod__greater__zero__iff__not__dvd,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% mod_greater_zero_iff_not_dvd
thf(fact_3633_mod__add__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3634_mod__add__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3635_mod__add__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_right_eq
thf(fact_3636_mod__add__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3637_mod__add__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3638_mod__add__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_left_eq
thf(fact_3639_mod__add__cong,axiom,
    ! [A: nat,C: nat,A6: nat,B: nat,B6: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A6 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B6 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3640_mod__add__cong,axiom,
    ! [A: int,C: int,A6: int,B: int,B6: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A6 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B6 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3641_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A6: code_integer,B: code_integer,B6: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A6 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B6 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A6 @ B6 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_3642_mod__add__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_3643_mod__add__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_3644_mod__add__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C ) ) ).

% mod_add_eq
thf(fact_3645_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_3646_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_3647_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_3648_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_3649_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_3650_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_3651_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A3: complex,B3: complex] :
          ( ( A3 = zero_zero_complex )
         => ( B3 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_3652_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A3: real,B3: real] :
          ( ( A3 = zero_zero_real )
         => ( B3 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_3653_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A3: rat,B3: rat] :
          ( ( A3 = zero_zero_rat )
         => ( B3 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_3654_mod__Suc__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ ( suc @ M ) ) @ N2 ) ) ).

% mod_Suc_Suc_eq
thf(fact_3655_mod__Suc__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) @ N2 )
      = ( modulo_modulo_nat @ ( suc @ M ) @ N2 ) ) ).

% mod_Suc_eq
thf(fact_3656_dvd__productE,axiom,
    ! [P4: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ P4 @ ( times_times_nat @ A @ B ) )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( P4
              = ( times_times_nat @ X4 @ Y5 ) )
           => ( ( dvd_dvd_nat @ X4 @ A )
             => ~ ( dvd_dvd_nat @ Y5 @ B ) ) ) ) ).

% dvd_productE
thf(fact_3657_dvd__productE,axiom,
    ! [P4: int,A: int,B: int] :
      ( ( dvd_dvd_int @ P4 @ ( times_times_int @ A @ B ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( P4
              = ( times_times_int @ X4 @ Y5 ) )
           => ( ( dvd_dvd_int @ X4 @ A )
             => ~ ( dvd_dvd_int @ Y5 @ B ) ) ) ) ).

% dvd_productE
thf(fact_3658_division__decomp,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
     => ? [B8: nat,C5: nat] :
          ( ( A
            = ( times_times_nat @ B8 @ C5 ) )
          & ( dvd_dvd_nat @ B8 @ B )
          & ( dvd_dvd_nat @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_3659_division__decomp,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
     => ? [B8: int,C5: int] :
          ( ( A
            = ( times_times_int @ B8 @ C5 ) )
          & ( dvd_dvd_int @ B8 @ B )
          & ( dvd_dvd_int @ C5 @ C ) ) ) ).

% division_decomp
thf(fact_3660_dvdE,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ~ ! [K2: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3661_dvdE,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ~ ! [K2: complex] :
            ( A
           != ( times_times_complex @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3662_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K2: real] :
            ( A
           != ( times_times_real @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3663_dvdE,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ~ ! [K2: rat] :
            ( A
           != ( times_times_rat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3664_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3665_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B @ K2 ) ) ) ).

% dvdE
thf(fact_3666_dvdI,axiom,
    ! [A: code_integer,B: code_integer,K: code_integer] :
      ( ( A
        = ( times_3573771949741848930nteger @ B @ K ) )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% dvdI
thf(fact_3667_dvdI,axiom,
    ! [A: complex,B: complex,K: complex] :
      ( ( A
        = ( times_times_complex @ B @ K ) )
     => ( dvd_dvd_complex @ B @ A ) ) ).

% dvdI
thf(fact_3668_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_3669_dvdI,axiom,
    ! [A: rat,B: rat,K: rat] :
      ( ( A
        = ( times_times_rat @ B @ K ) )
     => ( dvd_dvd_rat @ B @ A ) ) ).

% dvdI
thf(fact_3670_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_3671_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_3672_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B3: code_integer,A3: code_integer] :
        ? [K3: code_integer] :
          ( A3
          = ( times_3573771949741848930nteger @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3673_dvd__def,axiom,
    ( dvd_dvd_complex
    = ( ^ [B3: complex,A3: complex] :
        ? [K3: complex] :
          ( A3
          = ( times_times_complex @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3674_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B3: real,A3: real] :
        ? [K3: real] :
          ( A3
          = ( times_times_real @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3675_dvd__def,axiom,
    ( dvd_dvd_rat
    = ( ^ [B3: rat,A3: rat] :
        ? [K3: rat] :
          ( A3
          = ( times_times_rat @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3676_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B3: nat,A3: nat] :
        ? [K3: nat] :
          ( A3
          = ( times_times_nat @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3677_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B3: int,A3: int] :
        ? [K3: int] :
          ( A3
          = ( times_times_int @ B3 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_3678_dvd__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3679_dvd__mult,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( dvd_dvd_complex @ A @ C )
     => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3680_dvd__mult,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3681_dvd__mult,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3682_dvd__mult,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3683_dvd__mult,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult
thf(fact_3684_dvd__mult2,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3685_dvd__mult2,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ A @ B )
     => ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3686_dvd__mult2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( dvd_dvd_real @ A @ ( times_times_real @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3687_dvd__mult2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3688_dvd__mult2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3689_dvd__mult2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) ) ) ).

% dvd_mult2
thf(fact_3690_dvd__mult__left,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3691_dvd__mult__left,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
     => ( dvd_dvd_complex @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3692_dvd__mult__left,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3693_dvd__mult__left,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3694_dvd__mult__left,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3695_dvd__mult__left,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ A @ C ) ) ).

% dvd_mult_left
thf(fact_3696_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3697_dvd__triv__left,axiom,
    ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3698_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3699_dvd__triv__left,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3700_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3701_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_3702_mult__dvd__mono,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ C @ D )
       => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3703_mult__dvd__mono,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] :
      ( ( dvd_dvd_complex @ A @ B )
     => ( ( dvd_dvd_complex @ C @ D )
       => ( dvd_dvd_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3704_mult__dvd__mono,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ C @ D )
       => ( dvd_dvd_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3705_mult__dvd__mono,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ C @ D )
       => ( dvd_dvd_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3706_mult__dvd__mono,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ C @ D )
       => ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3707_mult__dvd__mono,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ C @ D )
       => ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ).

% mult_dvd_mono
thf(fact_3708_dvd__mult__right,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
     => ( dvd_dvd_Code_integer @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3709_dvd__mult__right,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( dvd_dvd_complex @ ( times_times_complex @ A @ B ) @ C )
     => ( dvd_dvd_complex @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3710_dvd__mult__right,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ ( times_times_real @ A @ B ) @ C )
     => ( dvd_dvd_real @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3711_dvd__mult__right,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ ( times_times_rat @ A @ B ) @ C )
     => ( dvd_dvd_rat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3712_dvd__mult__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
     => ( dvd_dvd_nat @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3713_dvd__mult__right,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
     => ( dvd_dvd_int @ B @ C ) ) ).

% dvd_mult_right
thf(fact_3714_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3715_dvd__triv__right,axiom,
    ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3716_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3717_dvd__triv__right,axiom,
    ! [A: rat,B: rat] : ( dvd_dvd_rat @ A @ ( times_times_rat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3718_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3719_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_3720_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_3721_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_3722_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_3723_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_3724_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_3725_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_3726_unit__imp__dvd,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( dvd_dvd_Code_integer @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_3727_unit__imp__dvd,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_3728_unit__imp__dvd,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( dvd_dvd_int @ B @ A ) ) ).

% unit_imp_dvd
thf(fact_3729_dvd__unit__imp__unit,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( dvd_dvd_Code_integer @ A @ one_one_Code_integer ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3730_dvd__unit__imp__unit,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( dvd_dvd_nat @ A @ one_one_nat ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3731_dvd__unit__imp__unit,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( dvd_dvd_int @ A @ one_one_int ) ) ) ).

% dvd_unit_imp_unit
thf(fact_3732_dvd__add__right__iff,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3733_dvd__add__right__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3734_dvd__add__right__iff,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3735_dvd__add__right__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3736_dvd__add__right__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_add_right_iff
thf(fact_3737_dvd__add__left__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ C )
     => ( ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3738_dvd__add__left__iff,axiom,
    ! [A: real,C: real,B: real] :
      ( ( dvd_dvd_real @ A @ C )
     => ( ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) )
        = ( dvd_dvd_real @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3739_dvd__add__left__iff,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( dvd_dvd_rat @ A @ C )
     => ( ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) )
        = ( dvd_dvd_rat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3740_dvd__add__left__iff,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3741_dvd__add__left__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ C )
     => ( ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ B ) ) ) ).

% dvd_add_left_iff
thf(fact_3742_dvd__add,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ B )
     => ( ( dvd_dvd_Code_integer @ A @ C )
       => ( dvd_dvd_Code_integer @ A @ ( plus_p5714425477246183910nteger @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3743_dvd__add,axiom,
    ! [A: real,B: real,C: real] :
      ( ( dvd_dvd_real @ A @ B )
     => ( ( dvd_dvd_real @ A @ C )
       => ( dvd_dvd_real @ A @ ( plus_plus_real @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3744_dvd__add,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( dvd_dvd_rat @ A @ B )
     => ( ( dvd_dvd_rat @ A @ C )
       => ( dvd_dvd_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3745_dvd__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3746_dvd__add,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ A @ C )
       => ( dvd_dvd_int @ A @ ( plus_plus_int @ B @ C ) ) ) ) ).

% dvd_add
thf(fact_3747_dvd__div__eq__iff,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( ( divide_divide_nat @ A @ C )
            = ( divide_divide_nat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3748_dvd__div__eq__iff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( ( divide_divide_int @ A @ C )
            = ( divide_divide_int @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3749_dvd__div__eq__iff,axiom,
    ! [C: real,A: real,B: real] :
      ( ( dvd_dvd_real @ C @ A )
     => ( ( dvd_dvd_real @ C @ B )
       => ( ( ( divide_divide_real @ A @ C )
            = ( divide_divide_real @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3750_dvd__div__eq__iff,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( dvd_dvd_rat @ C @ A )
     => ( ( dvd_dvd_rat @ C @ B )
       => ( ( ( divide_divide_rat @ A @ C )
            = ( divide_divide_rat @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3751_dvd__div__eq__iff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( ( divide6298287555418463151nteger @ A @ C )
            = ( divide6298287555418463151nteger @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_3752_dvd__div__eq__cancel,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( divide_divide_nat @ A @ C )
        = ( divide_divide_nat @ B @ C ) )
     => ( ( dvd_dvd_nat @ C @ A )
       => ( ( dvd_dvd_nat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3753_dvd__div__eq__cancel,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( divide_divide_int @ A @ C )
        = ( divide_divide_int @ B @ C ) )
     => ( ( dvd_dvd_int @ C @ A )
       => ( ( dvd_dvd_int @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3754_dvd__div__eq__cancel,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ( divide_divide_real @ A @ C )
        = ( divide_divide_real @ B @ C ) )
     => ( ( dvd_dvd_real @ C @ A )
       => ( ( dvd_dvd_real @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3755_dvd__div__eq__cancel,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ C )
        = ( divide_divide_rat @ B @ C ) )
     => ( ( dvd_dvd_rat @ C @ A )
       => ( ( dvd_dvd_rat @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3756_dvd__div__eq__cancel,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ C )
        = ( divide6298287555418463151nteger @ B @ C ) )
     => ( ( dvd_dvd_Code_integer @ C @ A )
       => ( ( dvd_dvd_Code_integer @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_3757_div__div__div__same,axiom,
    ! [D: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ D @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ ( divide_divide_nat @ A @ D ) @ ( divide_divide_nat @ B @ D ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_3758_div__div__div__same,axiom,
    ! [D: int,B: int,A: int] :
      ( ( dvd_dvd_int @ D @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ ( divide_divide_int @ A @ D ) @ ( divide_divide_int @ B @ D ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_3759_div__div__div__same,axiom,
    ! [D: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ D ) @ ( divide6298287555418463151nteger @ B @ D ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% div_div_div_same
thf(fact_3760_mod__less__eq__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ M ) ).

% mod_less_eq_dividend
thf(fact_3761_gcd__nat_Oextremum__uniqueI,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_uniqueI
thf(fact_3762_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_3763_gcd__nat_Oextremum__unique,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% gcd_nat.extremum_unique
thf(fact_3764_gcd__nat_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ zero_zero_nat @ A )
        & ( zero_zero_nat != A ) ) ).

% gcd_nat.extremum_strict
thf(fact_3765_gcd__nat_Oextremum,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% gcd_nat.extremum
thf(fact_3766_zdvd__antisym__nonneg,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ N2 )
       => ( ( dvd_dvd_int @ M @ N2 )
         => ( ( dvd_dvd_int @ N2 @ M )
           => ( M = N2 ) ) ) ) ) ).

% zdvd_antisym_nonneg
thf(fact_3767_zdvd__mult__cancel,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) )
     => ( ( K != zero_zero_int )
       => ( dvd_dvd_int @ M @ N2 ) ) ) ).

% zdvd_mult_cancel
thf(fact_3768_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_3769_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_3770_even__iff__mod__2__eq__zero,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% even_iff_mod_2_eq_zero
thf(fact_3771_subset__decode__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ ( nat_set_decode @ M ) @ ( nat_set_decode @ N2 ) )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% subset_decode_imp_le
thf(fact_3772_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ord_le3102999989581377725nteger @ ( modulo364778990260209775nteger @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3773_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3774_unique__euclidean__semiring__numeral__class_Omod__less__eq__dividend,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ A ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
thf(fact_3775_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_nat @ ( modulo_modulo_nat @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3776_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3777_unique__euclidean__semiring__numeral__class_Opos__mod__bound,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_bound
thf(fact_3778_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: nat,B: nat] :
      ( ( ( modulo_modulo_nat @ A @ B )
        = A )
      = ( ( divide_divide_nat @ A @ B )
        = zero_zero_nat ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_3779_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = A )
      = ( ( divide_divide_int @ A @ B )
        = zero_zero_int ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_3780_mod__eq__self__iff__div__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = A )
      = ( ( divide6298287555418463151nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% mod_eq_self_iff_div_eq_0
thf(fact_3781_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D4: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D4 ) ) ) ) ).

% mod_eqE
thf(fact_3782_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D4: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D4 ) ) ) ) ).

% mod_eqE
thf(fact_3783_div__add1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3784_div__add1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( plus_plus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3785_div__add1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_add1_eq
thf(fact_3786_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_3787_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_3788_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_3789_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3790_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3791_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3792_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3793_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3794_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X2 @ Z3 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3795_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z3 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3796_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z3 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3797_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z3 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3798_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z3 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3799_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3800_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3801_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3802_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3803_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3804_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z3: code_integer] :
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z3 @ X2 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3805_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z3: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z3 @ X2 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3806_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z3: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z3 @ X2 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3807_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z3: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z3 @ X2 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3808_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z3: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z3 @ X2 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3809_dvd__div__eq__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( ( divide1717551699836669952omplex @ A @ B )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3810_dvd__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3811_dvd__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3812_dvd__div__eq__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( ( divide_divide_real @ A @ B )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3813_dvd__div__eq__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( ( divide_divide_rat @ A @ B )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3814_dvd__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% dvd_div_eq_0_iff
thf(fact_3815_unit__mult__right__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ B @ A )
          = ( times_3573771949741848930nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3816_unit__mult__right__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ B @ A )
          = ( times_times_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3817_unit__mult__right__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ B @ A )
          = ( times_times_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_mult_right_cancel
thf(fact_3818_unit__mult__left__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( times_3573771949741848930nteger @ A @ B )
          = ( times_3573771949741848930nteger @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3819_unit__mult__left__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( times_times_nat @ A @ B )
          = ( times_times_nat @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3820_unit__mult__left__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( times_times_int @ A @ B )
          = ( times_times_int @ A @ C ) )
        = ( B = C ) ) ) ).

% unit_mult_left_cancel
thf(fact_3821_mult__unit__dvd__iff_H,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3822_mult__unit__dvd__iff_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3823_mult__unit__dvd__iff_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ B @ C ) ) ) ).

% mult_unit_dvd_iff'
thf(fact_3824_dvd__mult__unit__iff_H,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3825_dvd__mult__unit__iff_H,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3826_dvd__mult__unit__iff_H,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ B @ C ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff'
thf(fact_3827_mult__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3828_mult__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3829_mult__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% mult_unit_dvd_iff
thf(fact_3830_dvd__mult__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3831_dvd__mult__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3832_dvd__mult__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_mult_unit_iff
thf(fact_3833_is__unit__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        & ( dvd_dvd_Code_integer @ B @ one_one_Code_integer ) ) ) ).

% is_unit_mult_iff
thf(fact_3834_is__unit__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ B ) @ one_one_nat )
      = ( ( dvd_dvd_nat @ A @ one_one_nat )
        & ( dvd_dvd_nat @ B @ one_one_nat ) ) ) ).

% is_unit_mult_iff
thf(fact_3835_is__unit__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ B ) @ one_one_int )
      = ( ( dvd_dvd_int @ A @ one_one_int )
        & ( dvd_dvd_int @ B @ one_one_int ) ) ) ).

% is_unit_mult_iff
thf(fact_3836_mod__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
          = N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = zero_zero_nat ) )
      & ( ( ( suc @ ( modulo_modulo_nat @ M @ N2 ) )
         != N2 )
       => ( ( modulo_modulo_nat @ ( suc @ M ) @ N2 )
          = ( suc @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ).

% mod_Suc
thf(fact_3837_finite__set__decode,axiom,
    ! [N2: nat] : ( finite_finite_nat @ ( nat_set_decode @ N2 ) ) ).

% finite_set_decode
thf(fact_3838_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P4: nat,M: nat] :
      ( ( P @ N2 )
     => ( ( ord_less_nat @ N2 @ P4 )
       => ( ( ord_less_nat @ M @ P4 )
         => ( ! [N3: nat] :
                ( ( ord_less_nat @ N3 @ P4 )
               => ( ( P @ N3 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P4 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_3839_dvd__div__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ ( divide_divide_nat @ B @ C ) @ A )
        = ( divide_divide_nat @ ( times_times_nat @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_3840_dvd__div__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ ( divide_divide_int @ B @ C ) @ A )
        = ( divide_divide_int @ ( times_times_int @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_3841_dvd__div__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ B @ C ) @ A )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ B @ A ) @ C ) ) ) ).

% dvd_div_mult
thf(fact_3842_div__mult__swap,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_3843_div__mult__swap,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_3844_div__mult__swap,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% div_mult_swap
thf(fact_3845_div__div__eq__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_3846_div__div__eq__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( times_times_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_3847_div__div__eq__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_div_eq_right
thf(fact_3848_dvd__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ B @ C ) @ A )
     => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_3849_dvd__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ B @ C ) @ A )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_3850_dvd__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ B @ C ) @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ).

% dvd_div_mult2_eq
thf(fact_3851_dvd__mult__imp__div,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B )
     => ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_3852_dvd__mult__imp__div,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B )
     => ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_3853_dvd__mult__imp__div,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B )
     => ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) ) ) ).

% dvd_mult_imp_div
thf(fact_3854_div__mult__div__if__dvd,axiom,
    ! [B: nat,A: nat,D: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( dvd_dvd_nat @ D @ C )
       => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ ( divide_divide_nat @ C @ D ) )
          = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_3855_div__mult__div__if__dvd,axiom,
    ! [B: int,A: int,D: int,C: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( dvd_dvd_int @ D @ C )
       => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ C @ D ) )
          = ( divide_divide_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_3856_div__mult__div__if__dvd,axiom,
    ! [B: code_integer,A: code_integer,D: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( dvd_dvd_Code_integer @ D @ C )
       => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ ( divide6298287555418463151nteger @ C @ D ) )
          = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ D ) ) ) ) ) ).

% div_mult_div_if_dvd
thf(fact_3857_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: 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 @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_3858_mod__less__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_less_divisor
thf(fact_3859_dvd__div__unit__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ C @ B ) )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3860_dvd__div__unit__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ C @ B ) )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3861_dvd__div__unit__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ C @ B ) )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% dvd_div_unit_iff
thf(fact_3862_div__unit__dvd__iff,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( dvd_dvd_nat @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3863_div__unit__dvd__iff,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( dvd_dvd_int @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3864_div__unit__dvd__iff,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( dvd_dvd_Code_integer @ A @ C ) ) ) ).

% div_unit_dvd_iff
thf(fact_3865_unit__div__cancel,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ( ( ( divide_divide_nat @ B @ A )
          = ( divide_divide_nat @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3866_unit__div__cancel,axiom,
    ! [A: int,B: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ( ( ( divide_divide_int @ B @ A )
          = ( divide_divide_int @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3867_unit__div__cancel,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ B @ A )
          = ( divide6298287555418463151nteger @ C @ A ) )
        = ( B = C ) ) ) ).

% unit_div_cancel
thf(fact_3868_div__plus__div__distrib__dvd__right,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3869_div__plus__div__distrib__dvd__right,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3870_div__plus__div__distrib__dvd__right,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_right
thf(fact_3871_div__plus__div__distrib__dvd__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ A )
     => ( ( divide_divide_nat @ ( plus_plus_nat @ A @ B ) @ C )
        = ( plus_plus_nat @ ( divide_divide_nat @ A @ C ) @ ( divide_divide_nat @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3872_div__plus__div__distrib__dvd__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ A @ B ) @ C )
        = ( plus_plus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3873_div__plus__div__distrib__dvd__left,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
        = ( plus_p5714425477246183910nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ).

% div_plus_div_distrib_dvd_left
thf(fact_3874_mod__Suc__le__divisor,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ ( suc @ N2 ) ) @ N2 ) ).

% mod_Suc_le_divisor
thf(fact_3875_mod__eq__0D,axiom,
    ! [M: nat,D: nat] :
      ( ( ( modulo_modulo_nat @ M @ D )
        = zero_zero_nat )
     => ? [Q2: nat] :
          ( M
          = ( times_times_nat @ D @ Q2 ) ) ) ).

% mod_eq_0D
thf(fact_3876_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3877_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3878_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3879_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3880_le__imp__power__dvd,axiom,
    ! [M: nat,N2: nat,A: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% le_imp_power_dvd
thf(fact_3881_power__le__dvd,axiom,
    ! [A: code_integer,N2: nat,B: code_integer,M: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3882_power__le__dvd,axiom,
    ! [A: nat,N2: nat,B: nat,M: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ ( power_power_nat @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3883_power__le__dvd,axiom,
    ! [A: real,N2: nat,B: real,M: nat] :
      ( ( dvd_dvd_real @ ( power_power_real @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_real @ ( power_power_real @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3884_power__le__dvd,axiom,
    ! [A: int,N2: nat,B: int,M: nat] :
      ( ( dvd_dvd_int @ ( power_power_int @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_int @ ( power_power_int @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3885_power__le__dvd,axiom,
    ! [A: complex,N2: nat,B: complex,M: nat] :
      ( ( dvd_dvd_complex @ ( power_power_complex @ A @ N2 ) @ B )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_complex @ ( power_power_complex @ A @ M ) @ B ) ) ) ).

% power_le_dvd
thf(fact_3886_dvd__power__le,axiom,
    ! [X3: code_integer,Y3: code_integer,N2: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ N2 ) @ ( power_8256067586552552935nteger @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3887_dvd__power__le,axiom,
    ! [X3: nat,Y3: nat,N2: nat,M: nat] :
      ( ( dvd_dvd_nat @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X3 @ N2 ) @ ( power_power_nat @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3888_dvd__power__le,axiom,
    ! [X3: real,Y3: real,N2: nat,M: nat] :
      ( ( dvd_dvd_real @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X3 @ N2 ) @ ( power_power_real @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3889_dvd__power__le,axiom,
    ! [X3: int,Y3: int,N2: nat,M: nat] :
      ( ( dvd_dvd_int @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X3 @ N2 ) @ ( power_power_int @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3890_dvd__power__le,axiom,
    ! [X3: complex,Y3: complex,N2: nat,M: nat] :
      ( ( dvd_dvd_complex @ X3 @ Y3 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X3 @ N2 ) @ ( power_power_complex @ Y3 @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3891_nat__mod__eq__iff,axiom,
    ! [X3: nat,N2: nat,Y3: nat] :
      ( ( ( modulo_modulo_nat @ X3 @ N2 )
        = ( modulo_modulo_nat @ Y3 @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X3 @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_3892_nat__dvd__not__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ).

% nat_dvd_not_less
thf(fact_3893_dvd__pos__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
       => ( ord_less_nat @ zero_zero_nat @ M ) ) ) ).

% dvd_pos_nat
thf(fact_3894_zdvd__imp__le,axiom,
    ! [Z2: int,N2: int] :
      ( ( dvd_dvd_int @ Z2 @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ Z2 @ N2 ) ) ) ).

% zdvd_imp_le
thf(fact_3895_zdvd__not__zless,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_int @ M @ N2 )
       => ~ ( dvd_dvd_int @ N2 @ M ) ) ) ).

% zdvd_not_zless
thf(fact_3896_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X4: nat,Y5: nat] :
      ( ( dvd_dvd_nat @ D4 @ A )
      & ( dvd_dvd_nat @ D4 @ B )
      & ( ( ( times_times_nat @ A @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D4 ) )
        | ( ( times_times_nat @ B @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D4 ) ) ) ) ).

% bezout_add_nat
thf(fact_3897_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X3 )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D ) )
            | ( ( times_times_nat @ B @ X3 )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) )
         => ? [X4: nat,Y5: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X4 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y5 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X4 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y5 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_3898_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_3899_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_3900_parity__cases,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
         != zero_z3403309356797280102nteger ) )
     => ~ ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
           != one_one_Code_integer ) ) ) ).

% parity_cases
thf(fact_3901_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_3902_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_3903_mod2__eq__if,axiom,
    ! [A: code_integer] :
      ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = zero_z3403309356797280102nteger ) )
      & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       => ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          = one_one_Code_integer ) ) ) ).

% mod2_eq_if
thf(fact_3904_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ A @ B )
       => ( ( modulo364778990260209775nteger @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3905_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ A @ B )
       => ( ( modulo_modulo_nat @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3906_unique__euclidean__semiring__numeral__class_Omod__less,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ A @ B )
       => ( ( modulo_modulo_int @ A @ B )
          = A ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_less
thf(fact_3907_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3908_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3909_unique__euclidean__semiring__numeral__class_Opos__mod__sign,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% unique_euclidean_semiring_numeral_class.pos_mod_sign
thf(fact_3910_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_3911_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_3912_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_3913_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_3914_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_3915_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_3916_mult__div__mod__eq,axiom,
    ! [B: nat,A: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3917_mult__div__mod__eq,axiom,
    ! [B: int,A: int] :
      ( ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3918_mult__div__mod__eq,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% mult_div_mod_eq
thf(fact_3919_mod__mult__div__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3920_mod__mult__div__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3921_mod__mult__div__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = A ) ).

% mod_mult_div_eq
thf(fact_3922_mod__div__mult__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( modulo_modulo_nat @ A @ B ) @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3923_mod__div__mult__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( modulo_modulo_int @ A @ B ) @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3924_mod__div__mult__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ B ) @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = A ) ).

% mod_div_mult_eq
thf(fact_3925_div__mult__mod__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3926_div__mult__mod__eq,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3927_div__mult__mod__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) )
      = A ) ).

% div_mult_mod_eq
thf(fact_3928_mod__div__decomp,axiom,
    ! [A: nat,B: nat] :
      ( A
      = ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_3929_mod__div__decomp,axiom,
    ! [A: int,B: int] :
      ( A
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_3930_mod__div__decomp,axiom,
    ! [A: code_integer,B: code_integer] :
      ( A
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_div_decomp
thf(fact_3931_cancel__div__mod__rules_I1_J,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3932_cancel__div__mod__rules_I1_J,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3933_cancel__div__mod__rules_I1_J,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(1)
thf(fact_3934_cancel__div__mod__rules_I2_J,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( plus_plus_nat @ ( plus_plus_nat @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) @ ( modulo_modulo_nat @ A @ B ) ) @ C )
      = ( plus_plus_nat @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3935_cancel__div__mod__rules_I2_J,axiom,
    ! [B: int,A: int,C: int] :
      ( ( plus_plus_int @ ( plus_plus_int @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) @ ( modulo_modulo_int @ A @ B ) ) @ C )
      = ( plus_plus_int @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3936_cancel__div__mod__rules_I2_J,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) @ C )
      = ( plus_p5714425477246183910nteger @ A @ C ) ) ).

% cancel_div_mod_rules(2)
thf(fact_3937_div__mult1__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C )
      = ( plus_plus_nat @ ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) ) @ ( divide_divide_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3938_div__mult1__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( divide_divide_int @ ( times_times_int @ A @ B ) @ C )
      = ( plus_plus_int @ ( times_times_int @ A @ ( divide_divide_int @ B @ C ) ) @ ( divide_divide_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3939_div__mult1__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C ) ) ) ).

% div_mult1_eq
thf(fact_3940_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C2: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_3941_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C2: nat] :
              ( B
             != ( times_times_nat @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_3942_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C2: int] :
              ( B
             != ( times_times_int @ A @ C2 ) ) ) ) ).

% unit_dvdE
thf(fact_3943_unity__coeff__ex,axiom,
    ! [P: code_integer > $o,L: code_integer] :
      ( ( ? [X: code_integer] : ( P @ ( times_3573771949741848930nteger @ L @ X ) ) )
      = ( ? [X: code_integer] :
            ( ( dvd_dvd_Code_integer @ L @ ( plus_p5714425477246183910nteger @ X @ zero_z3403309356797280102nteger ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3944_unity__coeff__ex,axiom,
    ! [P: complex > $o,L: complex] :
      ( ( ? [X: complex] : ( P @ ( times_times_complex @ L @ X ) ) )
      = ( ? [X: complex] :
            ( ( dvd_dvd_complex @ L @ ( plus_plus_complex @ X @ zero_zero_complex ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3945_unity__coeff__ex,axiom,
    ! [P: real > $o,L: real] :
      ( ( ? [X: real] : ( P @ ( times_times_real @ L @ X ) ) )
      = ( ? [X: real] :
            ( ( dvd_dvd_real @ L @ ( plus_plus_real @ X @ zero_zero_real ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3946_unity__coeff__ex,axiom,
    ! [P: rat > $o,L: rat] :
      ( ( ? [X: rat] : ( P @ ( times_times_rat @ L @ X ) ) )
      = ( ? [X: rat] :
            ( ( dvd_dvd_rat @ L @ ( plus_plus_rat @ X @ zero_zero_rat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3947_unity__coeff__ex,axiom,
    ! [P: nat > $o,L: nat] :
      ( ( ? [X: nat] : ( P @ ( times_times_nat @ L @ X ) ) )
      = ( ? [X: nat] :
            ( ( dvd_dvd_nat @ L @ ( plus_plus_nat @ X @ zero_zero_nat ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3948_unity__coeff__ex,axiom,
    ! [P: int > $o,L: int] :
      ( ( ? [X: int] : ( P @ ( times_times_int @ L @ X ) ) )
      = ( ? [X: int] :
            ( ( dvd_dvd_int @ L @ ( plus_plus_int @ X @ zero_zero_int ) )
            & ( P @ X ) ) ) ) ).

% unity_coeff_ex
thf(fact_3949_dvd__div__eq__mult,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( ( divide_divide_nat @ B @ A )
            = C )
          = ( B
            = ( times_times_nat @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_3950_dvd__div__eq__mult,axiom,
    ! [A: int,B: int,C: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ A @ B )
       => ( ( ( divide_divide_int @ B @ A )
            = C )
          = ( B
            = ( times_times_int @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_3951_dvd__div__eq__mult,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ A @ B )
       => ( ( ( divide6298287555418463151nteger @ B @ A )
            = C )
          = ( B
            = ( times_3573771949741848930nteger @ C @ A ) ) ) ) ) ).

% dvd_div_eq_mult
thf(fact_3952_div__dvd__iff__mult,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( B != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( dvd_dvd_nat @ ( divide_divide_nat @ A @ B ) @ C )
          = ( dvd_dvd_nat @ A @ ( times_times_nat @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_3953_div__dvd__iff__mult,axiom,
    ! [B: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( dvd_dvd_int @ ( divide_divide_int @ A @ B ) @ C )
          = ( dvd_dvd_int @ A @ ( times_times_int @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_3954_div__dvd__iff__mult,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( B != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( dvd_dvd_Code_integer @ ( divide6298287555418463151nteger @ A @ B ) @ C )
          = ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ C @ B ) ) ) ) ) ).

% div_dvd_iff_mult
thf(fact_3955_dvd__div__iff__mult,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( C != zero_zero_nat )
     => ( ( dvd_dvd_nat @ C @ B )
       => ( ( dvd_dvd_nat @ A @ ( divide_divide_nat @ B @ C ) )
          = ( dvd_dvd_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_3956_dvd__div__iff__mult,axiom,
    ! [C: int,B: int,A: int] :
      ( ( C != zero_zero_int )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( dvd_dvd_int @ A @ ( divide_divide_int @ B @ C ) )
          = ( dvd_dvd_int @ ( times_times_int @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_3957_dvd__div__iff__mult,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( C != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( dvd_dvd_Code_integer @ A @ ( divide6298287555418463151nteger @ B @ C ) )
          = ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ) ).

% dvd_div_iff_mult
thf(fact_3958_dvd__div__div__eq__mult,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( A != zero_zero_nat )
     => ( ( C != zero_zero_nat )
       => ( ( dvd_dvd_nat @ A @ B )
         => ( ( dvd_dvd_nat @ C @ D )
           => ( ( ( divide_divide_nat @ B @ A )
                = ( divide_divide_nat @ D @ C ) )
              = ( ( times_times_nat @ B @ C )
                = ( times_times_nat @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_3959_dvd__div__div__eq__mult,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( A != zero_zero_int )
     => ( ( C != zero_zero_int )
       => ( ( dvd_dvd_int @ A @ B )
         => ( ( dvd_dvd_int @ C @ D )
           => ( ( ( divide_divide_int @ B @ A )
                = ( divide_divide_int @ D @ C ) )
              = ( ( times_times_int @ B @ C )
                = ( times_times_int @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_3960_dvd__div__div__eq__mult,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( C != zero_z3403309356797280102nteger )
       => ( ( dvd_dvd_Code_integer @ A @ B )
         => ( ( dvd_dvd_Code_integer @ C @ D )
           => ( ( ( divide6298287555418463151nteger @ B @ A )
                = ( divide6298287555418463151nteger @ D @ C ) )
              = ( ( times_3573771949741848930nteger @ B @ C )
                = ( times_3573771949741848930nteger @ A @ D ) ) ) ) ) ) ) ).

% dvd_div_div_eq_mult
thf(fact_3961_unit__div__eq__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = zero_zero_nat )
        = ( A = zero_zero_nat ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3962_unit__div__eq__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3963_unit__div__eq__0__iff,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = zero_z3403309356797280102nteger )
        = ( A = zero_z3403309356797280102nteger ) ) ) ).

% unit_div_eq_0_iff
thf(fact_3964_is__unit__div__mult2__eq,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( dvd_dvd_nat @ C @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3965_is__unit__div__mult2__eq,axiom,
    ! [B: int,C: int,A: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( dvd_dvd_int @ C @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3966_is__unit__div__mult2__eq,axiom,
    ! [B: code_integer,C: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% is_unit_div_mult2_eq
thf(fact_3967_unit__div__mult__swap,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( times_times_nat @ A @ ( divide_divide_nat @ B @ C ) )
        = ( divide_divide_nat @ ( times_times_nat @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3968_unit__div__mult__swap,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( times_times_int @ A @ ( divide_divide_int @ B @ C ) )
        = ( divide_divide_int @ ( times_times_int @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3969_unit__div__mult__swap,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ A @ ( divide6298287555418463151nteger @ B @ C ) )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ) ).

% unit_div_mult_swap
thf(fact_3970_unit__div__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ C )
        = ( divide_divide_nat @ ( times_times_nat @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_3971_unit__div__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( times_times_int @ ( divide_divide_int @ A @ B ) @ C )
        = ( divide_divide_int @ ( times_times_int @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_3972_unit__div__commute,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C )
        = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ C ) @ B ) ) ) ).

% unit_div_commute
thf(fact_3973_div__mult__unit2,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ one_one_nat )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ C ) )
          = ( divide_divide_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3974_div__mult__unit2,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ one_one_int )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
          = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3975_div__mult__unit2,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ one_one_Code_integer )
     => ( ( dvd_dvd_Code_integer @ B @ A )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
          = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) ) ) ).

% div_mult_unit2
thf(fact_3976_unit__eq__div2,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( A
          = ( divide_divide_nat @ C @ B ) )
        = ( ( times_times_nat @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3977_unit__eq__div2,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( A
          = ( divide_divide_int @ C @ B ) )
        = ( ( times_times_int @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3978_unit__eq__div2,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( A
          = ( divide6298287555418463151nteger @ C @ B ) )
        = ( ( times_3573771949741848930nteger @ A @ B )
          = C ) ) ) ).

% unit_eq_div2
thf(fact_3979_unit__eq__div1,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( dvd_dvd_nat @ B @ one_one_nat )
     => ( ( ( divide_divide_nat @ A @ B )
          = C )
        = ( A
          = ( times_times_nat @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_3980_unit__eq__div1,axiom,
    ! [B: int,A: int,C: int] :
      ( ( dvd_dvd_int @ B @ one_one_int )
     => ( ( ( divide_divide_int @ A @ B )
          = C )
        = ( A
          = ( times_times_int @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_3981_unit__eq__div1,axiom,
    ! [B: code_integer,A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
     => ( ( ( divide6298287555418463151nteger @ A @ B )
          = C )
        = ( A
          = ( times_3573771949741848930nteger @ C @ B ) ) ) ) ).

% unit_eq_div1
thf(fact_3982_mod__le__divisor,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_eq_nat @ ( modulo_modulo_nat @ M @ N2 ) @ N2 ) ) ).

% mod_le_divisor
thf(fact_3983_is__unit__power__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) @ one_one_Code_integer )
      = ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
        | ( N2 = zero_zero_nat ) ) ) ).

% is_unit_power_iff
thf(fact_3984_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_3985_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_3986_div__less__mono,axiom,
    ! [A2: nat,B2: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A2 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B2 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_3987_mod__eq__nat1E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q4 )
        = ( modulo_modulo_nat @ N2 @ Q4 ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ~ ! [S2: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_3988_mod__eq__nat2E,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q4 )
        = ( modulo_modulo_nat @ N2 @ Q4 ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ~ ! [S2: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q4 @ S2 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_3989_nat__mod__eq__lemma,axiom,
    ! [X3: nat,N2: nat,Y3: nat] :
      ( ( ( modulo_modulo_nat @ X3 @ N2 )
        = ( modulo_modulo_nat @ Y3 @ N2 ) )
     => ( ( ord_less_eq_nat @ Y3 @ X3 )
       => ? [Q2: nat] :
            ( X3
            = ( plus_plus_nat @ Y3 @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_3990_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_3991_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_3992_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q4 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q4 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_3993_dvd__mult__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_mult_cancel
thf(fact_3994_nat__mult__dvd__cancel1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) )
        = ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% nat_mult_dvd_cancel1
thf(fact_3995_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D4: nat,X4: nat,Y5: nat] :
          ( ( dvd_dvd_nat @ D4 @ A )
          & ( dvd_dvd_nat @ D4 @ B )
          & ( ( times_times_nat @ A @ X4 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ D4 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_3996_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_3997_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_3998_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_3999_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_4000_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_4001_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B4: code_integer] :
              ( ( B4 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B4 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B4 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B4 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B4 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B4 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_4002_is__unit__div__mult__cancel__left,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ A @ B ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4003_is__unit__div__mult__cancel__left,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ A @ B ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4004_is__unit__div__mult__cancel__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ A @ B ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_left
thf(fact_4005_is__unit__div__mult__cancel__right,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ( dvd_dvd_nat @ B @ one_one_nat )
       => ( ( divide_divide_nat @ A @ ( times_times_nat @ B @ A ) )
          = ( divide_divide_nat @ one_one_nat @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4006_is__unit__div__mult__cancel__right,axiom,
    ! [A: int,B: int] :
      ( ( A != zero_zero_int )
     => ( ( dvd_dvd_int @ B @ one_one_int )
       => ( ( divide_divide_int @ A @ ( times_times_int @ B @ A ) )
          = ( divide_divide_int @ one_one_int @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4007_is__unit__div__mult__cancel__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ B @ one_one_Code_integer )
       => ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ B @ A ) )
          = ( divide6298287555418463151nteger @ one_one_Code_integer @ B ) ) ) ) ).

% is_unit_div_mult_cancel_right
thf(fact_4008_odd__even__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_4009_odd__even__add,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_4010_odd__even__add,axiom,
    ! [A: int,B: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B )
       => ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ) ).

% odd_even_add
thf(fact_4011_dvd__power__iff,axiom,
    ! [X3: code_integer,M: nat,N2: nat] :
      ( ( X3 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X3 @ M ) @ ( power_8256067586552552935nteger @ X3 @ N2 ) )
        = ( ( dvd_dvd_Code_integer @ X3 @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_4012_dvd__power__iff,axiom,
    ! [X3: nat,M: nat,N2: nat] :
      ( ( X3 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X3 @ M ) @ ( power_power_nat @ X3 @ N2 ) )
        = ( ( dvd_dvd_nat @ X3 @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_4013_dvd__power__iff,axiom,
    ! [X3: int,M: nat,N2: nat] :
      ( ( X3 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ N2 ) )
        = ( ( dvd_dvd_int @ X3 @ one_one_int )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_4014_dvd__power,axiom,
    ! [N2: nat,X3: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X3 = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X3 @ ( power_8256067586552552935nteger @ X3 @ N2 ) ) ) ).

% dvd_power
thf(fact_4015_dvd__power,axiom,
    ! [N2: nat,X3: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X3 = one_one_rat ) )
     => ( dvd_dvd_rat @ X3 @ ( power_power_rat @ X3 @ N2 ) ) ) ).

% dvd_power
thf(fact_4016_dvd__power,axiom,
    ! [N2: nat,X3: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X3 = one_one_nat ) )
     => ( dvd_dvd_nat @ X3 @ ( power_power_nat @ X3 @ N2 ) ) ) ).

% dvd_power
thf(fact_4017_dvd__power,axiom,
    ! [N2: nat,X3: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X3 = one_one_real ) )
     => ( dvd_dvd_real @ X3 @ ( power_power_real @ X3 @ N2 ) ) ) ).

% dvd_power
thf(fact_4018_dvd__power,axiom,
    ! [N2: nat,X3: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X3 = one_one_int ) )
     => ( dvd_dvd_int @ X3 @ ( power_power_int @ X3 @ N2 ) ) ) ).

% dvd_power
thf(fact_4019_dvd__power,axiom,
    ! [N2: nat,X3: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X3 = one_one_complex ) )
     => ( dvd_dvd_complex @ X3 @ ( power_power_complex @ X3 @ N2 ) ) ) ).

% dvd_power
thf(fact_4020_split__mod,axiom,
    ! [P: nat > $o,M: nat,N2: nat] :
      ( ( P @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ M ) )
        & ( ( N2 != zero_zero_nat )
         => ! [I2: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I2 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_4021_dvd__mult__cancel1,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ M @ N2 ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel1
thf(fact_4022_dvd__mult__cancel2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( dvd_dvd_nat @ ( times_times_nat @ N2 @ M ) @ M )
        = ( N2 = one_one_nat ) ) ) ).

% dvd_mult_cancel2
thf(fact_4023_power__dvd__imp__le,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I @ M ) @ ( power_power_nat @ I @ N2 ) )
     => ( ( ord_less_nat @ one_one_nat @ I )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_4024_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ C )
     => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ B @ C ) )
        = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ B @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ B ) @ C ) ) @ ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4025_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ C )
     => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ B @ C ) )
        = ( plus_plus_nat @ ( times_times_nat @ B @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ B ) @ C ) ) @ ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4026_unique__euclidean__semiring__numeral__class_Omod__mult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% unique_euclidean_semiring_numeral_class.mod_mult2_eq
thf(fact_4027_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_4028_product__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr4606735188037164562VEBT_o @ ( product_VEBT_VEBT_o @ Xs @ Ys ) @ N2 )
        = ( produc8721562602347293563VEBT_o @ ( nth_VEBT_VEBT @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4029_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_4030_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_4031_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys: list_VEBT_VEBT] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_s6755466524823107622T_VEBT @ Ys ) ) )
     => ( ( nth_Pr6777367263587873994T_VEBT @ ( product_o_VEBT_VEBT @ Xs @ Ys ) @ N2 )
        = ( produc2982872950893828659T_VEBT @ ( nth_o @ 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_4032_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Product_prod_o_o @ ( product_o_o @ Xs @ Ys ) @ N2 )
        = ( product_Pair_o_o @ ( nth_o @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4033_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys: list_nat] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_nat @ Ys ) ) )
     => ( ( nth_Pr5826913651314560976_o_nat @ ( product_o_nat @ Xs @ Ys ) @ N2 )
        = ( product_Pair_o_nat @ ( nth_o @ 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_4034_product__nth,axiom,
    ! [N2: nat,Xs: list_o,Ys: list_int] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_o @ Xs ) @ ( size_size_list_int @ Ys ) ) )
     => ( ( nth_Pr1649062631805364268_o_int @ ( product_o_int @ Xs @ Ys ) @ N2 )
        = ( product_Pair_o_int @ ( nth_o @ 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_4035_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_4036_product__nth,axiom,
    ! [N2: nat,Xs: list_nat,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_nat @ Xs ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr112076138515278198_nat_o @ ( product_nat_o @ Xs @ Ys ) @ N2 )
        = ( product_Pair_nat_o @ ( nth_nat @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) @ ( nth_o @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_o @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_4037_power__mono__odd,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_4038_power__mono__odd,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ A @ B )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_4039_power__mono__odd,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ A @ B )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_odd
thf(fact_4040_Suc__times__mod__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ M )
     => ( ( modulo_modulo_nat @ ( suc @ ( times_times_nat @ M @ N2 ) ) @ M )
        = one_one_nat ) ) ).

% Suc_times_mod_eq
thf(fact_4041_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_4042_dvd__power__iff__le,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) )
        = ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% dvd_power_iff_le
thf(fact_4043_even__unset__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4044_even__unset__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se4205575877204974255it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4045_even__unset__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se4203085406695923979it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( M = zero_zero_nat ) ) ) ).

% even_unset_bit_iff
thf(fact_4046_even__set__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4047_even__set__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se7882103937844011126it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4048_even__set__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se7879613467334960850it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        & ( M != zero_zero_nat ) ) ) ).

% even_set_bit_iff
thf(fact_4049_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_4050_divmod__digit__0_I2_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4051_divmod__digit__0_I2_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) )
          = ( modulo_modulo_int @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4052_divmod__digit__0_I2_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) )
          = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(2)
thf(fact_4053_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_4054_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_4055_bits__stable__imp__add__self,axiom,
    ! [A: code_integer] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( plus_p5714425477246183910nteger @ A @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = zero_z3403309356797280102nteger ) ) ).

% bits_stable_imp_add_self
thf(fact_4056_div__exp__mod__exp__eq,axiom,
    ! [A: nat,N2: nat,M: 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 ) ) @ M ) )
      = ( divide_divide_nat @ ( modulo_modulo_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_4057_div__exp__mod__exp__eq,axiom,
    ! [A: int,N2: nat,M: 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 ) ) @ M ) )
      = ( divide_divide_int @ ( modulo_modulo_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_4058_div__exp__mod__exp__eq,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
      = ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_plus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% div_exp_mod_exp_eq
thf(fact_4059_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B4: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B4 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_4060_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_4061_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_4062_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_4063_zero__le__even__power,axiom,
    ! [N2: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% zero_le_even_power
thf(fact_4064_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_4065_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_4066_zero__le__odd__power,axiom,
    ! [N2: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
        = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ).

% zero_le_odd_power
thf(fact_4067_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_4068_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_4069_zero__le__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq
thf(fact_4070_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_4071_verit__le__mono__div,axiom,
    ! [A2: nat,B2: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B2 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B2 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B2 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_4072_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_4073_divmod__digit__0_I1_J,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ B )
     => ( ( ord_less_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_nat @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4074_divmod__digit__0_I1_J,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide_divide_int @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4075_divmod__digit__0_I1_J,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
     => ( ( ord_le6747313008572928689nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
       => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
          = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% divmod_digit_0(1)
thf(fact_4076_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_4077_zero__less__power__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ N2 ) )
      = ( ( N2 = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq
thf(fact_4078_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_4079_mod__double__modulus,axiom,
    ! [M: code_integer,X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
       => ( ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X3 @ M ) )
          | ( ( modulo364778990260209775nteger @ X3 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X3 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4080_mod__double__modulus,axiom,
    ! [M: nat,X3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X3 )
       => ( ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X3 @ M ) )
          | ( ( modulo_modulo_nat @ X3 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X3 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4081_mod__double__modulus,axiom,
    ! [M: int,X3: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X3 )
       => ( ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X3 @ M ) )
          | ( ( modulo_modulo_int @ X3 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X3 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_4082_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_4083_unset__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se8260200283734997820nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se8260200283734997820nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% unset_bit_Suc
thf(fact_4084_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_4085_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_4086_set__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se2793503036327961859nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2793503036327961859nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% set_bit_Suc
thf(fact_4087_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_4088_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_4089_power__le__zero__eq,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ N2 )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq
thf(fact_4090_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_4091_divmod__digit__1_I1_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_Code_integer )
            = ( divide6298287555418463151nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_4092_divmod__digit__1_I1_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_nat )
            = ( divide_divide_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_4093_divmod__digit__1_I1_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) @ one_one_int )
            = ( divide_divide_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(1)
thf(fact_4094_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_4095_flip__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% flip_bit_Suc
thf(fact_4096_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_4097_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_4098_signed__take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% signed_take_bit_Suc
thf(fact_4099_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_4100_even__mult__exp__div__exp__iff,axiom,
    ! [A: nat,M: 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 ) ) @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_nat )
        | ( ( ord_less_eq_nat @ M @ 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 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4101_even__mult__exp__div__exp__iff,axiom,
    ! [A: int,M: 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 ) ) @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
          = zero_zero_int )
        | ( ( ord_less_eq_nat @ M @ 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 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4102_even__mult__exp__div__exp__iff,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% even_mult_exp_div_exp_iff
thf(fact_4103_even__flip__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1345352211410354436nteger @ M @ A ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4104_even__flip__bit__iff,axiom,
    ! [M: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se2161824704523386999it_nat @ M @ A ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4105_even__flip__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se2159334234014336723it_int @ M @ A ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
       != ( M = zero_zero_nat ) ) ) ).

% even_flip_bit_iff
thf(fact_4106_one__mod__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% one_mod_2_pow_eq
thf(fact_4107_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_4108_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_4109_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_4110_insert__absorb2,axiom,
    ! [X3: nat,A2: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ X3 @ A2 ) )
      = ( insert_nat @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_4111_insert__absorb2,axiom,
    ! [X3: int,A2: set_int] :
      ( ( insert_int @ X3 @ ( insert_int @ X3 @ A2 ) )
      = ( insert_int @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_4112_insert__absorb2,axiom,
    ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ X3 @ A2 ) )
      = ( insert8211810215607154385at_nat @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_4113_insert__absorb2,axiom,
    ! [X3: real,A2: set_real] :
      ( ( insert_real @ X3 @ ( insert_real @ X3 @ A2 ) )
      = ( insert_real @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_4114_insert__absorb2,axiom,
    ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ X3 @ A2 ) )
      = ( insert9069300056098147895at_nat @ X3 @ A2 ) ) ).

% insert_absorb2
thf(fact_4115_insert__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4116_insert__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4117_insert__iff,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
      = ( ( A = B )
        | ( member_real @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4118_insert__iff,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member_set_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4119_insert__iff,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
      = ( ( A = B )
        | ( member_nat @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4120_insert__iff,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
      = ( ( A = B )
        | ( member_int @ A @ A2 ) ) ) ).

% insert_iff
thf(fact_4121_insertCI,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ~ ( member8757157785044589968at_nat @ A @ B2 )
       => ( A = B ) )
     => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4122_insertCI,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ~ ( member8440522571783428010at_nat @ A @ B2 )
       => ( A = B ) )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4123_insertCI,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( ~ ( member_real @ A @ B2 )
       => ( A = B ) )
     => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4124_insertCI,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( ~ ( member_set_nat @ A @ B2 )
       => ( A = B ) )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4125_insertCI,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( ~ ( member_nat @ A @ B2 )
       => ( A = B ) )
     => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4126_insertCI,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( ~ ( member_int @ A @ B2 )
       => ( A = B ) )
     => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).

% insertCI
thf(fact_4127_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_4128_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_4129_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_4130_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_4131_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_4132_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_4133_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_4134_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_4135_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_4136_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_4137_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_4138_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_4139_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_4140_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_4141_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_4142_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_4143_cancel__comm__monoid__add__class_Odiff__cancel,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% cancel_comm_monoid_add_class.diff_cancel
thf(fact_4144_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_4145_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_4146_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_4147_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_4148_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_4149_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_4150_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_4151_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_4152_add__diff__cancel__left,axiom,
    ! [C: real,A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ C @ A ) @ ( plus_plus_real @ C @ B ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_4153_add__diff__cancel__left,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ C @ A ) @ ( plus_plus_rat @ C @ B ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_4154_add__diff__cancel__left,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ ( plus_plus_nat @ C @ B ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_4155_add__diff__cancel__left,axiom,
    ! [C: int,A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ C @ A ) @ ( plus_plus_int @ C @ B ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_left
thf(fact_4156_add__diff__cancel__left_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_4157_add__diff__cancel__left_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_4158_add__diff__cancel__left_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_4159_add__diff__cancel__left_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ A )
      = B ) ).

% add_diff_cancel_left'
thf(fact_4160_add__diff__cancel__right,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_4161_add__diff__cancel__right,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_4162_add__diff__cancel__right,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ C ) @ ( plus_plus_nat @ B @ C ) )
      = ( minus_minus_nat @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_4163_add__diff__cancel__right,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ A @ B ) ) ).

% add_diff_cancel_right
thf(fact_4164_add__diff__cancel__right_H,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_4165_add__diff__cancel__right_H,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_4166_add__diff__cancel__right_H,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_4167_add__diff__cancel__right_H,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel_right'
thf(fact_4168_singletonI,axiom,
    ! [A: produc3843707927480180839at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singletonI
thf(fact_4169_singletonI,axiom,
    ! [A: set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singletonI
thf(fact_4170_singletonI,axiom,
    ! [A: product_prod_nat_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singletonI
thf(fact_4171_singletonI,axiom,
    ! [A: real] : ( member_real @ A @ ( insert_real @ A @ bot_bot_set_real ) ) ).

% singletonI
thf(fact_4172_singletonI,axiom,
    ! [A: nat] : ( member_nat @ A @ ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singletonI
thf(fact_4173_singletonI,axiom,
    ! [A: int] : ( member_int @ A @ ( insert_int @ A @ bot_bot_set_int ) ) ).

% singletonI
thf(fact_4174_finite__insert,axiom,
    ! [A: real,A2: set_real] :
      ( ( finite_finite_real @ ( insert_real @ A @ A2 ) )
      = ( finite_finite_real @ A2 ) ) ).

% finite_insert
thf(fact_4175_finite__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) )
      = ( finite4343798906461161616at_nat @ A2 ) ) ).

% finite_insert
thf(fact_4176_finite__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( finite_finite_nat @ ( insert_nat @ A @ A2 ) )
      = ( finite_finite_nat @ A2 ) ) ).

% finite_insert
thf(fact_4177_finite__insert,axiom,
    ! [A: int,A2: set_int] :
      ( ( finite_finite_int @ ( insert_int @ A @ A2 ) )
      = ( finite_finite_int @ A2 ) ) ).

% finite_insert
thf(fact_4178_finite__insert,axiom,
    ! [A: complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) )
      = ( finite3207457112153483333omplex @ A2 ) ) ).

% finite_insert
thf(fact_4179_finite__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) )
      = ( finite6177210948735845034at_nat @ A2 ) ) ).

% finite_insert
thf(fact_4180_insert__subset,axiom,
    ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
      = ( ( member8757157785044589968at_nat @ X3 @ B2 )
        & ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4181_insert__subset,axiom,
    ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
      = ( ( member8440522571783428010at_nat @ X3 @ B2 )
        & ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4182_insert__subset,axiom,
    ! [X3: real,A2: set_real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
      = ( ( member_real @ X3 @ B2 )
        & ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4183_insert__subset,axiom,
    ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
      = ( ( member_set_nat @ X3 @ B2 )
        & ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4184_insert__subset,axiom,
    ! [X3: nat,A2: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
      = ( ( member_nat @ X3 @ B2 )
        & ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4185_insert__subset,axiom,
    ! [X3: int,A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
      = ( ( member_int @ X3 @ B2 )
        & ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).

% insert_subset
thf(fact_4186_diff__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_Suc_Suc
thf(fact_4187_Suc__diff__diff,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( suc @ K ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ N2 ) @ K ) ) ).

% Suc_diff_diff
thf(fact_4188_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_4189_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_4190_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_4191_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_4192_flip__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% flip_bit_negative_int_iff
thf(fact_4193_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_4194_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_4195_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_4196_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_4197_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_4198_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_4199_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_4200_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_4201_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = zero_zero_complex )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4202_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4203_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4204_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4205_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_4206_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_4207_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_4208_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_4209_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_4210_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = one_one_complex )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4211_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4212_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4213_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4214_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_4215_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_4216_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_4217_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_4218_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_4219_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_4220_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_4221_diff__ge__0__iff__ge,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_eq_real @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_4222_diff__ge__0__iff__ge,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_eq_rat @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_4223_diff__ge__0__iff__ge,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_eq_int @ B @ A ) ) ).

% diff_ge_0_iff_ge
thf(fact_4224_diff__gt__0__iff__gt,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ ( minus_minus_real @ A @ B ) )
      = ( ord_less_real @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_4225_diff__gt__0__iff__gt,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( minus_minus_rat @ A @ B ) )
      = ( ord_less_rat @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_4226_diff__gt__0__iff__gt,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ ( minus_minus_int @ A @ B ) )
      = ( ord_less_int @ B @ A ) ) ).

% diff_gt_0_iff_gt
thf(fact_4227_le__add__diff__inverse2,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_4228_le__add__diff__inverse2,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_4229_le__add__diff__inverse2,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_4230_le__add__diff__inverse2,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
        = A ) ) ).

% le_add_diff_inverse2
thf(fact_4231_le__add__diff__inverse,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_4232_le__add__diff__inverse,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_4233_le__add__diff__inverse,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_eq_nat @ B @ A )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_4234_le__add__diff__inverse,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% le_add_diff_inverse
thf(fact_4235_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_4236_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_4237_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_4238_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_4239_diff__add__zero,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( plus_plus_nat @ A @ B ) )
      = zero_zero_nat ) ).

% diff_add_zero
thf(fact_4240_div__diff,axiom,
    ! [C: int,A: int,B: int] :
      ( ( dvd_dvd_int @ C @ A )
     => ( ( dvd_dvd_int @ C @ B )
       => ( ( divide_divide_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( minus_minus_int @ ( divide_divide_int @ A @ C ) @ ( divide_divide_int @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_4241_div__diff,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ A )
     => ( ( dvd_dvd_Code_integer @ C @ B )
       => ( ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( minus_8373710615458151222nteger @ ( divide6298287555418463151nteger @ A @ C ) @ ( divide6298287555418463151nteger @ B @ C ) ) ) ) ) ).

% div_diff
thf(fact_4242_singleton__insert__inj__eq_H,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ A2 )
        = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
      = ( ( A = B )
        & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4243_singleton__insert__inj__eq_H,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ A2 )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4244_singleton__insert__inj__eq_H,axiom,
    ! [A: real,A2: set_real,B: real] :
      ( ( ( insert_real @ A @ A2 )
        = ( insert_real @ B @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4245_singleton__insert__inj__eq_H,axiom,
    ! [A: nat,A2: set_nat,B: nat] :
      ( ( ( insert_nat @ A @ A2 )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4246_singleton__insert__inj__eq_H,axiom,
    ! [A: int,A2: set_int,B: int] :
      ( ( ( insert_int @ A @ A2 )
        = ( insert_int @ B @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq'
thf(fact_4247_singleton__insert__inj__eq,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4248_singleton__insert__inj__eq,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4249_singleton__insert__inj__eq,axiom,
    ! [B: real,A: real,A2: set_real] :
      ( ( ( insert_real @ B @ bot_bot_set_real )
        = ( insert_real @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ bot_bot_set_real ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4250_singleton__insert__inj__eq,axiom,
    ! [B: nat,A: nat,A2: set_nat] :
      ( ( ( insert_nat @ B @ bot_bot_set_nat )
        = ( insert_nat @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ bot_bot_set_nat ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4251_singleton__insert__inj__eq,axiom,
    ! [B: int,A: int,A2: set_int] :
      ( ( ( insert_int @ B @ bot_bot_set_int )
        = ( insert_int @ A @ A2 ) )
      = ( ( A = B )
        & ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ bot_bot_set_int ) ) ) ) ).

% singleton_insert_inj_eq
thf(fact_4252_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_4253_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_4254_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_4255_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_4256_zero__less__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( minus_minus_nat @ N2 @ M ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% zero_less_diff
thf(fact_4257_diff__is__0__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% diff_is_0_eq
thf(fact_4258_diff__is__0__eq_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% diff_is_0_eq'
thf(fact_4259_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_4260_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_less_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_4261_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_4262_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_4263_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_4264_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_4265_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n2052037380579107095ol_rat @ ~ P )
      = ( minus_minus_rat @ one_one_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ) ).

% of_bool_not_iff
thf(fact_4266_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_4267_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_4268_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_4269_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_4270_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_4271_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_4272_mod__pos__pos__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ L )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_pos_pos_trivial
thf(fact_4273_mod__neg__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ K @ zero_zero_int )
     => ( ( ord_less_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = K ) ) ) ).

% mod_neg_neg_trivial
thf(fact_4274_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_4275_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_4276_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_4277_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_4278_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_4279_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_4280_set__replicate,axiom,
    ! [N2: nat,X3: produc3843707927480180839at_nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N2 @ X3 ) )
        = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% set_replicate
thf(fact_4281_set__replicate,axiom,
    ! [N2: nat,X3: vEBT_VEBT] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
        = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ).

% set_replicate
thf(fact_4282_set__replicate,axiom,
    ! [N2: nat,X3: product_prod_nat_nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ X3 ) )
        = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% set_replicate
thf(fact_4283_set__replicate,axiom,
    ! [N2: nat,X3: real] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_real2 @ ( replicate_real @ N2 @ X3 ) )
        = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ).

% set_replicate
thf(fact_4284_set__replicate,axiom,
    ! [N2: nat,X3: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_nat2 @ ( replicate_nat @ N2 @ X3 ) )
        = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% set_replicate
thf(fact_4285_set__replicate,axiom,
    ! [N2: nat,X3: int] :
      ( ( N2 != zero_zero_nat )
     => ( ( set_int2 @ ( replicate_int @ N2 @ X3 ) )
        = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% set_replicate
thf(fact_4286_even__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( plus_p5714425477246183910nteger @ A @ B ) ) ) ).

% even_diff
thf(fact_4287_even__diff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ A @ B ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ A @ B ) ) ) ).

% even_diff
thf(fact_4288_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_4289_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = zero_zero_nat ) ).

% of_bool_half_eq_0
thf(fact_4290_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide_divide_int @ ( zero_n2684676970156552555ol_int @ B ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = zero_zero_int ) ).

% of_bool_half_eq_0
thf(fact_4291_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_4292_even__diff__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) )
      = ( ( ord_less_nat @ M @ N2 )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ).

% even_diff_nat
thf(fact_4293_semiring__parity__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_parity_class.even_mask_iff
thf(fact_4294_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_4295_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_4296_one__div__2__pow__eq,axiom,
    ! [N2: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).

% one_div_2_pow_eq
thf(fact_4297_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_4298_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_4299_bits__1__div__exp,axiom,
    ! [N2: nat] :
      ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( N2 = zero_zero_nat ) ) ) ).

% bits_1_div_exp
thf(fact_4300_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_4301_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_4302_flip__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1345352211410354436nteger @ zero_zero_nat @ A )
      = ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ).

% flip_bit_0
thf(fact_4303_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_4304_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_4305_mod__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R2 ) ) ).

% mod_int_unique
thf(fact_4306_gcd__nat_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ~ ( ( dvd_dvd_nat @ B @ A )
          & ( B != A ) ) ) ).

% gcd_nat.asym
thf(fact_4307_gcd__nat_Orefl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% gcd_nat.refl
thf(fact_4308_gcd__nat_Otrans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( dvd_dvd_nat @ A @ C ) ) ) ).

% gcd_nat.trans
thf(fact_4309_gcd__nat_Oeq__iff,axiom,
    ( ( ^ [Y4: nat,Z: nat] : Y4 = Z )
    = ( ^ [A3: nat,B3: nat] :
          ( ( dvd_dvd_nat @ A3 @ B3 )
          & ( dvd_dvd_nat @ B3 @ A3 ) ) ) ) ).

% gcd_nat.eq_iff
thf(fact_4310_gcd__nat_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ( dvd_dvd_nat @ A @ A )
        & ( A != A ) ) ).

% gcd_nat.irrefl
thf(fact_4311_gcd__nat_Oantisym,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ B @ A )
       => ( A = B ) ) ) ).

% gcd_nat.antisym
thf(fact_4312_gcd__nat_Ostrict__trans,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( ( ( dvd_dvd_nat @ B @ C )
          & ( B != C ) )
       => ( ( dvd_dvd_nat @ A @ C )
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans
thf(fact_4313_gcd__nat_Ostrict__trans1,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( ( dvd_dvd_nat @ B @ C )
          & ( B != C ) )
       => ( ( dvd_dvd_nat @ A @ C )
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans1
thf(fact_4314_gcd__nat_Ostrict__trans2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( ( dvd_dvd_nat @ B @ C )
       => ( ( dvd_dvd_nat @ A @ C )
          & ( A != C ) ) ) ) ).

% gcd_nat.strict_trans2
thf(fact_4315_gcd__nat_Ostrict__iff__not,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% gcd_nat.strict_iff_not
thf(fact_4316_gcd__nat_Oorder__iff__strict,axiom,
    ( dvd_dvd_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( ( dvd_dvd_nat @ A3 @ B3 )
            & ( A3 != B3 ) )
          | ( A3 = B3 ) ) ) ) ).

% gcd_nat.order_iff_strict
thf(fact_4317_gcd__nat_Ostrict__iff__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) ) ) ).

% gcd_nat.strict_iff_order
thf(fact_4318_gcd__nat_Ostrict__implies__order,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( dvd_dvd_nat @ A @ B ) ) ).

% gcd_nat.strict_implies_order
thf(fact_4319_gcd__nat_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( A != B ) ) ).

% gcd_nat.strict_implies_not_eq
thf(fact_4320_gcd__nat_Onot__eq__order__implies__strict,axiom,
    ! [A: nat,B: nat] :
      ( ( A != B )
     => ( ( dvd_dvd_nat @ A @ B )
       => ( ( dvd_dvd_nat @ A @ B )
          & ( A != B ) ) ) ) ).

% gcd_nat.not_eq_order_implies_strict
thf(fact_4321_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_4322_dvd__diff__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% dvd_diff_nat
thf(fact_4323_eucl__rel__int,axiom,
    ! [K: int,L: int] : ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ ( divide_divide_int @ K @ L ) @ ( modulo_modulo_int @ K @ L ) ) ) ).

% eucl_rel_int
thf(fact_4324_mk__disjoint__insert,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A2 )
     => ? [B9: set_Pr4329608150637261639at_nat] :
          ( ( A2
            = ( insert9069300056098147895at_nat @ A @ B9 ) )
          & ~ ( member8757157785044589968at_nat @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_4325_mk__disjoint__insert,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ? [B9: set_Pr1261947904930325089at_nat] :
          ( ( A2
            = ( insert8211810215607154385at_nat @ A @ B9 ) )
          & ~ ( member8440522571783428010at_nat @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_4326_mk__disjoint__insert,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ? [B9: set_real] :
          ( ( A2
            = ( insert_real @ A @ B9 ) )
          & ~ ( member_real @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_4327_mk__disjoint__insert,axiom,
    ! [A: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ A2 )
     => ? [B9: set_set_nat] :
          ( ( A2
            = ( insert_set_nat @ A @ B9 ) )
          & ~ ( member_set_nat @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_4328_mk__disjoint__insert,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ? [B9: set_nat] :
          ( ( A2
            = ( insert_nat @ A @ B9 ) )
          & ~ ( member_nat @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_4329_mk__disjoint__insert,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ? [B9: set_int] :
          ( ( A2
            = ( insert_int @ A @ B9 ) )
          & ~ ( member_int @ A @ B9 ) ) ) ).

% mk_disjoint_insert
thf(fact_4330_insert__commute,axiom,
    ! [X3: nat,Y3: nat,A2: set_nat] :
      ( ( insert_nat @ X3 @ ( insert_nat @ Y3 @ A2 ) )
      = ( insert_nat @ Y3 @ ( insert_nat @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_4331_insert__commute,axiom,
    ! [X3: int,Y3: int,A2: set_int] :
      ( ( insert_int @ X3 @ ( insert_int @ Y3 @ A2 ) )
      = ( insert_int @ Y3 @ ( insert_int @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_4332_insert__commute,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ X3 @ ( insert8211810215607154385at_nat @ Y3 @ A2 ) )
      = ( insert8211810215607154385at_nat @ Y3 @ ( insert8211810215607154385at_nat @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_4333_insert__commute,axiom,
    ! [X3: real,Y3: real,A2: set_real] :
      ( ( insert_real @ X3 @ ( insert_real @ Y3 @ A2 ) )
      = ( insert_real @ Y3 @ ( insert_real @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_4334_insert__commute,axiom,
    ! [X3: produc3843707927480180839at_nat,Y3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ X3 @ ( insert9069300056098147895at_nat @ Y3 @ A2 ) )
      = ( insert9069300056098147895at_nat @ Y3 @ ( insert9069300056098147895at_nat @ X3 @ A2 ) ) ) ).

% insert_commute
thf(fact_4335_insert__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ A @ A2 )
     => ( ~ ( member8757157785044589968at_nat @ B @ B2 )
       => ( ( ( insert9069300056098147895at_nat @ A @ A2 )
            = ( insert9069300056098147895at_nat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_Pr4329608150637261639at_nat] :
                  ( ( A2
                    = ( insert9069300056098147895at_nat @ B @ C6 ) )
                  & ~ ( member8757157785044589968at_nat @ B @ C6 )
                  & ( B2
                    = ( insert9069300056098147895at_nat @ A @ C6 ) )
                  & ~ ( member8757157785044589968at_nat @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4336_insert__eq__iff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ A @ A2 )
     => ( ~ ( member8440522571783428010at_nat @ B @ B2 )
       => ( ( ( insert8211810215607154385at_nat @ A @ A2 )
            = ( insert8211810215607154385at_nat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_Pr1261947904930325089at_nat] :
                  ( ( A2
                    = ( insert8211810215607154385at_nat @ B @ C6 ) )
                  & ~ ( member8440522571783428010at_nat @ B @ C6 )
                  & ( B2
                    = ( insert8211810215607154385at_nat @ A @ C6 ) )
                  & ~ ( member8440522571783428010at_nat @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4337_insert__eq__iff,axiom,
    ! [A: real,A2: set_real,B: real,B2: set_real] :
      ( ~ ( member_real @ A @ A2 )
     => ( ~ ( member_real @ B @ B2 )
       => ( ( ( insert_real @ A @ A2 )
            = ( insert_real @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_real] :
                  ( ( A2
                    = ( insert_real @ B @ C6 ) )
                  & ~ ( member_real @ B @ C6 )
                  & ( B2
                    = ( insert_real @ A @ C6 ) )
                  & ~ ( member_real @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4338_insert__eq__iff,axiom,
    ! [A: set_nat,A2: set_set_nat,B: set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ A @ A2 )
     => ( ~ ( member_set_nat @ B @ B2 )
       => ( ( ( insert_set_nat @ A @ A2 )
            = ( insert_set_nat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_set_nat] :
                  ( ( A2
                    = ( insert_set_nat @ B @ C6 ) )
                  & ~ ( member_set_nat @ B @ C6 )
                  & ( B2
                    = ( insert_set_nat @ A @ C6 ) )
                  & ~ ( member_set_nat @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4339_insert__eq__iff,axiom,
    ! [A: nat,A2: set_nat,B: nat,B2: set_nat] :
      ( ~ ( member_nat @ A @ A2 )
     => ( ~ ( member_nat @ B @ B2 )
       => ( ( ( insert_nat @ A @ A2 )
            = ( insert_nat @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_nat] :
                  ( ( A2
                    = ( insert_nat @ B @ C6 ) )
                  & ~ ( member_nat @ B @ C6 )
                  & ( B2
                    = ( insert_nat @ A @ C6 ) )
                  & ~ ( member_nat @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4340_insert__eq__iff,axiom,
    ! [A: int,A2: set_int,B: int,B2: set_int] :
      ( ~ ( member_int @ A @ A2 )
     => ( ~ ( member_int @ B @ B2 )
       => ( ( ( insert_int @ A @ A2 )
            = ( insert_int @ B @ B2 ) )
          = ( ( ( A = B )
             => ( A2 = B2 ) )
            & ( ( A != B )
             => ? [C6: set_int] :
                  ( ( A2
                    = ( insert_int @ B @ C6 ) )
                  & ~ ( member_int @ B @ C6 )
                  & ( B2
                    = ( insert_int @ A @ C6 ) )
                  & ~ ( member_int @ A @ C6 ) ) ) ) ) ) ) ).

% insert_eq_iff
thf(fact_4341_insert__absorb,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A2 )
     => ( ( insert9069300056098147895at_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4342_insert__absorb,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( ( insert8211810215607154385at_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4343_insert__absorb,axiom,
    ! [A: real,A2: set_real] :
      ( ( member_real @ A @ A2 )
     => ( ( insert_real @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4344_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_4345_insert__absorb,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( ( insert_nat @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4346_insert__absorb,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( ( insert_int @ A @ A2 )
        = A2 ) ) ).

% insert_absorb
thf(fact_4347_insert__ident,axiom,
    ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
     => ( ~ ( member8757157785044589968at_nat @ X3 @ B2 )
       => ( ( ( insert9069300056098147895at_nat @ X3 @ A2 )
            = ( insert9069300056098147895at_nat @ X3 @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4348_insert__ident,axiom,
    ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
     => ( ~ ( member8440522571783428010at_nat @ X3 @ B2 )
       => ( ( ( insert8211810215607154385at_nat @ X3 @ A2 )
            = ( insert8211810215607154385at_nat @ X3 @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4349_insert__ident,axiom,
    ! [X3: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ X3 @ A2 )
     => ( ~ ( member_real @ X3 @ B2 )
       => ( ( ( insert_real @ X3 @ A2 )
            = ( insert_real @ X3 @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4350_insert__ident,axiom,
    ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X3 @ A2 )
     => ( ~ ( member_set_nat @ X3 @ B2 )
       => ( ( ( insert_set_nat @ X3 @ A2 )
            = ( insert_set_nat @ X3 @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4351_insert__ident,axiom,
    ! [X3: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X3 @ A2 )
     => ( ~ ( member_nat @ X3 @ B2 )
       => ( ( ( insert_nat @ X3 @ A2 )
            = ( insert_nat @ X3 @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4352_insert__ident,axiom,
    ! [X3: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ X3 @ A2 )
     => ( ~ ( member_int @ X3 @ B2 )
       => ( ( ( insert_int @ X3 @ A2 )
            = ( insert_int @ X3 @ B2 ) )
          = ( A2 = B2 ) ) ) ) ).

% insert_ident
thf(fact_4353_Set_Oset__insert,axiom,
    ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X3 @ A2 )
     => ~ ! [B9: set_Pr4329608150637261639at_nat] :
            ( ( A2
              = ( insert9069300056098147895at_nat @ X3 @ B9 ) )
           => ( member8757157785044589968at_nat @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_4354_Set_Oset__insert,axiom,
    ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ A2 )
     => ~ ! [B9: set_Pr1261947904930325089at_nat] :
            ( ( A2
              = ( insert8211810215607154385at_nat @ X3 @ B9 ) )
           => ( member8440522571783428010at_nat @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_4355_Set_Oset__insert,axiom,
    ! [X3: real,A2: set_real] :
      ( ( member_real @ X3 @ A2 )
     => ~ ! [B9: set_real] :
            ( ( A2
              = ( insert_real @ X3 @ B9 ) )
           => ( member_real @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_4356_Set_Oset__insert,axiom,
    ! [X3: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X3 @ A2 )
     => ~ ! [B9: set_set_nat] :
            ( ( A2
              = ( insert_set_nat @ X3 @ B9 ) )
           => ( member_set_nat @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_4357_Set_Oset__insert,axiom,
    ! [X3: nat,A2: set_nat] :
      ( ( member_nat @ X3 @ A2 )
     => ~ ! [B9: set_nat] :
            ( ( A2
              = ( insert_nat @ X3 @ B9 ) )
           => ( member_nat @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_4358_Set_Oset__insert,axiom,
    ! [X3: int,A2: set_int] :
      ( ( member_int @ X3 @ A2 )
     => ~ ! [B9: set_int] :
            ( ( A2
              = ( insert_int @ X3 @ B9 ) )
           => ( member_int @ X3 @ B9 ) ) ) ).

% Set.set_insert
thf(fact_4359_insertI2,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ A @ B2 )
     => ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4360_insertI2,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ A @ B2 )
     => ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4361_insertI2,axiom,
    ! [A: real,B2: set_real,B: real] :
      ( ( member_real @ A @ B2 )
     => ( member_real @ A @ ( insert_real @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4362_insertI2,axiom,
    ! [A: set_nat,B2: set_set_nat,B: set_nat] :
      ( ( member_set_nat @ A @ B2 )
     => ( member_set_nat @ A @ ( insert_set_nat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4363_insertI2,axiom,
    ! [A: nat,B2: set_nat,B: nat] :
      ( ( member_nat @ A @ B2 )
     => ( member_nat @ A @ ( insert_nat @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4364_insertI2,axiom,
    ! [A: int,B2: set_int,B: int] :
      ( ( member_int @ A @ B2 )
     => ( member_int @ A @ ( insert_int @ B @ B2 ) ) ) ).

% insertI2
thf(fact_4365_insertI1,axiom,
    ! [A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] : ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).

% insertI1
thf(fact_4366_insertI1,axiom,
    ! [A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] : ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).

% insertI1
thf(fact_4367_insertI1,axiom,
    ! [A: real,B2: set_real] : ( member_real @ A @ ( insert_real @ A @ B2 ) ) ).

% insertI1
thf(fact_4368_insertI1,axiom,
    ! [A: set_nat,B2: set_set_nat] : ( member_set_nat @ A @ ( insert_set_nat @ A @ B2 ) ) ).

% insertI1
thf(fact_4369_insertI1,axiom,
    ! [A: nat,B2: set_nat] : ( member_nat @ A @ ( insert_nat @ A @ B2 ) ) ).

% insertI1
thf(fact_4370_insertI1,axiom,
    ! [A: int,B2: set_int] : ( member_int @ A @ ( insert_int @ A @ B2 ) ) ).

% insertI1
thf(fact_4371_insertE,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ ( insert9069300056098147895at_nat @ B @ A2 ) )
     => ( ( A != B )
       => ( member8757157785044589968at_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_4372_insertE,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ ( insert8211810215607154385at_nat @ B @ A2 ) )
     => ( ( A != B )
       => ( member8440522571783428010at_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_4373_insertE,axiom,
    ! [A: real,B: real,A2: set_real] :
      ( ( member_real @ A @ ( insert_real @ B @ A2 ) )
     => ( ( A != B )
       => ( member_real @ A @ A2 ) ) ) ).

% insertE
thf(fact_4374_insertE,axiom,
    ! [A: set_nat,B: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ A @ ( insert_set_nat @ B @ A2 ) )
     => ( ( A != B )
       => ( member_set_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_4375_insertE,axiom,
    ! [A: nat,B: nat,A2: set_nat] :
      ( ( member_nat @ A @ ( insert_nat @ B @ A2 ) )
     => ( ( A != B )
       => ( member_nat @ A @ A2 ) ) ) ).

% insertE
thf(fact_4376_insertE,axiom,
    ! [A: int,B: int,A2: set_int] :
      ( ( member_int @ A @ ( insert_int @ B @ A2 ) )
     => ( ( A != B )
       => ( member_int @ A @ A2 ) ) ) ).

% insertE
thf(fact_4377_of__bool__eq__iff,axiom,
    ! [P4: $o,Q4: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P4 )
        = ( zero_n2687167440665602831ol_nat @ Q4 ) )
      = ( P4 = Q4 ) ) ).

% of_bool_eq_iff
thf(fact_4378_of__bool__eq__iff,axiom,
    ! [P4: $o,Q4: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P4 )
        = ( zero_n2684676970156552555ol_int @ Q4 ) )
      = ( P4 = Q4 ) ) ).

% of_bool_eq_iff
thf(fact_4379_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_4380_diff__right__commute,axiom,
    ! [A: real,C: real,B: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B )
      = ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_4381_diff__right__commute,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_4382_diff__right__commute,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ C ) @ B )
      = ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_4383_diff__right__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B )
      = ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% diff_right_commute
thf(fact_4384_diff__eq__diff__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_4385_diff__eq__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_4386_diff__eq__diff__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( A = B )
        = ( C = D ) ) ) ).

% diff_eq_diff_eq
thf(fact_4387_full__exhaustive__int_H_Ocases,axiom,
    ! [X3: produc2285326912895808259nt_int] :
      ~ ! [F2: produc8551481072490612790e_term > option6357759511663192854e_term,D4: int,I3: int] :
          ( X3
         != ( produc5700946648718959541nt_int @ F2 @ ( product_Pair_int_int @ D4 @ I3 ) ) ) ).

% full_exhaustive_int'.cases
thf(fact_4388_exhaustive__int_H_Ocases,axiom,
    ! [X3: produc7773217078559923341nt_int] :
      ~ ! [F2: int > option6357759511663192854e_term,D4: int,I3: int] :
          ( X3
         != ( produc4305682042979456191nt_int @ F2 @ ( product_Pair_int_int @ D4 @ I3 ) ) ) ).

% exhaustive_int'.cases
thf(fact_4389_small__lazy_H_Ocases,axiom,
    ! [X3: product_prod_int_int] :
      ~ ! [D4: int,I3: int] :
          ( X3
         != ( product_Pair_int_int @ D4 @ I3 ) ) ).

% small_lazy'.cases
thf(fact_4390_unique__quotient,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q6: int,R3: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
       => ( Q4 = Q6 ) ) ) ).

% unique_quotient
thf(fact_4391_unique__remainder,axiom,
    ! [A: int,B: int,Q4: int,R2: int,Q6: int,R3: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q6 @ R3 ) )
       => ( R2 = R3 ) ) ) ).

% unique_remainder
thf(fact_4392_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_4393_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_4394_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n2052037380579107095ol_rat
        @ ( P
          & Q ) )
      = ( times_times_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ ( zero_n2052037380579107095ol_rat @ Q ) ) ) ).

% of_bool_conj
thf(fact_4395_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_4396_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_4397_diff__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ D @ C )
       => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_4398_diff__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ D @ C )
       => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_4399_diff__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ D @ C )
       => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_mono
thf(fact_4400_diff__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_eq_real @ B @ A )
     => ( ord_less_eq_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_4401_diff__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_eq_rat @ B @ A )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_4402_diff__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_eq_int @ B @ A )
     => ( ord_less_eq_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_left_mono
thf(fact_4403_diff__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_4404_diff__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_4405_diff__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_right_mono
thf(fact_4406_diff__eq__diff__less__eq,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_eq_real @ A @ B )
        = ( ord_less_eq_real @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4407_diff__eq__diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_eq_rat @ A @ B )
        = ( ord_less_eq_rat @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4408_diff__eq__diff__less__eq,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_eq_int @ A @ B )
        = ( ord_less_eq_int @ C @ D ) ) ) ).

% diff_eq_diff_less_eq
thf(fact_4409_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: complex,Z: complex] : Y4 = Z )
    = ( ^ [A3: complex,B3: complex] :
          ( ( minus_minus_complex @ A3 @ B3 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_4410_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: real,Z: real] : Y4 = Z )
    = ( ^ [A3: real,B3: real] :
          ( ( minus_minus_real @ A3 @ B3 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_4411_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: rat,Z: rat] : Y4 = Z )
    = ( ^ [A3: rat,B3: rat] :
          ( ( minus_minus_rat @ A3 @ B3 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_4412_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y4: int,Z: int] : Y4 = Z )
    = ( ^ [A3: int,B3: int] :
          ( ( minus_minus_int @ A3 @ B3 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_4413_diff__strict__mono,axiom,
    ! [A: real,B: real,D: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ D @ C )
       => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_4414_diff__strict__mono,axiom,
    ! [A: rat,B: rat,D: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ D @ C )
       => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_4415_diff__strict__mono,axiom,
    ! [A: int,B: int,D: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ( ord_less_int @ D @ C )
       => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ D ) ) ) ) ).

% diff_strict_mono
thf(fact_4416_diff__eq__diff__less,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( minus_minus_real @ A @ B )
        = ( minus_minus_real @ C @ D ) )
     => ( ( ord_less_real @ A @ B )
        = ( ord_less_real @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_4417_diff__eq__diff__less,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = ( minus_minus_rat @ C @ D ) )
     => ( ( ord_less_rat @ A @ B )
        = ( ord_less_rat @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_4418_diff__eq__diff__less,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( minus_minus_int @ A @ B )
        = ( minus_minus_int @ C @ D ) )
     => ( ( ord_less_int @ A @ B )
        = ( ord_less_int @ C @ D ) ) ) ).

% diff_eq_diff_less
thf(fact_4419_diff__strict__left__mono,axiom,
    ! [B: real,A: real,C: real] :
      ( ( ord_less_real @ B @ A )
     => ( ord_less_real @ ( minus_minus_real @ C @ A ) @ ( minus_minus_real @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_4420_diff__strict__left__mono,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ord_less_rat @ ( minus_minus_rat @ C @ A ) @ ( minus_minus_rat @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_4421_diff__strict__left__mono,axiom,
    ! [B: int,A: int,C: int] :
      ( ( ord_less_int @ B @ A )
     => ( ord_less_int @ ( minus_minus_int @ C @ A ) @ ( minus_minus_int @ C @ B ) ) ) ).

% diff_strict_left_mono
thf(fact_4422_diff__strict__right__mono,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( minus_minus_real @ A @ C ) @ ( minus_minus_real @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_4423_diff__strict__right__mono,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( minus_minus_rat @ A @ C ) @ ( minus_minus_rat @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_4424_diff__strict__right__mono,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( minus_minus_int @ A @ C ) @ ( minus_minus_int @ B @ C ) ) ) ).

% diff_strict_right_mono
thf(fact_4425_right__diff__distrib_H,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_4426_right__diff__distrib_H,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_4427_right__diff__distrib_H,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_4428_right__diff__distrib_H,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( times_times_nat @ A @ ( minus_minus_nat @ B @ C ) )
      = ( minus_minus_nat @ ( times_times_nat @ A @ B ) @ ( times_times_nat @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_4429_right__diff__distrib_H,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib'
thf(fact_4430_left__diff__distrib_H,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ B @ C ) @ A )
      = ( minus_minus_complex @ ( times_times_complex @ B @ A ) @ ( times_times_complex @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_4431_left__diff__distrib_H,axiom,
    ! [B: real,C: real,A: real] :
      ( ( times_times_real @ ( minus_minus_real @ B @ C ) @ A )
      = ( minus_minus_real @ ( times_times_real @ B @ A ) @ ( times_times_real @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_4432_left__diff__distrib_H,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ B @ C ) @ A )
      = ( minus_minus_rat @ ( times_times_rat @ B @ A ) @ ( times_times_rat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_4433_left__diff__distrib_H,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ B @ C ) @ A )
      = ( minus_minus_nat @ ( times_times_nat @ B @ A ) @ ( times_times_nat @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_4434_left__diff__distrib_H,axiom,
    ! [B: int,C: int,A: int] :
      ( ( times_times_int @ ( minus_minus_int @ B @ C ) @ A )
      = ( minus_minus_int @ ( times_times_int @ B @ A ) @ ( times_times_int @ C @ A ) ) ) ).

% left_diff_distrib'
thf(fact_4435_right__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ A @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ B ) @ ( times_times_complex @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_4436_right__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ A @ B ) @ ( times_times_real @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_4437_right__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ B ) @ ( times_times_rat @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_4438_right__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ A @ B ) @ ( times_times_int @ A @ C ) ) ) ).

% right_diff_distrib
thf(fact_4439_left__diff__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( times_times_complex @ A @ C ) @ ( times_times_complex @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_4440_left__diff__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( times_times_real @ A @ C ) @ ( times_times_real @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_4441_left__diff__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( times_times_rat @ A @ C ) @ ( times_times_rat @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_4442_left__diff__distrib,axiom,
    ! [A: int,B: int,C: int] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) ) ) ).

% left_diff_distrib
thf(fact_4443_add__diff__add,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ ( plus_plus_real @ B @ D ) )
      = ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ ( minus_minus_real @ C @ D ) ) ) ).

% add_diff_add
thf(fact_4444_add__diff__add,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ ( plus_plus_rat @ B @ D ) )
      = ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ ( minus_minus_rat @ C @ D ) ) ) ).

% add_diff_add
thf(fact_4445_add__diff__add,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ ( plus_plus_int @ B @ D ) )
      = ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ ( minus_minus_int @ C @ D ) ) ) ).

% add_diff_add
thf(fact_4446_group__cancel_Osub1,axiom,
    ! [A2: real,K: real,A: real,B: real] :
      ( ( A2
        = ( plus_plus_real @ K @ A ) )
     => ( ( minus_minus_real @ A2 @ B )
        = ( plus_plus_real @ K @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_4447_group__cancel_Osub1,axiom,
    ! [A2: rat,K: rat,A: rat,B: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( minus_minus_rat @ A2 @ B )
        = ( plus_plus_rat @ K @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_4448_group__cancel_Osub1,axiom,
    ! [A2: int,K: int,A: int,B: int] :
      ( ( A2
        = ( plus_plus_int @ K @ A ) )
     => ( ( minus_minus_int @ A2 @ B )
        = ( plus_plus_int @ K @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub1
thf(fact_4449_diff__eq__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( minus_minus_real @ A @ B )
        = C )
      = ( A
        = ( plus_plus_real @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_4450_diff__eq__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ( minus_minus_rat @ A @ B )
        = C )
      = ( A
        = ( plus_plus_rat @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_4451_diff__eq__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( minus_minus_int @ A @ B )
        = C )
      = ( A
        = ( plus_plus_int @ C @ B ) ) ) ).

% diff_eq_eq
thf(fact_4452_eq__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( A
        = ( minus_minus_real @ C @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_4453_eq__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( A
        = ( minus_minus_rat @ C @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_4454_eq__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( A
        = ( minus_minus_int @ C @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = C ) ) ).

% eq_diff_eq
thf(fact_4455_add__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_4456_add__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_4457_add__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% add_diff_eq
thf(fact_4458_diff__diff__eq2,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_4459_diff__diff__eq2,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_4460_diff__diff__eq2,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_diff_eq2
thf(fact_4461_diff__add__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( plus_plus_real @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_4462_diff__add__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( plus_plus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_4463_diff__add__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ ( plus_plus_int @ A @ C ) @ B ) ) ).

% diff_add_eq
thf(fact_4464_diff__add__eq__diff__diff__swap,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) )
      = ( minus_minus_real @ ( minus_minus_real @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_4465_diff__add__eq__diff__diff__swap,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( minus_minus_rat @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_4466_diff__add__eq__diff__diff__swap,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) )
      = ( minus_minus_int @ ( minus_minus_int @ A @ C ) @ B ) ) ).

% diff_add_eq_diff_diff_swap
thf(fact_4467_add__implies__diff,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ( plus_plus_real @ C @ B )
        = A )
     => ( C
        = ( minus_minus_real @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_4468_add__implies__diff,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ( plus_plus_rat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_rat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_4469_add__implies__diff,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( ( plus_plus_nat @ C @ B )
        = A )
     => ( C
        = ( minus_minus_nat @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_4470_add__implies__diff,axiom,
    ! [C: int,B: int,A: int] :
      ( ( ( plus_plus_int @ C @ B )
        = A )
     => ( C
        = ( minus_minus_int @ A @ B ) ) ) ).

% add_implies_diff
thf(fact_4471_diff__diff__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( minus_minus_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ A @ ( plus_plus_real @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_4472_diff__diff__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( minus_minus_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ A @ ( plus_plus_rat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_4473_diff__diff__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ A @ B ) @ C )
      = ( minus_minus_nat @ A @ ( plus_plus_nat @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_4474_diff__diff__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( minus_minus_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( minus_minus_int @ A @ ( plus_plus_int @ B @ C ) ) ) ).

% diff_diff_eq
thf(fact_4475_diff__divide__distrib,axiom,
    ! [A: real,B: real,C: real] :
      ( ( divide_divide_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( minus_minus_real @ ( divide_divide_real @ A @ C ) @ ( divide_divide_real @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_4476_diff__divide__distrib,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( divide_divide_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( minus_minus_rat @ ( divide_divide_rat @ A @ C ) @ ( divide_divide_rat @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_4477_singletonD,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4478_singletonD,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4479_singletonD,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4480_singletonD,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4481_singletonD,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4482_singletonD,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
     => ( B = A ) ) ).

% singletonD
thf(fact_4483_singleton__iff,axiom,
    ! [B: produc3843707927480180839at_nat,A: produc3843707927480180839at_nat] :
      ( ( member8757157785044589968at_nat @ B @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4484_singleton__iff,axiom,
    ! [B: set_nat,A: set_nat] :
      ( ( member_set_nat @ B @ ( insert_set_nat @ A @ bot_bot_set_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4485_singleton__iff,axiom,
    ! [B: product_prod_nat_nat,A: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ B @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4486_singleton__iff,axiom,
    ! [B: real,A: real] :
      ( ( member_real @ B @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4487_singleton__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( member_nat @ B @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4488_singleton__iff,axiom,
    ! [B: int,A: int] :
      ( ( member_int @ B @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( B = A ) ) ).

% singleton_iff
thf(fact_4489_doubleton__eq__iff,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat,C: produc3843707927480180839at_nat,D: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
        = ( insert9069300056098147895at_nat @ C @ ( insert9069300056098147895at_nat @ D @ bot_bo228742789529271731at_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4490_doubleton__eq__iff,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat,C: product_prod_nat_nat,D: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
        = ( insert8211810215607154385at_nat @ C @ ( insert8211810215607154385at_nat @ D @ bot_bo2099793752762293965at_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4491_doubleton__eq__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ( insert_real @ A @ ( insert_real @ B @ bot_bot_set_real ) )
        = ( insert_real @ C @ ( insert_real @ D @ bot_bot_set_real ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4492_doubleton__eq__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ( insert_nat @ A @ ( insert_nat @ B @ bot_bot_set_nat ) )
        = ( insert_nat @ C @ ( insert_nat @ D @ bot_bot_set_nat ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4493_doubleton__eq__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ( insert_int @ A @ ( insert_int @ B @ bot_bot_set_int ) )
        = ( insert_int @ C @ ( insert_int @ D @ bot_bot_set_int ) ) )
      = ( ( ( A = C )
          & ( B = D ) )
        | ( ( A = D )
          & ( B = C ) ) ) ) ).

% doubleton_eq_iff
thf(fact_4494_insert__not__empty,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A @ A2 )
     != bot_bo228742789529271731at_nat ) ).

% insert_not_empty
thf(fact_4495_insert__not__empty,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ A2 )
     != bot_bo2099793752762293965at_nat ) ).

% insert_not_empty
thf(fact_4496_insert__not__empty,axiom,
    ! [A: real,A2: set_real] :
      ( ( insert_real @ A @ A2 )
     != bot_bot_set_real ) ).

% insert_not_empty
thf(fact_4497_insert__not__empty,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( insert_nat @ A @ A2 )
     != bot_bot_set_nat ) ).

% insert_not_empty
thf(fact_4498_insert__not__empty,axiom,
    ! [A: int,A2: set_int] :
      ( ( insert_int @ A @ A2 )
     != bot_bot_set_int ) ).

% insert_not_empty
thf(fact_4499_singleton__inject,axiom,
    ! [A: produc3843707927480180839at_nat,B: produc3843707927480180839at_nat] :
      ( ( ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat )
        = ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_4500_singleton__inject,axiom,
    ! [A: product_prod_nat_nat,B: product_prod_nat_nat] :
      ( ( ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat )
        = ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_4501_singleton__inject,axiom,
    ! [A: real,B: real] :
      ( ( ( insert_real @ A @ bot_bot_set_real )
        = ( insert_real @ B @ bot_bot_set_real ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_4502_singleton__inject,axiom,
    ! [A: nat,B: nat] :
      ( ( ( insert_nat @ A @ bot_bot_set_nat )
        = ( insert_nat @ B @ bot_bot_set_nat ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_4503_singleton__inject,axiom,
    ! [A: int,B: int] :
      ( ( ( insert_int @ A @ bot_bot_set_int )
        = ( insert_int @ B @ bot_bot_set_int ) )
     => ( A = B ) ) ).

% singleton_inject
thf(fact_4504_finite_OinsertI,axiom,
    ! [A2: set_real,A: real] :
      ( ( finite_finite_real @ A2 )
     => ( finite_finite_real @ ( insert_real @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4505_finite_OinsertI,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( finite4343798906461161616at_nat @ A2 )
     => ( finite4343798906461161616at_nat @ ( insert9069300056098147895at_nat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4506_finite_OinsertI,axiom,
    ! [A2: set_nat,A: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( insert_nat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4507_finite_OinsertI,axiom,
    ! [A2: set_int,A: int] :
      ( ( finite_finite_int @ A2 )
     => ( finite_finite_int @ ( insert_int @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4508_finite_OinsertI,axiom,
    ! [A2: set_complex,A: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite3207457112153483333omplex @ ( insert_complex @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4509_finite_OinsertI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite6177210948735845034at_nat @ ( insert8211810215607154385at_nat @ A @ A2 ) ) ) ).

% finite.insertI
thf(fact_4510_insert__subsetI,axiom,
    ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,X7: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X3 @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ X7 @ A2 )
       => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4511_insert__subsetI,axiom,
    ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ X7 @ A2 )
       => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4512_insert__subsetI,axiom,
    ! [X3: real,A2: set_real,X7: set_real] :
      ( ( member_real @ X3 @ A2 )
     => ( ( ord_less_eq_set_real @ X7 @ A2 )
       => ( ord_less_eq_set_real @ ( insert_real @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4513_insert__subsetI,axiom,
    ! [X3: set_nat,A2: set_set_nat,X7: set_set_nat] :
      ( ( member_set_nat @ X3 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ X7 @ A2 )
       => ( ord_le6893508408891458716et_nat @ ( insert_set_nat @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4514_insert__subsetI,axiom,
    ! [X3: nat,A2: set_nat,X7: set_nat] :
      ( ( member_nat @ X3 @ A2 )
     => ( ( ord_less_eq_set_nat @ X7 @ A2 )
       => ( ord_less_eq_set_nat @ ( insert_nat @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4515_insert__subsetI,axiom,
    ! [X3: int,A2: set_int,X7: set_int] :
      ( ( member_int @ X3 @ A2 )
     => ( ( ord_less_eq_set_int @ X7 @ A2 )
       => ( ord_less_eq_set_int @ ( insert_int @ X3 @ X7 ) @ A2 ) ) ) ).

% insert_subsetI
thf(fact_4516_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_4517_insert__mono,axiom,
    ! [C4: set_Pr1261947904930325089at_nat,D6: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ C4 @ D6 )
     => ( ord_le3146513528884898305at_nat @ ( insert8211810215607154385at_nat @ A @ C4 ) @ ( insert8211810215607154385at_nat @ A @ D6 ) ) ) ).

% insert_mono
thf(fact_4518_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_4519_insert__mono,axiom,
    ! [C4: set_Pr4329608150637261639at_nat,D6: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ C4 @ D6 )
     => ( ord_le1268244103169919719at_nat @ ( insert9069300056098147895at_nat @ A @ C4 ) @ ( insert9069300056098147895at_nat @ A @ D6 ) ) ) ).

% insert_mono
thf(fact_4520_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_4521_subset__insert,axiom,
    ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
     => ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
        = ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4522_subset__insert,axiom,
    ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
     => ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
        = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4523_subset__insert,axiom,
    ! [X3: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ X3 @ A2 )
     => ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
        = ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4524_subset__insert,axiom,
    ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X3 @ A2 )
     => ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
        = ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4525_subset__insert,axiom,
    ! [X3: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X3 @ A2 )
     => ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
        = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4526_subset__insert,axiom,
    ! [X3: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ X3 @ A2 )
     => ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
        = ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ).

% subset_insert
thf(fact_4527_subset__insertI,axiom,
    ! [B2: set_nat,A: nat] : ( ord_less_eq_set_nat @ B2 @ ( insert_nat @ A @ B2 ) ) ).

% subset_insertI
thf(fact_4528_subset__insertI,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ B2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) ).

% subset_insertI
thf(fact_4529_subset__insertI,axiom,
    ! [B2: set_real,A: real] : ( ord_less_eq_set_real @ B2 @ ( insert_real @ A @ B2 ) ) ).

% subset_insertI
thf(fact_4530_subset__insertI,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ B2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) ).

% subset_insertI
thf(fact_4531_subset__insertI,axiom,
    ! [B2: set_int,A: int] : ( ord_less_eq_set_int @ B2 @ ( insert_int @ A @ B2 ) ) ).

% subset_insertI
thf(fact_4532_subset__insertI2,axiom,
    ! [A2: set_nat,B2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_4533_subset__insertI2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B2 )
     => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_4534_subset__insertI2,axiom,
    ! [A2: set_real,B2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ B2 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_4535_subset__insertI2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ B2 )
     => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_4536_subset__insertI2,axiom,
    ! [A2: set_int,B2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ B @ B2 ) ) ) ).

% subset_insertI2
thf(fact_4537_dvd__diff,axiom,
    ! [X3: code_integer,Y3: code_integer,Z2: code_integer] :
      ( ( dvd_dvd_Code_integer @ X3 @ Y3 )
     => ( ( dvd_dvd_Code_integer @ X3 @ Z2 )
       => ( dvd_dvd_Code_integer @ X3 @ ( minus_8373710615458151222nteger @ Y3 @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_4538_dvd__diff,axiom,
    ! [X3: real,Y3: real,Z2: real] :
      ( ( dvd_dvd_real @ X3 @ Y3 )
     => ( ( dvd_dvd_real @ X3 @ Z2 )
       => ( dvd_dvd_real @ X3 @ ( minus_minus_real @ Y3 @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_4539_dvd__diff,axiom,
    ! [X3: rat,Y3: rat,Z2: rat] :
      ( ( dvd_dvd_rat @ X3 @ Y3 )
     => ( ( dvd_dvd_rat @ X3 @ Z2 )
       => ( dvd_dvd_rat @ X3 @ ( minus_minus_rat @ Y3 @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_4540_dvd__diff,axiom,
    ! [X3: int,Y3: int,Z2: int] :
      ( ( dvd_dvd_int @ X3 @ Y3 )
     => ( ( dvd_dvd_int @ X3 @ Z2 )
       => ( dvd_dvd_int @ X3 @ ( minus_minus_int @ Y3 @ Z2 ) ) ) ) ).

% dvd_diff
thf(fact_4541_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_4542_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_4543_diffs0__imp__equal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( minus_minus_nat @ M @ N2 )
        = zero_zero_nat )
     => ( ( ( minus_minus_nat @ N2 @ M )
          = zero_zero_nat )
       => ( M = N2 ) ) ) ).

% diffs0_imp_equal
thf(fact_4544_diff__less__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( ord_less_nat @ M @ L )
       => ( ord_less_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ) ).

% diff_less_mono2
thf(fact_4545_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_4546_dvd__minus__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) )
      = ( ( ord_less_nat @ N2 @ M )
        | ( dvd_dvd_nat @ M @ N2 ) ) ) ).

% dvd_minus_self
thf(fact_4547_eq__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ( minus_minus_nat @ M @ K )
            = ( minus_minus_nat @ N2 @ K ) )
          = ( M = N2 ) ) ) ) ).

% eq_diff_iff
thf(fact_4548_le__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% le_diff_iff
thf(fact_4549_Nat_Odiff__diff__eq,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( minus_minus_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Nat.diff_diff_eq
thf(fact_4550_diff__le__mono,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ L ) @ ( minus_minus_nat @ N2 @ L ) ) ) ).

% diff_le_mono
thf(fact_4551_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_4552_le__diff__iff_H,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ C )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ( ord_less_eq_nat @ ( minus_minus_nat @ C @ A ) @ ( minus_minus_nat @ C @ B ) )
          = ( ord_less_eq_nat @ B @ A ) ) ) ) ).

% le_diff_iff'
thf(fact_4553_diff__le__mono2,axiom,
    ! [M: nat,N2: nat,L: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ L @ N2 ) @ ( minus_minus_nat @ L @ M ) ) ) ).

% diff_le_mono2
thf(fact_4554_dvd__diffD,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ M ) ) ) ) ).

% dvd_diffD
thf(fact_4555_dvd__diffD1,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
     => ( ( dvd_dvd_nat @ K @ M )
       => ( ( ord_less_eq_nat @ N2 @ M )
         => ( dvd_dvd_nat @ K @ N2 ) ) ) ) ).

% dvd_diffD1
thf(fact_4556_less__eq__dvd__minus,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ M @ N2 )
        = ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% less_eq_dvd_minus
thf(fact_4557_Nat_Odiff__cancel,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ K @ M ) @ ( plus_plus_nat @ K @ N2 ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% Nat.diff_cancel
thf(fact_4558_diff__cancel2,axiom,
    ! [M: nat,K: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) )
      = ( minus_minus_nat @ M @ N2 ) ) ).

% diff_cancel2
thf(fact_4559_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_4560_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_4561_diff__mult__distrib2,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( minus_minus_nat @ M @ N2 ) )
      = ( minus_minus_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% diff_mult_distrib2
thf(fact_4562_diff__mult__distrib,axiom,
    ! [M: nat,N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ M @ N2 ) @ K )
      = ( minus_minus_nat @ ( times_times_nat @ M @ K ) @ ( times_times_nat @ N2 @ K ) ) ) ).

% diff_mult_distrib
thf(fact_4563_bezout1__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D4: nat,X4: nat,Y5: nat] :
      ( ( dvd_dvd_nat @ D4 @ A )
      & ( dvd_dvd_nat @ D4 @ B )
      & ( ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y5 ) )
          = D4 )
        | ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y5 ) )
          = D4 ) ) ) ).

% bezout1_nat
thf(fact_4564_eucl__rel__int__by0,axiom,
    ! [K: int] : ( eucl_rel_int @ K @ zero_zero_int @ ( product_Pair_int_int @ zero_zero_int @ K ) ) ).

% eucl_rel_int_by0
thf(fact_4565_div__int__unique,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( divide_divide_int @ K @ L )
        = Q4 ) ) ).

% div_int_unique
thf(fact_4566_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_4567_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ zero_zero_rat @ ( zero_n2052037380579107095ol_rat @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_4568_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_4569_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_4570_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_4571_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_less_eq_rat @ ( zero_n2052037380579107095ol_rat @ P ) @ one_one_rat ) ).

% of_bool_less_eq_one
thf(fact_4572_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_4573_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_4574_split__of__bool__asm,axiom,
    ! [P: complex > $o,P4: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_complex ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4575_split__of__bool__asm,axiom,
    ! [P: real > $o,P4: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_real ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4576_split__of__bool__asm,axiom,
    ! [P: rat > $o,P4: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_rat ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4577_split__of__bool__asm,axiom,
    ! [P: nat > $o,P4: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_nat ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4578_split__of__bool__asm,axiom,
    ! [P: int > $o,P4: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ~ ( ( P4
              & ~ ( P @ one_one_int ) )
            | ( ~ P4
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_4579_split__of__bool,axiom,
    ! [P: complex > $o,P4: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P4 ) )
      = ( ( P4
         => ( P @ one_one_complex ) )
        & ( ~ P4
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_4580_split__of__bool,axiom,
    ! [P: real > $o,P4: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P4 ) )
      = ( ( P4
         => ( P @ one_one_real ) )
        & ( ~ P4
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_4581_split__of__bool,axiom,
    ! [P: rat > $o,P4: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P4 ) )
      = ( ( P4
         => ( P @ one_one_rat ) )
        & ( ~ P4
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_4582_split__of__bool,axiom,
    ! [P: nat > $o,P4: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P4 ) )
      = ( ( P4
         => ( P @ one_one_nat ) )
        & ( ~ P4
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_4583_split__of__bool,axiom,
    ! [P: int > $o,P4: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P4 ) )
      = ( ( P4
         => ( P @ one_one_int ) )
        & ( ~ P4
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_4584_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P6: $o] : ( if_complex @ P6 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_4585_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P6: $o] : ( if_real @ P6 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_4586_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P6: $o] : ( if_rat @ P6 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_4587_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P6: $o] : ( if_nat @ P6 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_4588_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P6: $o] : ( if_int @ P6 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_4589_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_4590_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_4591_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_4592_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_4593_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B3: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B3 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_4594_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_4595_ordered__cancel__comm__monoid__diff__class_Ole__imp__diff__is__add,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ A @ B )
       => ( ( ( minus_minus_nat @ B @ A )
            = C )
          = ( B
            = ( plus_plus_nat @ C @ A ) ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_imp_diff_is_add
thf(fact_4596_ordered__cancel__comm__monoid__diff__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ A @ ( minus_minus_nat @ B @ A ) )
        = B ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_inverse
thf(fact_4597_ordered__cancel__comm__monoid__diff__class_Odiff__diff__right,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_diff_right
thf(fact_4598_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A )
        = ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc2
thf(fact_4599_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ C )
        = ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc2
thf(fact_4600_ordered__cancel__comm__monoid__diff__class_Odiff__add__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A )
        = ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.diff_add_assoc
thf(fact_4601_ordered__cancel__comm__monoid__diff__class_Oadd__diff__assoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ C @ B ) @ A ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.add_diff_assoc
thf(fact_4602_ordered__cancel__comm__monoid__diff__class_Ole__diff__conv2,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ ( minus_minus_nat @ B @ A ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ C @ A ) @ B ) ) ) ).

% ordered_cancel_comm_monoid_diff_class.le_diff_conv2
thf(fact_4603_le__add__diff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ord_less_eq_nat @ C @ ( minus_minus_nat @ ( plus_plus_nat @ B @ C ) @ A ) ) ) ).

% le_add_diff
thf(fact_4604_diff__add,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ B @ A ) @ A )
        = B ) ) ).

% diff_add
thf(fact_4605_le__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_4606_le__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_4607_le__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% le_diff_eq
thf(fact_4608_diff__le__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_eq_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_4609_diff__le__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_eq_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_4610_diff__le__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_eq_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_eq_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_le_eq
thf(fact_4611_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_4612_add__le__add__imp__diff__le,axiom,
    ! [I: rat,K: rat,N2: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
         => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
           => ( ord_less_eq_rat @ ( minus_minus_rat @ N2 @ K ) @ J ) ) ) ) ) ).

% add_le_add_imp_diff_le
thf(fact_4613_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_4614_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_4615_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_4616_add__le__imp__le__diff,axiom,
    ! [I: rat,K: rat,N2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I @ K ) @ N2 )
     => ( ord_less_eq_rat @ I @ ( minus_minus_rat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4617_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_4618_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_4619_less__diff__eq,axiom,
    ! [A: real,C: real,B: real] :
      ( ( ord_less_real @ A @ ( minus_minus_real @ C @ B ) )
      = ( ord_less_real @ ( plus_plus_real @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_4620_less__diff__eq,axiom,
    ! [A: rat,C: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( minus_minus_rat @ C @ B ) )
      = ( ord_less_rat @ ( plus_plus_rat @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_4621_less__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ord_less_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( ord_less_int @ ( plus_plus_int @ A @ B ) @ C ) ) ).

% less_diff_eq
thf(fact_4622_diff__less__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ ( minus_minus_real @ A @ B ) @ C )
      = ( ord_less_real @ A @ ( plus_plus_real @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_4623_diff__less__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ A @ B ) @ C )
      = ( ord_less_rat @ A @ ( plus_plus_rat @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_4624_diff__less__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ord_less_int @ ( minus_minus_int @ A @ B ) @ C )
      = ( ord_less_int @ A @ ( plus_plus_int @ C @ B ) ) ) ).

% diff_less_eq
thf(fact_4625_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ~ ( ord_less_real @ A @ B )
     => ( ( plus_plus_real @ B @ ( minus_minus_real @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_4626_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ~ ( ord_less_rat @ A @ B )
     => ( ( plus_plus_rat @ B @ ( minus_minus_rat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_4627_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: nat,B: nat] :
      ( ~ ( ord_less_nat @ A @ B )
     => ( ( plus_plus_nat @ B @ ( minus_minus_nat @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_4628_linordered__semidom__class_Oadd__diff__inverse,axiom,
    ! [A: int,B: int] :
      ( ~ ( ord_less_int @ A @ B )
     => ( ( plus_plus_int @ B @ ( minus_minus_int @ A @ B ) )
        = A ) ) ).

% linordered_semidom_class.add_diff_inverse
thf(fact_4629_mult__diff__mult,axiom,
    ! [X3: complex,Y3: complex,A: complex,B: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X3 @ Y3 ) @ ( times_times_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ X3 @ ( minus_minus_complex @ Y3 @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X3 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_4630_mult__diff__mult,axiom,
    ! [X3: real,Y3: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X3 @ Y3 ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X3 @ ( minus_minus_real @ Y3 @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X3 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_4631_mult__diff__mult,axiom,
    ! [X3: rat,Y3: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X3 @ Y3 ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X3 @ ( minus_minus_rat @ Y3 @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X3 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_4632_mult__diff__mult,axiom,
    ! [X3: int,Y3: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X3 @ Y3 ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X3 @ ( minus_minus_int @ Y3 @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X3 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_4633_square__diff__square__factored,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ ( times_times_complex @ Y3 @ Y3 ) )
      = ( times_times_complex @ ( plus_plus_complex @ X3 @ Y3 ) @ ( minus_minus_complex @ X3 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_4634_square__diff__square__factored,axiom,
    ! [X3: real,Y3: real] :
      ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ ( times_times_real @ Y3 @ Y3 ) )
      = ( times_times_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ X3 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_4635_square__diff__square__factored,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ ( times_times_rat @ Y3 @ Y3 ) )
      = ( times_times_rat @ ( plus_plus_rat @ X3 @ Y3 ) @ ( minus_minus_rat @ X3 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_4636_square__diff__square__factored,axiom,
    ! [X3: int,Y3: int] :
      ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ ( times_times_int @ Y3 @ Y3 ) )
      = ( times_times_int @ ( plus_plus_int @ X3 @ Y3 ) @ ( minus_minus_int @ X3 @ Y3 ) ) ) ).

% square_diff_square_factored
thf(fact_4637_eq__add__iff2,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_4638_eq__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_4639_eq__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_4640_eq__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( C
        = ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% eq_add_iff2
thf(fact_4641_eq__add__iff1,axiom,
    ! [A: complex,E2: complex,C: complex,B: complex,D: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ A @ E2 ) @ C )
        = ( plus_plus_complex @ ( times_times_complex @ B @ E2 ) @ D ) )
      = ( ( plus_plus_complex @ ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_4642_eq__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C )
        = ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_4643_eq__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C )
        = ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_4644_eq__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C )
        = ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C )
        = D ) ) ).

% eq_add_iff1
thf(fact_4645_finite_Ocases,axiom,
    ! [A: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ A )
     => ( ( A != bot_bo228742789529271731at_nat )
       => ~ ! [A7: set_Pr4329608150637261639at_nat] :
              ( ? [A4: produc3843707927480180839at_nat] :
                  ( A
                  = ( insert9069300056098147895at_nat @ A4 @ A7 ) )
             => ~ ( finite4343798906461161616at_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_4646_finite_Ocases,axiom,
    ! [A: set_complex] :
      ( ( finite3207457112153483333omplex @ A )
     => ( ( A != bot_bot_set_complex )
       => ~ ! [A7: set_complex] :
              ( ? [A4: complex] :
                  ( A
                  = ( insert_complex @ A4 @ A7 ) )
             => ~ ( finite3207457112153483333omplex @ A7 ) ) ) ) ).

% finite.cases
thf(fact_4647_finite_Ocases,axiom,
    ! [A: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A )
     => ( ( A != bot_bo2099793752762293965at_nat )
       => ~ ! [A7: set_Pr1261947904930325089at_nat] :
              ( ? [A4: product_prod_nat_nat] :
                  ( A
                  = ( insert8211810215607154385at_nat @ A4 @ A7 ) )
             => ~ ( finite6177210948735845034at_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_4648_finite_Ocases,axiom,
    ! [A: set_real] :
      ( ( finite_finite_real @ A )
     => ( ( A != bot_bot_set_real )
       => ~ ! [A7: set_real] :
              ( ? [A4: real] :
                  ( A
                  = ( insert_real @ A4 @ A7 ) )
             => ~ ( finite_finite_real @ A7 ) ) ) ) ).

% finite.cases
thf(fact_4649_finite_Ocases,axiom,
    ! [A: set_nat] :
      ( ( finite_finite_nat @ A )
     => ( ( A != bot_bot_set_nat )
       => ~ ! [A7: set_nat] :
              ( ? [A4: nat] :
                  ( A
                  = ( insert_nat @ A4 @ A7 ) )
             => ~ ( finite_finite_nat @ A7 ) ) ) ) ).

% finite.cases
thf(fact_4650_finite_Ocases,axiom,
    ! [A: set_int] :
      ( ( finite_finite_int @ A )
     => ( ( A != bot_bot_set_int )
       => ~ ! [A7: set_int] :
              ( ? [A4: int] :
                  ( A
                  = ( insert_int @ A4 @ A7 ) )
             => ~ ( finite_finite_int @ A7 ) ) ) ) ).

% finite.cases
thf(fact_4651_finite_Osimps,axiom,
    ( finite4343798906461161616at_nat
    = ( ^ [A3: set_Pr4329608150637261639at_nat] :
          ( ( A3 = bot_bo228742789529271731at_nat )
          | ? [A5: set_Pr4329608150637261639at_nat,B3: produc3843707927480180839at_nat] :
              ( ( A3
                = ( insert9069300056098147895at_nat @ B3 @ A5 ) )
              & ( finite4343798906461161616at_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4652_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_4653_finite_Osimps,axiom,
    ( finite6177210948735845034at_nat
    = ( ^ [A3: set_Pr1261947904930325089at_nat] :
          ( ( A3 = bot_bo2099793752762293965at_nat )
          | ? [A5: set_Pr1261947904930325089at_nat,B3: product_prod_nat_nat] :
              ( ( A3
                = ( insert8211810215607154385at_nat @ B3 @ A5 ) )
              & ( finite6177210948735845034at_nat @ A5 ) ) ) ) ) ).

% finite.simps
thf(fact_4654_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_4655_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_4656_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_4657_finite__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F3 )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ F4 )
             => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4658_finite__induct,axiom,
    ! [F3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4659_finite__induct,axiom,
    ! [F3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4660_finite__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F3 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F4 )
             => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4661_finite__induct,axiom,
    ! [F3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4662_finite__induct,axiom,
    ! [F3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4663_finite__induct,axiom,
    ! [F3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
         => ( P @ F3 ) ) ) ) ).

% finite_induct
thf(fact_4664_finite__ne__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F3 )
     => ( ( F3 != bot_bo228742789529271731at_nat )
       => ( ! [X4: produc3843707927480180839at_nat] : ( P @ ( insert9069300056098147895at_nat @ X4 @ bot_bo228742789529271731at_nat ) )
         => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F4 )
               => ( ( F4 != bot_bo228742789529271731at_nat )
                 => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4665_finite__ne__induct,axiom,
    ! [F3: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ F3 )
     => ( ( F3 != bot_bot_set_set_nat )
       => ( ! [X4: set_nat] : ( P @ ( insert_set_nat @ X4 @ bot_bot_set_set_nat ) )
         => ( ! [X4: set_nat,F4: set_set_nat] :
                ( ( finite1152437895449049373et_nat @ F4 )
               => ( ( F4 != bot_bot_set_set_nat )
                 => ( ~ ( member_set_nat @ X4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4666_finite__ne__induct,axiom,
    ! [F3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( F3 != bot_bot_set_complex )
       => ( ! [X4: complex] : ( P @ ( insert_complex @ X4 @ bot_bot_set_complex ) )
         => ( ! [X4: complex,F4: set_complex] :
                ( ( finite3207457112153483333omplex @ F4 )
               => ( ( F4 != bot_bot_set_complex )
                 => ( ~ ( member_complex @ X4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4667_finite__ne__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F3 )
     => ( ( F3 != bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat] : ( P @ ( insert8211810215607154385at_nat @ X4 @ bot_bo2099793752762293965at_nat ) )
         => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F4 )
               => ( ( F4 != bot_bo2099793752762293965at_nat )
                 => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4668_finite__ne__induct,axiom,
    ! [F3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ F3 )
     => ( ( F3 != bot_bot_set_real )
       => ( ! [X4: real] : ( P @ ( insert_real @ X4 @ bot_bot_set_real ) )
         => ( ! [X4: real,F4: set_real] :
                ( ( finite_finite_real @ F4 )
               => ( ( F4 != bot_bot_set_real )
                 => ( ~ ( member_real @ X4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4669_finite__ne__induct,axiom,
    ! [F3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ F3 )
     => ( ( F3 != bot_bot_set_nat )
       => ( ! [X4: nat] : ( P @ ( insert_nat @ X4 @ bot_bot_set_nat ) )
         => ( ! [X4: nat,F4: set_nat] :
                ( ( finite_finite_nat @ F4 )
               => ( ( F4 != bot_bot_set_nat )
                 => ( ~ ( member_nat @ X4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4670_finite__ne__induct,axiom,
    ! [F3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ F3 )
     => ( ( F3 != bot_bot_set_int )
       => ( ! [X4: int] : ( P @ ( insert_int @ X4 @ bot_bot_set_int ) )
         => ( ! [X4: int,F4: set_int] :
                ( ( finite_finite_int @ F4 )
               => ( ( F4 != bot_bot_set_int )
                 => ( ~ ( member_int @ X4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_ne_induct
thf(fact_4671_infinite__finite__induct,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
      ( ! [A7: set_Pr4329608150637261639at_nat] :
          ( ~ ( finite4343798906461161616at_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [X4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ F4 )
             => ( ~ ( member8757157785044589968at_nat @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert9069300056098147895at_nat @ X4 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4672_infinite__finite__induct,axiom,
    ! [P: set_set_nat > $o,A2: set_set_nat] :
      ( ! [A7: set_set_nat] :
          ( ~ ( finite1152437895449049373et_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [X4: set_nat,F4: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ F4 )
             => ( ~ ( member_set_nat @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_set_nat @ X4 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4673_infinite__finite__induct,axiom,
    ! [P: set_complex > $o,A2: set_complex] :
      ( ! [A7: set_complex] :
          ( ~ ( finite3207457112153483333omplex @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,F4: set_complex] :
              ( ( finite3207457112153483333omplex @ F4 )
             => ( ~ ( member_complex @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_complex @ X4 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4674_infinite__finite__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
      ( ! [A7: set_Pr1261947904930325089at_nat] :
          ( ~ ( finite6177210948735845034at_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [X4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ F4 )
             => ( ~ ( member8440522571783428010at_nat @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert8211810215607154385at_nat @ X4 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4675_infinite__finite__induct,axiom,
    ! [P: set_real > $o,A2: set_real] :
      ( ! [A7: set_real] :
          ( ~ ( finite_finite_real @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,F4: set_real] :
              ( ( finite_finite_real @ F4 )
             => ( ~ ( member_real @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_real @ X4 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4676_infinite__finite__induct,axiom,
    ! [P: set_nat > $o,A2: set_nat] :
      ( ! [A7: set_nat] :
          ( ~ ( finite_finite_nat @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,F4: set_nat] :
              ( ( finite_finite_nat @ F4 )
             => ( ~ ( member_nat @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_nat @ X4 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4677_infinite__finite__induct,axiom,
    ! [P: set_int > $o,A2: set_int] :
      ( ! [A7: set_int] :
          ( ~ ( finite_finite_int @ A7 )
         => ( P @ A7 ) )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,F4: set_int] :
              ( ( finite_finite_int @ F4 )
             => ( ~ ( member_int @ X4 @ F4 )
               => ( ( P @ F4 )
                 => ( P @ ( insert_int @ X4 @ F4 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% infinite_finite_induct
thf(fact_4678_Euclidean__Division_Opos__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_int @ ( modulo_modulo_int @ K @ L ) @ L ) ) ).

% Euclidean_Division.pos_mod_bound
thf(fact_4679_neg__mod__bound,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_int @ L @ ( modulo_modulo_int @ K @ L ) ) ) ).

% neg_mod_bound
thf(fact_4680_subset__singleton__iff,axiom,
    ! [X7: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ X7 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) )
      = ( ( X7 = bot_bo228742789529271731at_nat )
        | ( X7
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_4681_subset__singleton__iff,axiom,
    ! [X7: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ X7 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) )
      = ( ( X7 = bot_bo2099793752762293965at_nat )
        | ( X7
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_4682_subset__singleton__iff,axiom,
    ! [X7: set_real,A: real] :
      ( ( ord_less_eq_set_real @ X7 @ ( insert_real @ A @ bot_bot_set_real ) )
      = ( ( X7 = bot_bot_set_real )
        | ( X7
          = ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% subset_singleton_iff
thf(fact_4683_subset__singleton__iff,axiom,
    ! [X7: set_nat,A: nat] :
      ( ( ord_less_eq_set_nat @ X7 @ ( insert_nat @ A @ bot_bot_set_nat ) )
      = ( ( X7 = bot_bot_set_nat )
        | ( X7
          = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% subset_singleton_iff
thf(fact_4684_subset__singleton__iff,axiom,
    ! [X7: set_int,A: int] :
      ( ( ord_less_eq_set_int @ X7 @ ( insert_int @ A @ bot_bot_set_int ) )
      = ( ( X7 = bot_bot_set_int )
        | ( X7
          = ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% subset_singleton_iff
thf(fact_4685_subset__singletonD,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
     => ( ( A2 = bot_bo228742789529271731at_nat )
        | ( A2
          = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_4686_subset__singletonD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
     => ( ( A2 = bot_bo2099793752762293965at_nat )
        | ( A2
          = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% subset_singletonD
thf(fact_4687_subset__singletonD,axiom,
    ! [A2: set_real,X3: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) )
     => ( ( A2 = bot_bot_set_real )
        | ( A2
          = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).

% subset_singletonD
thf(fact_4688_subset__singletonD,axiom,
    ! [A2: set_nat,X3: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
     => ( ( A2 = bot_bot_set_nat )
        | ( A2
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% subset_singletonD
thf(fact_4689_subset__singletonD,axiom,
    ! [A2: set_int,X3: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) )
     => ( ( A2 = bot_bot_set_int )
        | ( A2
          = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).

% subset_singletonD
thf(fact_4690_dvd__minus__mod,axiom,
    ! [B: nat,A: nat] : ( dvd_dvd_nat @ B @ ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_4691_dvd__minus__mod,axiom,
    ! [B: int,A: int] : ( dvd_dvd_int @ B @ ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_4692_dvd__minus__mod,axiom,
    ! [B: code_integer,A: code_integer] : ( dvd_dvd_Code_integer @ B @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% dvd_minus_mod
thf(fact_4693_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_4694_Suc__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( suc @ ( minus_minus_nat @ M @ ( suc @ N2 ) ) )
        = ( minus_minus_nat @ M @ N2 ) ) ) ).

% Suc_diff_Suc
thf(fact_4695_diff__less,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ) ) ).

% diff_less
thf(fact_4696_Suc__diff__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( suc @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% Suc_diff_le
thf(fact_4697_diff__less__mono,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( ( ord_less_eq_nat @ C @ A )
       => ( ord_less_nat @ ( minus_minus_nat @ A @ C ) @ ( minus_minus_nat @ B @ C ) ) ) ) ).

% diff_less_mono
thf(fact_4698_less__diff__iff,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( ord_less_nat @ ( minus_minus_nat @ M @ K ) @ ( minus_minus_nat @ N2 @ K ) )
          = ( ord_less_nat @ M @ N2 ) ) ) ) ).

% less_diff_iff
thf(fact_4699_diff__add__0,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) )
      = zero_zero_nat ) ).

% diff_add_0
thf(fact_4700_add__diff__inverse__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( plus_plus_nat @ N2 @ ( minus_minus_nat @ M @ N2 ) )
        = M ) ) ).

% add_diff_inverse_nat
thf(fact_4701_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_4702_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_4703_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_4704_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_4705_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_4706_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_4707_diff__Suc__eq__diff__pred,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( minus_minus_nat @ ( minus_minus_nat @ M @ one_one_nat ) @ N2 ) ) ).

% diff_Suc_eq_diff_pred
thf(fact_4708_mod__geq,axiom,
    ! [M: nat,N2: nat] :
      ( ~ ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% mod_geq
thf(fact_4709_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M2 @ N ) @ M2 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ).

% mod_if
thf(fact_4710_le__mod__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = ( modulo_modulo_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ).

% le_mod_geq
thf(fact_4711_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q4: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q4 )
          = ( modulo_modulo_nat @ N2 @ Q4 ) )
        = ( dvd_dvd_nat @ Q4 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_4712_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_4713_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_4714_ordered__ring__class_Ole__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% ordered_ring_class.le_add_iff2
thf(fact_4715_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_4716_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_4717_ordered__ring__class_Ole__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% ordered_ring_class.le_add_iff1
thf(fact_4718_less__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ C @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_4719_less__add__iff2,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ C @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_4720_less__add__iff2,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ C @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ B @ A ) @ E2 ) @ D ) ) ) ).

% less_add_iff2
thf(fact_4721_less__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( times_times_real @ B @ E2 ) @ D ) )
      = ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_4722_less__add__iff1,axiom,
    ! [A: rat,E2: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ E2 ) @ C ) @ ( plus_plus_rat @ ( times_times_rat @ B @ E2 ) @ D ) )
      = ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_4723_less__add__iff1,axiom,
    ! [A: int,E2: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ A @ E2 ) @ C ) @ ( plus_plus_int @ ( times_times_int @ B @ E2 ) @ D ) )
      = ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ ( minus_minus_int @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% less_add_iff1
thf(fact_4724_divide__diff__eq__iff,axiom,
    ! [Z2: complex,X3: complex,Y3: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X3 @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_4725_divide__diff__eq__iff,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Z2 ) @ Y3 )
        = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_4726_divide__diff__eq__iff,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Z2 ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ X3 @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% divide_diff_eq_iff
thf(fact_4727_diff__divide__eq__iff,axiom,
    ! [Z2: complex,X3: complex,Y3: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( minus_minus_complex @ X3 @ ( divide1717551699836669952omplex @ Y3 @ Z2 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_4728_diff__divide__eq__iff,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( minus_minus_real @ X3 @ ( divide_divide_real @ Y3 @ Z2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_4729_diff__divide__eq__iff,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ X3 @ ( divide_divide_rat @ Y3 @ Z2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ Y3 ) @ Z2 ) ) ) ).

% diff_divide_eq_iff
thf(fact_4730_diff__frac__eq,axiom,
    ! [Y3: complex,Z2: complex,X3: complex,W2: complex] :
      ( ( Y3 != zero_zero_complex )
     => ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X3 @ Y3 ) @ ( divide1717551699836669952omplex @ W2 @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X3 @ Z2 ) @ ( times_times_complex @ W2 @ Y3 ) ) @ ( times_times_complex @ Y3 @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4731_diff__frac__eq,axiom,
    ! [Y3: real,Z2: real,X3: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4732_diff__frac__eq,axiom,
    ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4733_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4734_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4735_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = A ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z2 ) @ B ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4736_square__diff__one__factored,axiom,
    ! [X3: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X3 @ X3 ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X3 @ one_one_complex ) @ ( minus_minus_complex @ X3 @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_4737_square__diff__one__factored,axiom,
    ! [X3: real] :
      ( ( minus_minus_real @ ( times_times_real @ X3 @ X3 ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X3 @ one_one_real ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_4738_square__diff__one__factored,axiom,
    ! [X3: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X3 @ X3 ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) @ ( minus_minus_rat @ X3 @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_4739_square__diff__one__factored,axiom,
    ! [X3: int] :
      ( ( minus_minus_int @ ( times_times_int @ X3 @ X3 ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X3 @ one_one_int ) @ ( minus_minus_int @ X3 @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_4740_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,S5: set_complex] :
              ( ( finite3207457112153483333omplex @ S5 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_complex @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4741_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,S5: set_real] :
              ( ( finite_finite_real @ S5 )
             => ( ! [Y6: real] :
                    ( ( member_real @ Y6 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_real @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4742_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,S5: set_nat] :
              ( ( finite_finite_nat @ S5 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_nat @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4743_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,S5: set_int] :
              ( ( finite_finite_int @ S5 )
             => ( ! [Y6: int] :
                    ( ( member_int @ Y6 @ S5 )
                   => ( ord_less_eq_rat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_int @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4744_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > num] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,S5: set_complex] :
              ( ( finite3207457112153483333omplex @ S5 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_complex @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4745_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > num] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,S5: set_real] :
              ( ( finite_finite_real @ S5 )
             => ( ! [Y6: real] :
                    ( ( member_real @ Y6 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_real @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4746_finite__ranking__induct,axiom,
    ! [S3: set_nat,P: set_nat > $o,F: nat > num] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [X4: nat,S5: set_nat] :
              ( ( finite_finite_nat @ S5 )
             => ( ! [Y6: nat] :
                    ( ( member_nat @ Y6 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_nat @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4747_finite__ranking__induct,axiom,
    ! [S3: set_int,P: set_int > $o,F: int > num] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [X4: int,S5: set_int] :
              ( ( finite_finite_int @ S5 )
             => ( ! [Y6: int] :
                    ( ( member_int @ Y6 @ S5 )
                   => ( ord_less_eq_num @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_int @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4748_finite__ranking__induct,axiom,
    ! [S3: set_complex,P: set_complex > $o,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [X4: complex,S5: set_complex] :
              ( ( finite3207457112153483333omplex @ S5 )
             => ( ! [Y6: complex] :
                    ( ( member_complex @ Y6 @ S5 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_complex @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4749_finite__ranking__induct,axiom,
    ! [S3: set_real,P: set_real > $o,F: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [X4: real,S5: set_real] :
              ( ( finite_finite_real @ S5 )
             => ( ! [Y6: real] :
                    ( ( member_real @ Y6 @ S5 )
                   => ( ord_less_eq_nat @ ( F @ Y6 ) @ ( F @ X4 ) ) )
               => ( ( P @ S5 )
                 => ( P @ ( insert_real @ X4 @ S5 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_ranking_induct
thf(fact_4750_finite__linorder__min__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B4: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X2: real] :
                    ( ( member_real @ X2 @ A7 )
                   => ( ord_less_real @ B4 @ X2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_real @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4751_finite__linorder__min__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B4: rat,A7: set_rat] :
              ( ( finite_finite_rat @ A7 )
             => ( ! [X2: rat] :
                    ( ( member_rat @ X2 @ A7 )
                   => ( ord_less_rat @ B4 @ X2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_rat @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4752_finite__linorder__min__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B4: num,A7: set_num] :
              ( ( finite_finite_num @ A7 )
             => ( ! [X2: num] :
                    ( ( member_num @ X2 @ A7 )
                   => ( ord_less_num @ B4 @ X2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_num @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4753_finite__linorder__min__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B4: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X2: nat] :
                    ( ( member_nat @ X2 @ A7 )
                   => ( ord_less_nat @ B4 @ X2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4754_finite__linorder__min__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B4: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X2: int] :
                    ( ( member_int @ X2 @ A7 )
                   => ( ord_less_int @ B4 @ X2 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_min_induct
thf(fact_4755_finite__linorder__max__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [B4: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ! [X2: real] :
                    ( ( member_real @ X2 @ A7 )
                   => ( ord_less_real @ X2 @ B4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_real @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4756_finite__linorder__max__induct,axiom,
    ! [A2: set_rat,P: set_rat > $o] :
      ( ( finite_finite_rat @ A2 )
     => ( ( P @ bot_bot_set_rat )
       => ( ! [B4: rat,A7: set_rat] :
              ( ( finite_finite_rat @ A7 )
             => ( ! [X2: rat] :
                    ( ( member_rat @ X2 @ A7 )
                   => ( ord_less_rat @ X2 @ B4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_rat @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4757_finite__linorder__max__induct,axiom,
    ! [A2: set_num,P: set_num > $o] :
      ( ( finite_finite_num @ A2 )
     => ( ( P @ bot_bot_set_num )
       => ( ! [B4: num,A7: set_num] :
              ( ( finite_finite_num @ A7 )
             => ( ! [X2: num] :
                    ( ( member_num @ X2 @ A7 )
                   => ( ord_less_num @ X2 @ B4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_num @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4758_finite__linorder__max__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [B4: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ! [X2: nat] :
                    ( ( member_nat @ X2 @ A7 )
                   => ( ord_less_nat @ X2 @ B4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_nat @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4759_finite__linorder__max__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [B4: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ! [X2: int] :
                    ( ( member_int @ X2 @ A7 )
                   => ( ord_less_int @ X2 @ B4 ) )
               => ( ( P @ A7 )
                 => ( P @ ( insert_int @ B4 @ A7 ) ) ) ) )
         => ( P @ A2 ) ) ) ) ).

% finite_linorder_max_induct
thf(fact_4760_inf__period_I4_J,axiom,
    ! [D: code_integer,D6: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D6 )
     => ! [X2: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K4 @ D6 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4761_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_4762_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_4763_inf__period_I4_J,axiom,
    ! [D: rat,D6: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D6 )
     => ! [X2: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ) ).

% inf_period(4)
thf(fact_4764_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_4765_inf__period_I3_J,axiom,
    ! [D: code_integer,D6: code_integer,T: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D6 )
     => ! [X2: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K4 @ D6 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4766_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_4767_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_4768_inf__period_I3_J,axiom,
    ! [D: rat,D6: rat,T: rat] :
      ( ( dvd_dvd_rat @ D @ D6 )
     => ! [X2: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D6 ) ) @ T ) ) ) ) ).

% inf_period(3)
thf(fact_4769_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_4770_finite__subset__induct,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F3 )
     => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F4 )
               => ( ( member8757157785044589968at_nat @ A4 @ A2 )
                 => ( ~ ( member8757157785044589968at_nat @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert9069300056098147895at_nat @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4771_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_4772_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_4773_finite__subset__induct,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F3 )
     => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F4 )
               => ( ( member8440522571783428010at_nat @ A4 @ A2 )
                 => ( ~ ( member8440522571783428010at_nat @ A4 @ F4 )
                   => ( ( P @ F4 )
                     => ( P @ ( insert8211810215607154385at_nat @ A4 @ F4 ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct
thf(fact_4774_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_4775_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_4776_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_4777_finite__subset__induct_H,axiom,
    ! [F3: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ F3 )
     => ( ( ord_le1268244103169919719at_nat @ F3 @ A2 )
       => ( ( P @ bot_bo228742789529271731at_nat )
         => ( ! [A4: produc3843707927480180839at_nat,F4: set_Pr4329608150637261639at_nat] :
                ( ( finite4343798906461161616at_nat @ F4 )
               => ( ( member8757157785044589968at_nat @ A4 @ A2 )
                 => ( ( ord_le1268244103169919719at_nat @ F4 @ A2 )
                   => ( ~ ( member8757157785044589968at_nat @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert9069300056098147895at_nat @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4778_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_4779_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_4780_finite__subset__induct_H,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ F3 )
     => ( ( ord_le3146513528884898305at_nat @ F3 @ A2 )
       => ( ( P @ bot_bo2099793752762293965at_nat )
         => ( ! [A4: product_prod_nat_nat,F4: set_Pr1261947904930325089at_nat] :
                ( ( finite6177210948735845034at_nat @ F4 )
               => ( ( member8440522571783428010at_nat @ A4 @ A2 )
                 => ( ( ord_le3146513528884898305at_nat @ F4 @ A2 )
                   => ( ~ ( member8440522571783428010at_nat @ A4 @ F4 )
                     => ( ( P @ F4 )
                       => ( P @ ( insert8211810215607154385at_nat @ A4 @ F4 ) ) ) ) ) ) )
           => ( P @ F3 ) ) ) ) ) ).

% finite_subset_induct'
thf(fact_4781_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_4782_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_4783_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_4784_minus__mult__div__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4785_minus__mult__div__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4786_minus__mult__div__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mult_div_eq_mod
thf(fact_4787_minus__mod__eq__mult__div,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ B @ ( divide_divide_nat @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4788_minus__mod__eq__mult__div,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ B @ ( divide_divide_int @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4789_minus__mod__eq__mult__div,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ B @ ( divide6298287555418463151nteger @ A @ B ) ) ) ).

% minus_mod_eq_mult_div
thf(fact_4790_minus__mod__eq__div__mult,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( modulo_modulo_nat @ A @ B ) )
      = ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4791_minus__mod__eq__div__mult,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( modulo_modulo_int @ A @ B ) )
      = ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4792_minus__mod__eq__div__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) ) ).

% minus_mod_eq_div_mult
thf(fact_4793_minus__div__mult__eq__mod,axiom,
    ! [A: nat,B: nat] :
      ( ( minus_minus_nat @ A @ ( times_times_nat @ ( divide_divide_nat @ A @ B ) @ B ) )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4794_minus__div__mult__eq__mod,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( times_times_int @ ( divide_divide_int @ A @ B ) @ B ) )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4795_minus__div__mult__eq__mod,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( times_3573771949741848930nteger @ ( divide6298287555418463151nteger @ A @ B ) @ B ) )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_div_mult_eq_mod
thf(fact_4796_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q4: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q4 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_4797_Euclidean__Division_Opos__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) ) ) ).

% Euclidean_Division.pos_mod_sign
thf(fact_4798_neg__mod__sign,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ K @ L ) @ zero_zero_int ) ) ).

% neg_mod_sign
thf(fact_4799_neg__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ zero_zero_int )
     => ( ( ord_less_eq_int @ ( modulo_modulo_int @ A @ B ) @ zero_zero_int )
        & ( ord_less_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% neg_mod_conj
thf(fact_4800_pos__mod__conj,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ A @ B ) )
        & ( ord_less_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ).

% pos_mod_conj
thf(fact_4801_zmod__trivial__iff,axiom,
    ! [I: int,K: int] :
      ( ( ( modulo_modulo_int @ I @ K )
        = I )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I )
          & ( ord_less_int @ I @ K ) )
        | ( ( ord_less_eq_int @ I @ zero_zero_int )
          & ( ord_less_int @ K @ I ) ) ) ) ).

% zmod_trivial_iff
thf(fact_4802_mod__int__pos__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( modulo_modulo_int @ K @ L ) )
      = ( ( dvd_dvd_int @ L @ K )
        | ( ( L = zero_zero_int )
          & ( ord_less_eq_int @ zero_zero_int @ K ) )
        | ( ord_less_int @ zero_zero_int @ L ) ) ) ).

% mod_int_pos_iff
thf(fact_4803_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_4804_zdiv__mono__strict,axiom,
    ! [A2: int,B2: int,N2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ( ( modulo_modulo_int @ A2 @ N2 )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B2 @ N2 )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B2 @ N2 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_4805_nat__diff__split__asm,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ~ ( ( ( ord_less_nat @ A @ B )
              & ~ ( P @ zero_zero_nat ) )
            | ? [D5: nat] :
                ( ( A
                  = ( plus_plus_nat @ B @ D5 ) )
                & ~ ( P @ D5 ) ) ) ) ) ).

% nat_diff_split_asm
thf(fact_4806_nat__diff__split,axiom,
    ! [P: nat > $o,A: nat,B: nat] :
      ( ( P @ ( minus_minus_nat @ A @ B ) )
      = ( ( ( ord_less_nat @ A @ B )
         => ( P @ zero_zero_nat ) )
        & ! [D5: nat] :
            ( ( A
              = ( plus_plus_nat @ B @ D5 ) )
           => ( P @ D5 ) ) ) ) ).

% nat_diff_split
thf(fact_4807_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_4808_nat__diff__add__eq2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_4809_nat__diff__add__eq1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_4810_nat__le__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_4811_nat__le__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_4812_nat__eq__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_4813_nat__eq__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_4814_dvd__minus__add,axiom,
    ! [Q4: nat,N2: nat,R2: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q4 @ N2 )
     => ( ( ord_less_eq_nat @ Q4 @ ( times_times_nat @ R2 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q4 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R2 @ M ) @ Q4 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_4815_set__update__subset__insert,axiom,
    ! [Xs: list_P6011104703257516679at_nat,I: nat,X3: product_prod_nat_nat] : ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ ( list_u6180841689913720943at_nat @ Xs @ I @ X3 ) ) @ ( insert8211810215607154385at_nat @ X3 @ ( set_Pr5648618587558075414at_nat @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_4816_set__update__subset__insert,axiom,
    ! [Xs: list_real,I: nat,X3: real] : ( ord_less_eq_set_real @ ( set_real2 @ ( list_update_real @ Xs @ I @ X3 ) ) @ ( insert_real @ X3 @ ( set_real2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_4817_set__update__subset__insert,axiom,
    ! [Xs: list_P5464809261938338413at_nat,I: nat,X3: produc3843707927480180839at_nat] : ( ord_le1268244103169919719at_nat @ ( set_Pr3765526544606949372at_nat @ ( list_u4696772448584712917at_nat @ Xs @ I @ X3 ) ) @ ( insert9069300056098147895at_nat @ X3 @ ( set_Pr3765526544606949372at_nat @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_4818_set__update__subset__insert,axiom,
    ! [Xs: list_VEBT_VEBT,I: nat,X3: vEBT_VEBT] : ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ ( list_u1324408373059187874T_VEBT @ Xs @ I @ X3 ) ) @ ( insert_VEBT_VEBT @ X3 @ ( set_VEBT_VEBT2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_4819_set__update__subset__insert,axiom,
    ! [Xs: list_nat,I: nat,X3: nat] : ( ord_less_eq_set_nat @ ( set_nat2 @ ( list_update_nat @ Xs @ I @ X3 ) ) @ ( insert_nat @ X3 @ ( set_nat2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_4820_set__update__subset__insert,axiom,
    ! [Xs: list_int,I: nat,X3: int] : ( ord_less_eq_set_int @ ( set_int2 @ ( list_update_int @ Xs @ I @ X3 ) ) @ ( insert_int @ X3 @ ( set_int2 @ Xs ) ) ) ).

% set_update_subset_insert
thf(fact_4821_mod__nat__eqI,axiom,
    ! [R2: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R2 @ N2 )
     => ( ( ord_less_eq_nat @ R2 @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R2 ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R2 ) ) ) ) ).

% mod_nat_eqI
thf(fact_4822_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_3573771949741848930nteger
        @ ( zero_n356916108424825756nteger
          @ ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M )
             != zero_z3403309356797280102nteger )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4823_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat
        @ ( zero_n2687167440665602831ol_nat
          @ ( ( ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M )
             != zero_zero_nat )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4824_exp__div__exp__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int
        @ ( zero_n2684676970156552555ol_int
          @ ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M )
             != zero_zero_int )
            & ( ord_less_eq_nat @ N2 @ M ) ) )
        @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% exp_div_exp_eq
thf(fact_4825_frac__le__eq,axiom,
    ! [Y3: real,Z2: real,X3: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_4826_frac__le__eq,axiom,
    ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_4827_frac__less__eq,axiom,
    ! [Y3: real,Z2: real,X3: real,W2: real] :
      ( ( Y3 != zero_zero_real )
     => ( ( Z2 != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X3 @ Y3 ) @ ( divide_divide_real @ W2 @ Z2 ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X3 @ Z2 ) @ ( times_times_real @ W2 @ Y3 ) ) @ ( times_times_real @ Y3 @ Z2 ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_4828_frac__less__eq,axiom,
    ! [Y3: rat,Z2: rat,X3: rat,W2: rat] :
      ( ( Y3 != zero_zero_rat )
     => ( ( Z2 != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X3 @ Y3 ) @ ( divide_divide_rat @ W2 @ Z2 ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X3 @ Z2 ) @ ( times_times_rat @ W2 @ Y3 ) ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_4829_signed__take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% signed_take_bit_int_less_exp
thf(fact_4830_power__diff,axiom,
    ! [A: complex,N2: nat,M: nat] :
      ( ( A != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_complex @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ M ) @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4831_power__diff,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4832_power__diff,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4833_power__diff,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( A != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_real @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_real @ ( power_power_real @ A @ M ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4834_power__diff,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( A != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_power_rat @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide_divide_rat @ ( power_power_rat @ A @ M ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4835_power__diff,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N2 ) )
          = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).

% power_diff
thf(fact_4836_mod__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( modulo_modulo_int @ K @ L )
          = ( plus_plus_int @ K @ L ) ) ) ) ).

% mod_pos_neg_trivial
thf(fact_4837_Suc__diff__eq__diff__pred,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( minus_minus_nat @ ( suc @ M ) @ N2 )
        = ( minus_minus_nat @ M @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% Suc_diff_eq_diff_pred
thf(fact_4838_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_4839_div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% div_geq
thf(fact_4840_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M2: nat,N: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M2 @ N )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% div_if
thf(fact_4841_set__replicate__Suc,axiom,
    ! [N2: nat,X3: produc3843707927480180839at_nat] :
      ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ ( suc @ N2 ) @ X3 ) )
      = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ).

% set_replicate_Suc
thf(fact_4842_set__replicate__Suc,axiom,
    ! [N2: nat,X3: vEBT_VEBT] :
      ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ ( suc @ N2 ) @ X3 ) )
      = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ).

% set_replicate_Suc
thf(fact_4843_set__replicate__Suc,axiom,
    ! [N2: nat,X3: product_prod_nat_nat] :
      ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ ( suc @ N2 ) @ X3 ) )
      = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ).

% set_replicate_Suc
thf(fact_4844_set__replicate__Suc,axiom,
    ! [N2: nat,X3: real] :
      ( ( set_real2 @ ( replicate_real @ ( suc @ N2 ) @ X3 ) )
      = ( insert_real @ X3 @ bot_bot_set_real ) ) ).

% set_replicate_Suc
thf(fact_4845_set__replicate__Suc,axiom,
    ! [N2: nat,X3: nat] :
      ( ( set_nat2 @ ( replicate_nat @ ( suc @ N2 ) @ X3 ) )
      = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ).

% set_replicate_Suc
thf(fact_4846_set__replicate__Suc,axiom,
    ! [N2: nat,X3: int] :
      ( ( set_int2 @ ( replicate_int @ ( suc @ N2 ) @ X3 ) )
      = ( insert_int @ X3 @ bot_bot_set_int ) ) ).

% set_replicate_Suc
thf(fact_4847_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).

% add_eq_if
thf(fact_4848_set__replicate__conv__if,axiom,
    ! [N2: nat,X3: produc3843707927480180839at_nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N2 @ X3 ) )
          = bot_bo228742789529271731at_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_Pr3765526544606949372at_nat @ ( replic2264142908078655527at_nat @ N2 @ X3 ) )
          = ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4849_set__replicate__conv__if,axiom,
    ! [N2: nat,X3: vEBT_VEBT] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
          = bot_bo8194388402131092736T_VEBT ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ X3 ) )
          = ( insert_VEBT_VEBT @ X3 @ bot_bo8194388402131092736T_VEBT ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4850_set__replicate__conv__if,axiom,
    ! [N2: nat,X3: product_prod_nat_nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ X3 ) )
          = bot_bo2099793752762293965at_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ X3 ) )
          = ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4851_set__replicate__conv__if,axiom,
    ! [N2: nat,X3: real] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X3 ) )
          = bot_bot_set_real ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_real2 @ ( replicate_real @ N2 @ X3 ) )
          = ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4852_set__replicate__conv__if,axiom,
    ! [N2: nat,X3: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X3 ) )
          = bot_bot_set_nat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_nat2 @ ( replicate_nat @ N2 @ X3 ) )
          = ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4853_set__replicate__conv__if,axiom,
    ! [N2: nat,X3: int] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X3 ) )
          = bot_bot_set_int ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( set_int2 @ ( replicate_int @ N2 @ X3 ) )
          = ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ).

% set_replicate_conv_if
thf(fact_4854_nat__less__add__iff2,axiom,
    ! [I: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_4855_nat__less__add__iff1,axiom,
    ! [J: nat,I: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_4856_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) @ N ) ) ) ) ) ).

% mult_eq_if
thf(fact_4857_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_4858_scaling__mono,axiom,
    ! [U: rat,V: rat,R2: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R2 )
       => ( ( ord_less_eq_rat @ R2 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R2 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4859_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: 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 @ M ) )
       != zero_zero_nat ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4860_exp__not__zero__imp__exp__diff__not__zero,axiom,
    ! [N2: nat,M: 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 @ M ) )
       != zero_zero_int ) ) ).

% exp_not_zero_imp_exp_diff_not_zero
thf(fact_4861_signed__take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% signed_take_bit_int_greater_eq_self_iff
thf(fact_4862_signed__take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% signed_take_bit_int_less_self_iff
thf(fact_4863_power__diff__power__eq,axiom,
    ! [A: nat,N2: nat,M: nat] :
      ( ( A != zero_zero_nat )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( power_power_nat @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_nat @ ( power_power_nat @ A @ M ) @ ( power_power_nat @ A @ N2 ) )
            = ( divide_divide_nat @ one_one_nat @ ( power_power_nat @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4864_power__diff__power__eq,axiom,
    ! [A: int,N2: nat,M: nat] :
      ( ( A != zero_zero_int )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( power_power_int @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide_divide_int @ ( power_power_int @ A @ M ) @ ( power_power_int @ A @ N2 ) )
            = ( divide_divide_int @ one_one_int @ ( power_power_int @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4865_power__diff__power__eq,axiom,
    ! [A: code_integer,N2: nat,M: nat] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
            = ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ M @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ N2 @ M )
         => ( ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ M ) @ ( power_8256067586552552935nteger @ A @ N2 ) )
            = ( divide6298287555418463151nteger @ one_one_Code_integer @ ( power_8256067586552552935nteger @ A @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ) ).

% power_diff_power_eq
thf(fact_4866_split__zmod,axiom,
    ! [P: int > $o,N2: int,K: int] :
      ( ( P @ ( modulo_modulo_int @ N2 @ K ) )
      = ( ( ( K = zero_zero_int )
         => ( P @ N2 ) )
        & ( ( ord_less_int @ zero_zero_int @ K )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_4867_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ R2 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R2 )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_4868_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R2 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
       => ( ( ord_less_int @ R2 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R2 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_4869_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P6: complex,M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P6 @ ( power_power_complex @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4870_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P6: real,M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P6 @ ( power_power_real @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4871_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P6: rat,M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P6 @ ( power_power_rat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4872_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P6: nat,M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P6 @ ( power_power_nat @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4873_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P6: int,M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P6 @ ( power_power_int @ P6 @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4874_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_4875_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_4876_power__minus__mult,axiom,
    ! [N2: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( power_power_rat @ A @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ A )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus_mult
thf(fact_4877_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_4878_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_4879_diff__le__diff__pow,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ ( minus_minus_nat @ ( power_power_nat @ K @ M ) @ ( power_power_nat @ K @ N2 ) ) ) ) ).

% diff_le_diff_pow
thf(fact_4880_le__div__geq,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( divide_divide_nat @ M @ N2 )
          = ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 ) ) ) ) ) ).

% le_div_geq
thf(fact_4881_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A4: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A4 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A4 )
         => ( P @ A4 ) )
     => ( ! [A4: code_integer,B4: $o] :
            ( ( P @ A4 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A4 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B4 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A4 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_4882_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_4883_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_4884_verit__le__mono__div__int,axiom,
    ! [A2: int,B2: int,N2: int] :
      ( ( ord_less_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B2 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B2 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_4885_split__pos__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_4886_split__neg__lemma,axiom,
    ! [K: int,P: int > int > $o,N2: int] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ( P @ ( divide_divide_int @ N2 @ K ) @ ( modulo_modulo_int @ N2 @ K ) )
        = ( ! [I2: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I2 ) @ J3 ) ) )
             => ( P @ I2 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_4887_int__power__div__base,axiom,
    ! [M: nat,K: int] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ( divide_divide_int @ ( power_power_int @ K @ M ) @ K )
          = ( power_power_int @ K @ ( minus_minus_nat @ M @ ( suc @ zero_zero_nat ) ) ) ) ) ) ).

% int_power_div_base
thf(fact_4888_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z2: nat] :
      ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z2 ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z2 ) )
        = ( insert_nat @ N2 @ ( nat_set_decode @ Z2 ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_4889_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ M @ N2 ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_4890_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_4891_exp__mod__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ M @ N2 ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ).

% exp_mod_exp
thf(fact_4892_power2__diff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_4893_power2__diff,axiom,
    ! [X3: real,Y3: real] :
      ( ( power_power_real @ ( minus_minus_real @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_4894_power2__diff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_4895_power2__diff,axiom,
    ! [X3: int,Y3: int] :
      ( ( power_power_int @ ( minus_minus_int @ X3 @ Y3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 ) @ Y3 ) ) ) ).

% power2_diff
thf(fact_4896_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) @ ( 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 @ M ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4897_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( times_times_int @ A @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) @ ( 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 @ M ) ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4898_mult__exp__mod__exp__eq,axiom,
    ! [M: nat,N2: nat,A: code_integer] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
        = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) ) ) ) ).

% mult_exp_mod_exp_eq
thf(fact_4899_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_4900_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q4 ) @ R2 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R2 )
            & ( ord_less_int @ R2 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R2 )
                & ( ord_less_eq_int @ R2 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q4 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_4901_divmod__digit__1_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ B )
       => ( ( ord_le3102999989581377725nteger @ B @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo364778990260209775nteger @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4902_divmod__digit__1_I2_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ A )
     => ( ( ord_less_nat @ zero_zero_nat @ B )
       => ( ( ord_less_eq_nat @ B @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_nat @ ( modulo_modulo_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_nat @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4903_divmod__digit__1_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ( ord_less_eq_int @ B @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) )
         => ( ( minus_minus_int @ ( modulo_modulo_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ B )
            = ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% divmod_digit_1(2)
thf(fact_4904_even__mask__div__iff_H,axiom,
    ! [M: 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 ) ) @ M ) @ one_one_nat ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_4905_even__mask__div__iff_H,axiom,
    ! [M: 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 ) ) @ M ) @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_4906_even__mask__div__iff_H,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% even_mask_div_iff'
thf(fact_4907_even__mask__div__iff,axiom,
    ! [M: 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 ) ) @ M ) @ 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 @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_4908_even__mask__div__iff,axiom,
    ! [M: 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 ) ) @ M ) @ 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 @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_4909_even__mask__div__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) @ one_one_Code_integer ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( ( ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 )
          = zero_z3403309356797280102nteger )
        | ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% even_mask_div_iff
thf(fact_4910_divmod__step__eq,axiom,
    ! [L: num,R2: nat,Q4: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ one_one_nat ) @ ( minus_minus_nat @ R2 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R2 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q4 @ R2 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4911_divmod__step__eq,axiom,
    ! [L: num,R2: int,Q4: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ one_one_int ) @ ( minus_minus_int @ R2 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R2 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q4 @ R2 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4912_divmod__step__eq,axiom,
    ! [L: num,R2: code_integer,Q4: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R2 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R2 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q4 @ R2 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q4 ) @ R2 ) ) ) ) ).

% divmod_step_eq
thf(fact_4913_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_4914_Divides_Oadjust__div__eq,axiom,
    ! [Q4: int,R2: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q4 @ R2 ) )
      = ( plus_plus_int @ Q4 @ ( zero_n2684676970156552555ol_int @ ( R2 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_4915_signed__take__bit__rec,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N: nat,A3: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% signed_take_bit_rec
thf(fact_4916_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_4917_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q4: int,R2: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
       => ( eucl_rel_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) @ ( product_Pair_int_int @ Q4 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R2 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_4918_diff__shunt__var,axiom,
    ! [X3: set_Pr1261947904930325089at_nat,Y3: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ X3 @ Y3 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ X3 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_4919_diff__shunt__var,axiom,
    ! [X3: set_real,Y3: set_real] :
      ( ( ( minus_minus_set_real @ X3 @ Y3 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ X3 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_4920_diff__shunt__var,axiom,
    ! [X3: set_nat,Y3: set_nat] :
      ( ( ( minus_minus_set_nat @ X3 @ Y3 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ X3 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_4921_diff__shunt__var,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ( minus_minus_set_int @ X3 @ Y3 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).

% diff_shunt_var
thf(fact_4922_take__bit__rec,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N: nat,A3: code_integer] : ( if_Code_integer @ ( N = zero_zero_nat ) @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ ( minus_minus_nat @ N @ one_one_nat ) @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% take_bit_rec
thf(fact_4923_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_4924_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_4925_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4926_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4927_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4928_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4929_neg__equal__iff__equal,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = ( uminus_uminus_int @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4930_neg__equal__iff__equal,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = ( uminus_uminus_real @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4931_neg__equal__iff__equal,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = ( uminus_uminus_rat @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4932_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4933_Diff__cancel,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ A2 )
      = bot_bo2099793752762293965at_nat ) ).

% Diff_cancel
thf(fact_4934_Diff__cancel,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ A2 )
      = bot_bot_set_real ) ).

% Diff_cancel
thf(fact_4935_Diff__cancel,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ A2 )
      = bot_bot_set_int ) ).

% Diff_cancel
thf(fact_4936_Diff__cancel,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ A2 )
      = bot_bot_set_nat ) ).

% Diff_cancel
thf(fact_4937_empty__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ bot_bo2099793752762293965at_nat @ A2 )
      = bot_bo2099793752762293965at_nat ) ).

% empty_Diff
thf(fact_4938_empty__Diff,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ bot_bot_set_real @ A2 )
      = bot_bot_set_real ) ).

% empty_Diff
thf(fact_4939_empty__Diff,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ bot_bot_set_int @ A2 )
      = bot_bot_set_int ) ).

% empty_Diff
thf(fact_4940_empty__Diff,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ bot_bot_set_nat @ A2 )
      = bot_bot_set_nat ) ).

% empty_Diff
thf(fact_4941_Diff__empty,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ bot_bo2099793752762293965at_nat )
      = A2 ) ).

% Diff_empty
thf(fact_4942_Diff__empty,axiom,
    ! [A2: set_real] :
      ( ( minus_minus_set_real @ A2 @ bot_bot_set_real )
      = A2 ) ).

% Diff_empty
thf(fact_4943_Diff__empty,axiom,
    ! [A2: set_int] :
      ( ( minus_minus_set_int @ A2 @ bot_bot_set_int )
      = A2 ) ).

% Diff_empty
thf(fact_4944_Diff__empty,axiom,
    ! [A2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ bot_bot_set_nat )
      = A2 ) ).

% Diff_empty
thf(fact_4945_finite__Diff2,axiom,
    ! [B2: set_int,A2: set_int] :
      ( ( finite_finite_int @ B2 )
     => ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) )
        = ( finite_finite_int @ A2 ) ) ) ).

% finite_Diff2
thf(fact_4946_finite__Diff2,axiom,
    ! [B2: set_complex,A2: set_complex] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
        = ( finite3207457112153483333omplex @ A2 ) ) ) ).

% finite_Diff2
thf(fact_4947_finite__Diff2,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
        = ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_4948_finite__Diff2,axiom,
    ! [B2: set_nat,A2: set_nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) )
        = ( finite_finite_nat @ A2 ) ) ) ).

% finite_Diff2
thf(fact_4949_finite__Diff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( finite_finite_int @ A2 )
     => ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% finite_Diff
thf(fact_4950_finite__Diff,axiom,
    ! [A2: set_complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).

% finite_Diff
thf(fact_4951_finite__Diff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).

% finite_Diff
thf(fact_4952_finite__Diff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% finite_Diff
thf(fact_4953_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B2 ) )
      = ( ord_less_eq_set_int @ B2 @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_4954_Compl__anti__mono,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B2 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_4955_Diff__insert0,axiom,
    ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
     => ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
        = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4956_Diff__insert0,axiom,
    ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
     => ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
        = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4957_Diff__insert0,axiom,
    ! [X3: real,A2: set_real,B2: set_real] :
      ( ~ ( member_real @ X3 @ A2 )
     => ( ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
        = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4958_Diff__insert0,axiom,
    ! [X3: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ~ ( member_set_nat @ X3 @ A2 )
     => ( ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
        = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4959_Diff__insert0,axiom,
    ! [X3: int,A2: set_int,B2: set_int] :
      ( ~ ( member_int @ X3 @ A2 )
     => ( ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
        = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4960_Diff__insert0,axiom,
    ! [X3: nat,A2: set_nat,B2: set_nat] :
      ( ~ ( member_nat @ X3 @ A2 )
     => ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
        = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% Diff_insert0
thf(fact_4961_insert__Diff1,axiom,
    ! [X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ X3 @ B2 )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
        = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4962_insert__Diff1,axiom,
    ! [X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ B2 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
        = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4963_insert__Diff1,axiom,
    ! [X3: real,B2: set_real,A2: set_real] :
      ( ( member_real @ X3 @ B2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
        = ( minus_minus_set_real @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4964_insert__Diff1,axiom,
    ! [X3: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ X3 @ B2 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
        = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4965_insert__Diff1,axiom,
    ! [X3: int,B2: set_int,A2: set_int] :
      ( ( member_int @ X3 @ B2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
        = ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4966_insert__Diff1,axiom,
    ! [X3: nat,B2: set_nat,A2: set_nat] :
      ( ( member_nat @ X3 @ B2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
        = ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% insert_Diff1
thf(fact_4967_compl__le__compl__iff,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ ( uminus1532241313380277803et_int @ Y3 ) )
      = ( ord_less_eq_set_int @ Y3 @ X3 ) ) ).

% compl_le_compl_iff
thf(fact_4968_neg__le__iff__le,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_eq_real @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4969_neg__le__iff__le,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4970_neg__le__iff__le,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4971_neg__le__iff__le,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_eq_int @ A @ B ) ) ).

% neg_le_iff_le
thf(fact_4972_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_4973_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_4974_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_4975_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_4976_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_4977_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_4978_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_4979_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_4980_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_4981_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_4982_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_4983_neg__equal__0__iff__equal,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_0_iff_equal
thf(fact_4984_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_4985_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4986_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_4987_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_4988_neg__0__equal__iff__equal,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( uminus_uminus_rat @ A ) )
      = ( zero_zero_rat = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4989_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4990_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_4991_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_4992_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_4993_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_4994_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_4995_neg__less__iff__less,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) )
      = ( ord_less_int @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4996_neg__less__iff__less,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) )
      = ( ord_less_real @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4997_neg__less__iff__less,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4998_neg__less__iff__less,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ B ) ) ).

% neg_less_iff_less
thf(fact_4999_mult__minus__left,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_5000_mult__minus__left,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_5001_mult__minus__left,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_5002_mult__minus__left,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_5003_mult__minus__left,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_left
thf(fact_5004_minus__mult__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( times_times_complex @ A @ B ) ) ).

% minus_mult_minus
thf(fact_5005_minus__mult__minus,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( times_times_int @ A @ B ) ) ).

% minus_mult_minus
thf(fact_5006_minus__mult__minus,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( times_times_real @ A @ B ) ) ).

% minus_mult_minus
thf(fact_5007_minus__mult__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( times_times_rat @ A @ B ) ) ).

% minus_mult_minus
thf(fact_5008_minus__mult__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( times_3573771949741848930nteger @ A @ B ) ) ).

% minus_mult_minus
thf(fact_5009_mult__minus__right,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( times_times_complex @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_5010_mult__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( times_times_int @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_5011_mult__minus__right,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ A @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( times_times_real @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_5012_mult__minus__right,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( times_times_rat @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_5013_mult__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( times_3573771949741848930nteger @ A @ B ) ) ) ).

% mult_minus_right
thf(fact_5014_minus__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) ) ) ).

% minus_add_distrib
thf(fact_5015_minus__add__distrib,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) ) ) ).

% minus_add_distrib
thf(fact_5016_minus__add__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_add_distrib
thf(fact_5017_minus__add__distrib,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_add_distrib
thf(fact_5018_minus__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ ( plus_plus_int @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5019_minus__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( plus_plus_real @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5020_minus__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( plus_plus_rat @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5021_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_5022_add__minus__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ A @ ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5023_add__minus__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ A @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5024_add__minus__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ A @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5025_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_5026_minus__diff__eq,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) )
      = ( minus_minus_int @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5027_minus__diff__eq,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) )
      = ( minus_minus_real @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5028_minus__diff__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) )
      = ( minus_minus_rat @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5029_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_5030_dvd__minus__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( dvd_dvd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
      = ( dvd_dvd_int @ X3 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_5031_dvd__minus__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( dvd_dvd_real @ X3 @ ( uminus_uminus_real @ Y3 ) )
      = ( dvd_dvd_real @ X3 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_5032_dvd__minus__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( dvd_dvd_rat @ X3 @ ( uminus_uminus_rat @ Y3 ) )
      = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_5033_dvd__minus__iff,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( dvd_dvd_Code_integer @ X3 @ ( uminus1351360451143612070nteger @ Y3 ) )
      = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).

% dvd_minus_iff
thf(fact_5034_minus__dvd__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
      = ( dvd_dvd_int @ X3 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_5035_minus__dvd__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X3 ) @ Y3 )
      = ( dvd_dvd_real @ X3 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_5036_minus__dvd__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X3 ) @ Y3 )
      = ( dvd_dvd_rat @ X3 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_5037_minus__dvd__iff,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X3 ) @ Y3 )
      = ( dvd_dvd_Code_integer @ X3 @ Y3 ) ) ).

% minus_dvd_iff
thf(fact_5038_Diff__eq__empty__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ( minus_1356011639430497352at_nat @ A2 @ B2 )
        = bot_bo2099793752762293965at_nat )
      = ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_5039_Diff__eq__empty__iff,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ( minus_minus_set_real @ A2 @ B2 )
        = bot_bot_set_real )
      = ( ord_less_eq_set_real @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_5040_Diff__eq__empty__iff,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ( minus_minus_set_nat @ A2 @ B2 )
        = bot_bot_set_nat )
      = ( ord_less_eq_set_nat @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_5041_Diff__eq__empty__iff,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ( minus_minus_set_int @ A2 @ B2 )
        = bot_bot_set_int )
      = ( ord_less_eq_set_int @ A2 @ B2 ) ) ).

% Diff_eq_empty_iff
thf(fact_5042_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_5043_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_5044_atLeastAtMost__iff,axiom,
    ! [I: rat,L: rat,U: rat] :
      ( ( member_rat @ I @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I )
        & ( ord_less_eq_rat @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5045_atLeastAtMost__iff,axiom,
    ! [I: num,L: num,U: num] :
      ( ( member_num @ I @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I )
        & ( ord_less_eq_num @ I @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_5046_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_5047_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_5048_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_5049_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_5050_Icc__eq__Icc,axiom,
    ! [L: rat,H2: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H2 )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H2 )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5051_Icc__eq__Icc,axiom,
    ! [L: num,H2: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H2 )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H2 = H3 ) )
        | ( ~ ( ord_less_eq_num @ L @ H2 )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_5052_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_5053_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_5054_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_5055_insert__Diff__single,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      = ( insert9069300056098147895at_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_5056_insert__Diff__single,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      = ( insert8211810215607154385at_nat @ A @ A2 ) ) ).

% insert_Diff_single
thf(fact_5057_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_5058_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_5059_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_5060_finite__Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( finite_finite_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) ) )
      = ( finite_finite_real @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_5061_finite__Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) ) )
      = ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_5062_finite__Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( finite_finite_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) ) )
      = ( finite_finite_int @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_5063_finite__Diff__insert,axiom,
    ! [A2: set_complex,A: complex,B2: set_complex] :
      ( ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ B2 ) ) )
      = ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_5064_finite__Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) ) )
      = ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_5065_finite__Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) ) )
      = ( finite_finite_nat @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ).

% finite_Diff_insert
thf(fact_5066_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_5067_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_5068_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% finite_atLeastAtMost
thf(fact_5069_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_5070_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_5071_neg__less__eq__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_5072_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_5073_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_5074_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_5075_less__eq__neg__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% less_eq_neg_nonpos
thf(fact_5076_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_5077_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_5078_neg__le__0__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_le_0_iff_le
thf(fact_5079_neg__le__0__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% neg_le_0_iff_le
thf(fact_5080_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_5081_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_5082_neg__0__le__iff__le,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_le_iff_le
thf(fact_5083_neg__0__le__iff__le,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% neg_0_le_iff_le
thf(fact_5084_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_5085_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_5086_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_5087_less__neg__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% less_neg_neg
thf(fact_5088_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_5089_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_5090_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_5091_neg__less__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ A )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_pos
thf(fact_5092_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_5093_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_5094_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_5095_neg__0__less__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ A ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% neg_0_less_iff_less
thf(fact_5096_neg__0__less__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% neg_0_less_iff_less
thf(fact_5097_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_5098_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_5099_neg__less__0__iff__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% neg_less_0_iff_less
thf(fact_5100_neg__less__0__iff__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_0_iff_less
thf(fact_5101_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_5102_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_5103_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_5104_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_5105_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_5106_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_5107_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_5108_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_5109_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_5110_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_5111_verit__minus__simplify_I3_J,axiom,
    ! [B: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_5112_verit__minus__simplify_I3_J,axiom,
    ! [B: int] :
      ( ( minus_minus_int @ zero_zero_int @ B )
      = ( uminus_uminus_int @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_5113_verit__minus__simplify_I3_J,axiom,
    ! [B: real] :
      ( ( minus_minus_real @ zero_zero_real @ B )
      = ( uminus_uminus_real @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_5114_verit__minus__simplify_I3_J,axiom,
    ! [B: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_5115_verit__minus__simplify_I3_J,axiom,
    ! [B: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% verit_minus_simplify(3)
thf(fact_5116_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_5117_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_5118_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_5119_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_5120_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_5121_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5122_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( plus_plus_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5123_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ M ) @ ( numeral_numeral_rat @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5124_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_5125_uminus__add__conv__diff,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ B )
      = ( minus_minus_int @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5126_uminus__add__conv__diff,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ B )
      = ( minus_minus_real @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5127_uminus__add__conv__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( minus_minus_rat @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5128_uminus__add__conv__diff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_5129_diff__minus__eq__add,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ A @ ( uminus_uminus_int @ B ) )
      = ( plus_plus_int @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_5130_diff__minus__eq__add,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ A @ ( uminus_uminus_real @ B ) )
      = ( plus_plus_real @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_5131_diff__minus__eq__add,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( plus_plus_rat @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_5132_diff__minus__eq__add,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( plus_p5714425477246183910nteger @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_5133_divide__minus1,axiom,
    ! [X3: complex] :
      ( ( divide1717551699836669952omplex @ X3 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X3 ) ) ).

% divide_minus1
thf(fact_5134_divide__minus1,axiom,
    ! [X3: real] :
      ( ( divide_divide_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X3 ) ) ).

% divide_minus1
thf(fact_5135_divide__minus1,axiom,
    ! [X3: rat] :
      ( ( divide_divide_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X3 ) ) ).

% divide_minus1
thf(fact_5136_atLeastatMost__empty__iff,axiom,
    ! [A: set_int,B: set_int] :
      ( ( ( set_or370866239135849197et_int @ A @ B )
        = bot_bot_set_set_int )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5137_atLeastatMost__empty__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5138_atLeastatMost__empty__iff,axiom,
    ! [A: num,B: num] :
      ( ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5139_atLeastatMost__empty__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5140_atLeastatMost__empty__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5141_atLeastatMost__empty__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff
thf(fact_5142_atLeastatMost__empty__iff2,axiom,
    ! [A: set_int,B: set_int] :
      ( ( bot_bot_set_set_int
        = ( set_or370866239135849197et_int @ A @ B ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5143_atLeastatMost__empty__iff2,axiom,
    ! [A: rat,B: rat] :
      ( ( bot_bot_set_rat
        = ( set_or633870826150836451st_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5144_atLeastatMost__empty__iff2,axiom,
    ! [A: num,B: num] :
      ( ( bot_bot_set_num
        = ( set_or7049704709247886629st_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5145_atLeastatMost__empty__iff2,axiom,
    ! [A: nat,B: nat] :
      ( ( bot_bot_set_nat
        = ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5146_atLeastatMost__empty__iff2,axiom,
    ! [A: int,B: int] :
      ( ( bot_bot_set_int
        = ( set_or1266510415728281911st_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5147_atLeastatMost__empty__iff2,axiom,
    ! [A: real,B: real] :
      ( ( bot_bot_set_real
        = ( set_or1222579329274155063t_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B ) ) ) ).

% atLeastatMost_empty_iff2
thf(fact_5148_atLeastatMost__subset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ~ ( ord_less_eq_set_int @ A @ B )
        | ( ( ord_less_eq_set_int @ C @ A )
          & ( ord_less_eq_set_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5149_atLeastatMost__subset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( ( ord_less_eq_rat @ C @ A )
          & ( ord_less_eq_rat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5150_atLeastatMost__subset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( ( ord_less_eq_num @ C @ A )
          & ( ord_less_eq_num @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5151_atLeastatMost__subset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( ( ord_less_eq_nat @ C @ A )
          & ( ord_less_eq_nat @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5152_atLeastatMost__subset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( ( ord_less_eq_int @ C @ A )
          & ( ord_less_eq_int @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5153_atLeastatMost__subset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( ( ord_less_eq_real @ C @ A )
          & ( ord_less_eq_real @ B @ D ) ) ) ) ).

% atLeastatMost_subset_iff
thf(fact_5154_atLeastatMost__empty,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ( ( set_or633870826150836451st_rat @ A @ B )
        = bot_bot_set_rat ) ) ).

% atLeastatMost_empty
thf(fact_5155_atLeastatMost__empty,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ( ( set_or7049704709247886629st_num @ A @ B )
        = bot_bot_set_num ) ) ).

% atLeastatMost_empty
thf(fact_5156_atLeastatMost__empty,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = bot_bot_set_nat ) ) ).

% atLeastatMost_empty
thf(fact_5157_atLeastatMost__empty,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = bot_bot_set_int ) ) ).

% atLeastatMost_empty
thf(fact_5158_atLeastatMost__empty,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = bot_bot_set_real ) ) ).

% atLeastatMost_empty
thf(fact_5159_infinite__Icc__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) )
      = ( ord_less_rat @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_5160_infinite__Icc__iff,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) )
      = ( ord_less_real @ A @ B ) ) ).

% infinite_Icc_iff
thf(fact_5161_subset__Compl__singleton,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B: produc3843707927480180839at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ B @ bot_bo228742789529271731at_nat ) ) )
      = ( ~ ( member8757157785044589968at_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_5162_subset__Compl__singleton,axiom,
    ! [A2: set_set_nat,B: set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( uminus613421341184616069et_nat @ ( insert_set_nat @ B @ bot_bot_set_set_nat ) ) )
      = ( ~ ( member_set_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_5163_subset__Compl__singleton,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ B @ bot_bo2099793752762293965at_nat ) ) )
      = ( ~ ( member8440522571783428010at_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_5164_subset__Compl__singleton,axiom,
    ! [A2: set_real,B: real] :
      ( ( ord_less_eq_set_real @ A2 @ ( uminus612125837232591019t_real @ ( insert_real @ B @ bot_bot_set_real ) ) )
      = ( ~ ( member_real @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_5165_subset__Compl__singleton,axiom,
    ! [A2: set_nat,B: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( uminus5710092332889474511et_nat @ ( insert_nat @ B @ bot_bot_set_nat ) ) )
      = ( ~ ( member_nat @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_5166_subset__Compl__singleton,axiom,
    ! [A2: set_int,B: int] :
      ( ( ord_less_eq_set_int @ A2 @ ( uminus1532241313380277803et_int @ ( insert_int @ B @ bot_bot_set_int ) ) )
      = ( ~ ( member_int @ B @ A2 ) ) ) ).

% subset_Compl_singleton
thf(fact_5167_take__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_5168_take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_5169_atLeastAtMost__singleton,axiom,
    ! [A: nat] :
      ( ( set_or1269000886237332187st_nat @ A @ A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% atLeastAtMost_singleton
thf(fact_5170_atLeastAtMost__singleton,axiom,
    ! [A: int] :
      ( ( set_or1266510415728281911st_int @ A @ A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% atLeastAtMost_singleton
thf(fact_5171_atLeastAtMost__singleton,axiom,
    ! [A: real] :
      ( ( set_or1222579329274155063t_real @ A @ A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% atLeastAtMost_singleton
thf(fact_5172_atLeastAtMost__singleton__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ C @ bot_bot_set_nat ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5173_atLeastAtMost__singleton__iff,axiom,
    ! [A: int,B: int,C: int] :
      ( ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ C @ bot_bot_set_int ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5174_atLeastAtMost__singleton__iff,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ C @ bot_bot_set_real ) )
      = ( ( A = B )
        & ( B = C ) ) ) ).

% atLeastAtMost_singleton_iff
thf(fact_5175_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_5176_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_5177_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_5178_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_5179_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_5180_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = zero_zero_rat ) ).

% add_neg_numeral_special(8)
thf(fact_5181_add__neg__numeral__special_I8_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(8)
thf(fact_5182_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_5183_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_5184_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_5185_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_plus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% add_neg_numeral_special(7)
thf(fact_5186_add__neg__numeral__special_I7_J,axiom,
    ( ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% add_neg_numeral_special(7)
thf(fact_5187_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_5188_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_5189_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_5190_diff__numeral__special_I12_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = zero_zero_rat ) ).

% diff_numeral_special(12)
thf(fact_5191_diff__numeral__special_I12_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% diff_numeral_special(12)
thf(fact_5192_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_5193_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_5194_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_5195_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_5196_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y3: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W2 ) ) @ Y3 ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_5197_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y3: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ Y3 ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_5198_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y3: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ Y3 ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_5199_semiring__norm_I168_J,axiom,
    ! [V: num,W2: num,Y3: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W2 ) ) @ Y3 ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W2 ) ) ) @ Y3 ) ) ).

% semiring_norm(168)
thf(fact_5200_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5201_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5202_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5203_neg__numeral__le__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_eq_num @ N2 @ M ) ) ).

% neg_numeral_le_iff
thf(fact_5204_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5205_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5206_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5207_neg__numeral__less__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( ord_less_num @ N2 @ M ) ) ).

% neg_numeral_less_iff
thf(fact_5208_zle__diff1__eq,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ W2 @ ( minus_minus_int @ Z2 @ one_one_int ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% zle_diff1_eq
thf(fact_5209_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_5210_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5211_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5212_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5213_not__neg__one__le__neg__numeral__iff,axiom,
    ! [M: num] :
      ( ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) )
      = ( M != one ) ) ).

% not_neg_one_le_neg_numeral_iff
thf(fact_5214_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5215_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_5216_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5217_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_5218_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W2: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5219_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5220_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_5221_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W2: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
            = B ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5222_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
            = B ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5223_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_5224_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5225_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5226_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5227_neg__numeral__less__neg__one__iff,axiom,
    ! [M: num] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( M != one ) ) ).

% neg_numeral_less_neg_one_iff
thf(fact_5228_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W2: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5229_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W2: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_5230_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W2: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5231_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W2: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_5232_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_5233_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_5234_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_5235_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_5236_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_5237_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_5238_add__neg__numeral__special_I9_J,axiom,
    ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% add_neg_numeral_special(9)
thf(fact_5239_even__take__bit__eq,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se1745604003318907178nteger @ N2 @ A ) )
      = ( ( N2 = zero_zero_nat )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_take_bit_eq
thf(fact_5240_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_5241_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_5242_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_5243_take__bit__Suc__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ zero_zero_nat ) @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_Suc_0
thf(fact_5244_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_5245_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_5246_signed__take__bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_ri6519982836138164636nteger @ zero_zero_nat @ A )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% signed_take_bit_0
thf(fact_5247_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_5248_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_nat @ ( zero_n2687167440665602831ol_nat @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_5249_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se2923211474154528505it_int @ M @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ N2 @ M ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_5250_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_5251_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_5252_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_5253_compl__mono,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_less_eq_set_int @ X3 @ Y3 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ ( uminus1532241313380277803et_int @ X3 ) ) ) ).

% compl_mono
thf(fact_5254_compl__le__swap1,axiom,
    ! [Y3: set_int,X3: set_int] :
      ( ( ord_less_eq_set_int @ Y3 @ ( uminus1532241313380277803et_int @ X3 ) )
     => ( ord_less_eq_set_int @ X3 @ ( uminus1532241313380277803et_int @ Y3 ) ) ) ).

% compl_le_swap1
thf(fact_5255_compl__le__swap2,axiom,
    ! [Y3: set_int,X3: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y3 ) @ X3 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X3 ) @ Y3 ) ) ).

% compl_le_swap2
thf(fact_5256_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_5257_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_5258_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_5259_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_5260_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5261_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5262_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5263_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_5264_Compl__insert,axiom,
    ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( uminus935396558254630718at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) )
      = ( minus_3314409938677909166at_nat @ ( uminus935396558254630718at_nat @ A2 ) @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) ) ).

% Compl_insert
thf(fact_5265_Compl__insert,axiom,
    ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( uminus6524753893492686040at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) )
      = ( minus_1356011639430497352at_nat @ ( uminus6524753893492686040at_nat @ A2 ) @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) ) ).

% Compl_insert
thf(fact_5266_Compl__insert,axiom,
    ! [X3: real,A2: set_real] :
      ( ( uminus612125837232591019t_real @ ( insert_real @ X3 @ A2 ) )
      = ( minus_minus_set_real @ ( uminus612125837232591019t_real @ A2 ) @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ).

% Compl_insert
thf(fact_5267_Compl__insert,axiom,
    ! [X3: int,A2: set_int] :
      ( ( uminus1532241313380277803et_int @ ( insert_int @ X3 @ A2 ) )
      = ( minus_minus_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ).

% Compl_insert
thf(fact_5268_Compl__insert,axiom,
    ! [X3: nat,A2: set_nat] :
      ( ( uminus5710092332889474511et_nat @ ( insert_nat @ X3 @ A2 ) )
      = ( minus_minus_set_nat @ ( uminus5710092332889474511et_nat @ A2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) ) ).

% Compl_insert
thf(fact_5269_take__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) )
      = ( bit_se2925701944663578781it_nat @ N2 @ ( plus_plus_nat @ A @ B ) ) ) ).

% take_bit_add
thf(fact_5270_take__bit__add,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ A @ B ) ) ) ).

% take_bit_add
thf(fact_5271_take__bit__tightened,axiom,
    ! [N2: nat,A: nat,B: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ A )
        = ( bit_se2925701944663578781it_nat @ N2 @ B ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( bit_se2925701944663578781it_nat @ M @ A )
          = ( bit_se2925701944663578781it_nat @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_5272_take__bit__tightened,axiom,
    ! [N2: nat,A: int,B: int,M: nat] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = ( bit_se2923211474154528505it_int @ N2 @ B ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( bit_se2923211474154528505it_int @ M @ A )
          = ( bit_se2923211474154528505it_int @ M @ B ) ) ) ) ).

% take_bit_tightened
thf(fact_5273_take__bit__nat__less__eq__self,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M ) ).

% take_bit_nat_less_eq_self
thf(fact_5274_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q4 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q4 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_5275_le__imp__neg__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5276_le__imp__neg__le,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5277_le__imp__neg__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5278_le__imp__neg__le,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% le_imp_neg_le
thf(fact_5279_minus__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5280_minus__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5281_minus__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5282_minus__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_le_iff
thf(fact_5283_le__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_eq_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% le_minus_iff
thf(fact_5284_le__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le3102999989581377725nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% le_minus_iff
thf(fact_5285_le__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_eq_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% le_minus_iff
thf(fact_5286_le__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_eq_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% le_minus_iff
thf(fact_5287_minus__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( uminus_uminus_int @ A ) @ B )
      = ( ord_less_int @ ( uminus_uminus_int @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5288_minus__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ A ) @ B )
      = ( ord_less_real @ ( uminus_uminus_real @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5289_minus__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5290_minus__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ A ) ) ).

% minus_less_iff
thf(fact_5291_less__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ ( uminus_uminus_int @ B ) )
      = ( ord_less_int @ B @ ( uminus_uminus_int @ A ) ) ) ).

% less_minus_iff
thf(fact_5292_less__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ B ) )
      = ( ord_less_real @ B @ ( uminus_uminus_real @ A ) ) ) ).

% less_minus_iff
thf(fact_5293_less__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ B ) )
      = ( ord_less_rat @ B @ ( uminus_uminus_rat @ A ) ) ) ).

% less_minus_iff
thf(fact_5294_less__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( ord_le6747313008572928689nteger @ B @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% less_minus_iff
thf(fact_5295_verit__negate__coefficient_I2_J,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( ord_less_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5296_verit__negate__coefficient_I2_J,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5297_verit__negate__coefficient_I2_J,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5298_verit__negate__coefficient_I2_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ B )
     => ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% verit_negate_coefficient(2)
thf(fact_5299_square__eq__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ A )
        = ( times_times_complex @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5300_square__eq__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( times_times_int @ A @ A )
        = ( times_times_int @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_int @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5301_square__eq__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ A )
        = ( times_times_real @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5302_square__eq__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ A )
        = ( times_times_rat @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5303_square__eq__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( times_3573771949741848930nteger @ A @ A )
        = ( times_3573771949741848930nteger @ B @ B ) )
      = ( ( A = B )
        | ( A
          = ( uminus1351360451143612070nteger @ B ) ) ) ) ).

% square_eq_iff
thf(fact_5304_minus__mult__commute,axiom,
    ! [A: complex,B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_mult_commute
thf(fact_5305_minus__mult__commute,axiom,
    ! [A: int,B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ A ) @ B )
      = ( times_times_int @ A @ ( uminus_uminus_int @ B ) ) ) ).

% minus_mult_commute
thf(fact_5306_minus__mult__commute,axiom,
    ! [A: real,B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ A ) @ B )
      = ( times_times_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_mult_commute
thf(fact_5307_minus__mult__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ A ) @ B )
      = ( times_times_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_mult_commute
thf(fact_5308_minus__mult__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
      = ( times_3573771949741848930nteger @ A @ ( uminus1351360451143612070nteger @ B ) ) ) ).

% minus_mult_commute
thf(fact_5309_is__num__normalize_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5310_is__num__normalize_I8_J,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5311_is__num__normalize_I8_J,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5312_is__num__normalize_I8_J,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5313_add_Oinverse__distrib__swap,axiom,
    ! [A: int,B: int] :
      ( ( uminus_uminus_int @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ B ) @ ( uminus_uminus_int @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5314_add_Oinverse__distrib__swap,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( plus_plus_real @ A @ B ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ B ) @ ( uminus_uminus_real @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5315_add_Oinverse__distrib__swap,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( plus_plus_rat @ A @ B ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ B ) @ ( uminus_uminus_rat @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5316_add_Oinverse__distrib__swap,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ B ) @ ( uminus1351360451143612070nteger @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5317_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_5318_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_5319_group__cancel_Oneg1,axiom,
    ! [A2: rat,K: rat,A: rat] :
      ( ( A2
        = ( plus_plus_rat @ K @ A ) )
     => ( ( uminus_uminus_rat @ A2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( uminus_uminus_rat @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5320_group__cancel_Oneg1,axiom,
    ! [A2: code_integer,K: code_integer,A: code_integer] :
      ( ( A2
        = ( plus_p5714425477246183910nteger @ K @ A ) )
     => ( ( uminus1351360451143612070nteger @ A2 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( uminus1351360451143612070nteger @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5321_minus__diff__commute,axiom,
    ! [B: int,A: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ B ) @ A )
      = ( minus_minus_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5322_minus__diff__commute,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ B ) @ A )
      = ( minus_minus_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5323_minus__diff__commute,axiom,
    ! [B: rat,A: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ B ) @ A )
      = ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5324_minus__diff__commute,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ B ) @ A )
      = ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5325_minus__divide__right,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ).

% minus_divide_right
thf(fact_5326_minus__divide__right,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ).

% minus_divide_right
thf(fact_5327_minus__divide__divide,axiom,
    ! [A: real,B: real] :
      ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( divide_divide_real @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5328_minus__divide__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( divide_divide_rat @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5329_minus__divide__left,axiom,
    ! [A: real,B: real] :
      ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5330_minus__divide__left,axiom,
    ! [A: rat,B: rat] :
      ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5331_Diff__infinite__finite,axiom,
    ! [T3: set_int,S3: set_int] :
      ( ( finite_finite_int @ T3 )
     => ( ~ ( finite_finite_int @ S3 )
       => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_5332_Diff__infinite__finite,axiom,
    ! [T3: set_complex,S3: set_complex] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ~ ( finite3207457112153483333omplex @ S3 )
       => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_5333_Diff__infinite__finite,axiom,
    ! [T3: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ T3 )
     => ( ~ ( finite6177210948735845034at_nat @ S3 )
       => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_5334_Diff__infinite__finite,axiom,
    ! [T3: set_nat,S3: set_nat] :
      ( ( finite_finite_nat @ T3 )
     => ( ~ ( finite_finite_nat @ S3 )
       => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ T3 ) ) ) ) ).

% Diff_infinite_finite
thf(fact_5335_double__diff,axiom,
    ! [A2: set_nat,B2: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ C4 )
       => ( ( minus_minus_set_nat @ B2 @ ( minus_minus_set_nat @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_5336_double__diff,axiom,
    ! [A2: set_int,B2: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ C4 )
       => ( ( minus_minus_set_int @ B2 @ ( minus_minus_set_int @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_5337_Diff__subset,axiom,
    ! [A2: set_nat,B2: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ A2 ) ).

% Diff_subset
thf(fact_5338_Diff__subset,axiom,
    ! [A2: set_int,B2: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ A2 ) ).

% Diff_subset
thf(fact_5339_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D6: set_nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C4 )
     => ( ( ord_less_eq_set_nat @ D6 @ B2 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( minus_minus_set_nat @ C4 @ D6 ) ) ) ) ).

% Diff_mono
thf(fact_5340_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D6: set_int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C4 )
     => ( ( ord_less_eq_set_int @ D6 @ B2 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( minus_minus_set_int @ C4 @ D6 ) ) ) ) ).

% Diff_mono
thf(fact_5341_insert__Diff__if,axiom,
    ! [X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( ( member8757157785044589968at_nat @ X3 @ B2 )
       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
          = ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) )
      & ( ~ ( member8757157785044589968at_nat @ X3 @ B2 )
       => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ B2 )
          = ( insert9069300056098147895at_nat @ X3 @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5342_insert__Diff__if,axiom,
    ! [X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( ( member8440522571783428010at_nat @ X3 @ B2 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
          = ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) )
      & ( ~ ( member8440522571783428010at_nat @ X3 @ B2 )
       => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ B2 )
          = ( insert8211810215607154385at_nat @ X3 @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5343_insert__Diff__if,axiom,
    ! [X3: real,B2: set_real,A2: set_real] :
      ( ( ( member_real @ X3 @ B2 )
       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
          = ( minus_minus_set_real @ A2 @ B2 ) ) )
      & ( ~ ( member_real @ X3 @ B2 )
       => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ B2 )
          = ( insert_real @ X3 @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5344_insert__Diff__if,axiom,
    ! [X3: set_nat,B2: set_set_nat,A2: set_set_nat] :
      ( ( ( member_set_nat @ X3 @ B2 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
          = ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) )
      & ( ~ ( member_set_nat @ X3 @ B2 )
       => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ B2 )
          = ( insert_set_nat @ X3 @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5345_insert__Diff__if,axiom,
    ! [X3: int,B2: set_int,A2: set_int] :
      ( ( ( member_int @ X3 @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
          = ( minus_minus_set_int @ A2 @ B2 ) ) )
      & ( ~ ( member_int @ X3 @ B2 )
       => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ B2 )
          = ( insert_int @ X3 @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5346_insert__Diff__if,axiom,
    ! [X3: nat,B2: set_nat,A2: set_nat] :
      ( ( ( member_nat @ X3 @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
          = ( minus_minus_set_nat @ A2 @ B2 ) ) )
      & ( ~ ( member_nat @ X3 @ B2 )
       => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ B2 )
          = ( insert_nat @ X3 @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ) ).

% insert_Diff_if
thf(fact_5347_minus__int__code_I1_J,axiom,
    ! [K: int] :
      ( ( minus_minus_int @ K @ zero_zero_int )
      = K ) ).

% minus_int_code(1)
thf(fact_5348_uminus__int__code_I1_J,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% uminus_int_code(1)
thf(fact_5349_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_5350_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_5351_psubset__imp__ex__mem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ B2 )
     => ? [B4: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B4 @ ( minus_1356011639430497352at_nat @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5352_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B2: set_real] :
      ( ( ord_less_set_real @ A2 @ B2 )
     => ? [B4: real] : ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5353_psubset__imp__ex__mem,axiom,
    ! [A2: set_set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ B2 )
     => ? [B4: set_nat] : ( member_set_nat @ B4 @ ( minus_2163939370556025621et_nat @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5354_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ B2 )
     => ? [B4: int] : ( member_int @ B4 @ ( minus_minus_set_int @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5355_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B2 )
     => ? [B4: nat] : ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5356_zdvd__zdiffD,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ K @ ( minus_minus_int @ M @ N2 ) )
     => ( ( dvd_dvd_int @ K @ N2 )
       => ( dvd_dvd_int @ K @ M ) ) ) ).

% zdvd_zdiffD
thf(fact_5357_zmod__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( minus_minus_int @ B @ one_one_int ) ) ) ).

% zmod_minus1
thf(fact_5358_take__bit__minus__small__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) )
          = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ) ) ).

% take_bit_minus_small_eq
thf(fact_5359_infinite__Icc,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( finite_finite_rat @ ( set_or633870826150836451st_rat @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_5360_infinite__Icc,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( finite_finite_real @ ( set_or1222579329274155063t_real @ A @ B ) ) ) ).

% infinite_Icc
thf(fact_5361_atLeastAtMost__singleton_H,axiom,
    ! [A: nat,B: nat] :
      ( ( A = B )
     => ( ( set_or1269000886237332187st_nat @ A @ B )
        = ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5362_atLeastAtMost__singleton_H,axiom,
    ! [A: int,B: int] :
      ( ( A = B )
     => ( ( set_or1266510415728281911st_int @ A @ B )
        = ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5363_atLeastAtMost__singleton_H,axiom,
    ! [A: real,B: real] :
      ( ( A = B )
     => ( ( set_or1222579329274155063t_real @ A @ B )
        = ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% atLeastAtMost_singleton'
thf(fact_5364_take__bit__tightened__less__eq__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ M @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_tightened_less_eq_int
thf(fact_5365_take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% take_bit_int_greater_self_iff
thf(fact_5366_not__take__bit__negative,axiom,
    ! [N2: nat,K: int] :
      ~ ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ zero_zero_int ) ).

% not_take_bit_negative
thf(fact_5367_signed__take__bit__eq__iff__take__bit__eq,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ A )
        = ( bit_ri631733984087533419it_int @ N2 @ B ) )
      = ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ A )
        = ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ B ) ) ) ).

% signed_take_bit_eq_iff_take_bit_eq
thf(fact_5368_signed__take__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_ri631733984087533419it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( if_int_int @ ( ord_less_eq_nat @ N2 @ M ) @ ( bit_se2923211474154528505it_int @ N2 ) @ ( bit_ri631733984087533419it_int @ M ) @ A ) ) ).

% signed_take_bit_take_bit
thf(fact_5369_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N2 )
            & ( P @ M2 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_5370_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_eq_nat @ M2 @ N2 )
           => ( P @ M2 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_5371_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5372_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5373_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5374_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5375_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5376_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5377_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5378_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5379_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5380_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5381_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5382_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5383_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5384_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5385_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5386_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5387_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5388_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5389_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5390_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5391_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5392_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_5393_le__minus__one__simps_I2_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% le_minus_one_simps(2)
thf(fact_5394_le__minus__one__simps_I2_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% le_minus_one_simps(2)
thf(fact_5395_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_5396_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_5397_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(4)
thf(fact_5398_le__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(4)
thf(fact_5399_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_5400_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_5401_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_5402_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_5403_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_5404_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_5405_add__eq__0__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5406_add__eq__0__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5407_add__eq__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5408_add__eq__0__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5409_add__eq__0__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% add_eq_0_iff
thf(fact_5410_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_5411_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_5412_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_5413_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5414_ab__group__add__class_Oab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_group_add_class.ab_left_minus
thf(fact_5415_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5416_add_Oinverse__unique,axiom,
    ! [A: int,B: int] :
      ( ( ( plus_plus_int @ A @ B )
        = zero_zero_int )
     => ( ( uminus_uminus_int @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5417_add_Oinverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( plus_plus_real @ A @ B )
        = zero_zero_real )
     => ( ( uminus_uminus_real @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5418_add_Oinverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat )
     => ( ( uminus_uminus_rat @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5419_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5420_eq__neg__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5421_eq__neg__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5422_eq__neg__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5423_eq__neg__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5424_eq__neg__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% eq_neg_iff_add_eq_0
thf(fact_5425_neg__eq__iff__add__eq__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5426_neg__eq__iff__add__eq__0,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( plus_plus_int @ A @ B )
        = zero_zero_int ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5427_neg__eq__iff__add__eq__0,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( plus_plus_real @ A @ B )
        = zero_zero_real ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5428_neg__eq__iff__add__eq__0,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( plus_plus_rat @ A @ B )
        = zero_zero_rat ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5429_neg__eq__iff__add__eq__0,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger ) ) ).

% neg_eq_iff_add_eq_0
thf(fact_5430_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_5431_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_5432_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(4)
thf(fact_5433_less__minus__one__simps_I4_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(4)
thf(fact_5434_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_5435_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_5436_less__minus__one__simps_I2_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat ).

% less_minus_one_simps(2)
thf(fact_5437_less__minus__one__simps_I2_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ).

% less_minus_one_simps(2)
thf(fact_5438_nonzero__minus__divide__right,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5439_nonzero__minus__divide__right,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5440_nonzero__minus__divide__right,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) ) ) ) ).

% nonzero_minus_divide_right
thf(fact_5441_nonzero__minus__divide__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
        = ( divide1717551699836669952omplex @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5442_nonzero__minus__divide__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
        = ( divide_divide_real @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5443_nonzero__minus__divide__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
        = ( divide_divide_rat @ A @ B ) ) ) ).

% nonzero_minus_divide_divide
thf(fact_5444_square__eq__1__iff,axiom,
    ! [X3: complex] :
      ( ( ( times_times_complex @ X3 @ X3 )
        = one_one_complex )
      = ( ( X3 = one_one_complex )
        | ( X3
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_5445_square__eq__1__iff,axiom,
    ! [X3: int] :
      ( ( ( times_times_int @ X3 @ X3 )
        = one_one_int )
      = ( ( X3 = one_one_int )
        | ( X3
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_5446_square__eq__1__iff,axiom,
    ! [X3: real] :
      ( ( ( times_times_real @ X3 @ X3 )
        = one_one_real )
      = ( ( X3 = one_one_real )
        | ( X3
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_5447_square__eq__1__iff,axiom,
    ! [X3: rat] :
      ( ( ( times_times_rat @ X3 @ X3 )
        = one_one_rat )
      = ( ( X3 = one_one_rat )
        | ( X3
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_5448_square__eq__1__iff,axiom,
    ! [X3: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X3 @ X3 )
        = one_one_Code_integer )
      = ( ( X3 = one_one_Code_integer )
        | ( X3
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_5449_group__cancel_Osub2,axiom,
    ! [B2: int,K: int,B: int,A: int] :
      ( ( B2
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B2 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5450_group__cancel_Osub2,axiom,
    ! [B2: real,K: real,B: real,A: real] :
      ( ( B2
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B2 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5451_group__cancel_Osub2,axiom,
    ! [B2: rat,K: rat,B: rat,A: rat] :
      ( ( B2
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B2 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5452_group__cancel_Osub2,axiom,
    ! [B2: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B2
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B2 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5453_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_5454_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_5455_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5456_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5457_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_5458_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_5459_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B3: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5460_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B3: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B3 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5461_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se4205575877204974255it_nat @ M @ A ) )
          = ( bit_se4205575877204974255it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_5462_take__bit__unset__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se4203085406695923979it_int @ M @ A ) )
          = ( bit_se4203085406695923979it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_unset_bit_eq
thf(fact_5463_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se7882103937844011126it_nat @ M @ A ) )
          = ( bit_se7882103937844011126it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_5464_take__bit__set__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se7879613467334960850it_int @ M @ A ) )
          = ( bit_se7879613467334960850it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_set_bit_eq
thf(fact_5465_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: nat] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2925701944663578781it_nat @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se2161824704523386999it_nat @ M @ A ) )
          = ( bit_se2161824704523386999it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_5466_take__bit__flip__bit__eq,axiom,
    ! [N2: nat,M: nat,A: int] :
      ( ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ N2 @ M )
       => ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se2159334234014336723it_int @ M @ A ) )
          = ( bit_se2159334234014336723it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ) ) ).

% take_bit_flip_bit_eq
thf(fact_5467_dvd__neg__div,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5468_dvd__neg__div,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ ( uminus_uminus_real @ A ) @ B )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5469_dvd__neg__div,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ ( uminus_uminus_rat @ A ) @ B )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5470_dvd__neg__div,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5471_dvd__div__neg,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
        = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5472_dvd__div__neg,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ( ( divide_divide_real @ A @ ( uminus_uminus_real @ B ) )
        = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5473_dvd__div__neg,axiom,
    ! [B: rat,A: rat] :
      ( ( dvd_dvd_rat @ B @ A )
     => ( ( divide_divide_rat @ A @ ( uminus_uminus_rat @ B ) )
        = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5474_dvd__div__neg,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
        = ( uminus1351360451143612070nteger @ ( divide6298287555418463151nteger @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5475_subset__Compl__self__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( uminus6524753893492686040at_nat @ A2 ) )
      = ( A2 = bot_bo2099793752762293965at_nat ) ) ).

% subset_Compl_self_eq
thf(fact_5476_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_5477_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_5478_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_5479_Diff__insert__absorb,axiom,
    ! [X3: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
     => ( ( minus_3314409938677909166at_nat @ ( insert9069300056098147895at_nat @ X3 @ A2 ) @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5480_Diff__insert__absorb,axiom,
    ! [X3: set_nat,A2: set_set_nat] :
      ( ~ ( member_set_nat @ X3 @ A2 )
     => ( ( minus_2163939370556025621et_nat @ ( insert_set_nat @ X3 @ A2 ) @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5481_Diff__insert__absorb,axiom,
    ! [X3: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
     => ( ( minus_1356011639430497352at_nat @ ( insert8211810215607154385at_nat @ X3 @ A2 ) @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5482_Diff__insert__absorb,axiom,
    ! [X3: real,A2: set_real] :
      ( ~ ( member_real @ X3 @ A2 )
     => ( ( minus_minus_set_real @ ( insert_real @ X3 @ A2 ) @ ( insert_real @ X3 @ bot_bot_set_real ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5483_Diff__insert__absorb,axiom,
    ! [X3: int,A2: set_int] :
      ( ~ ( member_int @ X3 @ A2 )
     => ( ( minus_minus_set_int @ ( insert_int @ X3 @ A2 ) @ ( insert_int @ X3 @ bot_bot_set_int ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5484_Diff__insert__absorb,axiom,
    ! [X3: nat,A2: set_nat] :
      ( ~ ( member_nat @ X3 @ A2 )
     => ( ( minus_minus_set_nat @ ( insert_nat @ X3 @ A2 ) @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
        = A2 ) ) ).

% Diff_insert_absorb
thf(fact_5485_Diff__insert2,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_5486_Diff__insert2,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_5487_Diff__insert2,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_5488_Diff__insert2,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_5489_Diff__insert2,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) @ B2 ) ) ).

% Diff_insert2
thf(fact_5490_insert__Diff,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat] :
      ( ( member8757157785044589968at_nat @ A @ A2 )
     => ( ( insert9069300056098147895at_nat @ A @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5491_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_5492_insert__Diff,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ A @ A2 )
     => ( ( insert8211810215607154385at_nat @ A @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
        = A2 ) ) ).

% insert_Diff
thf(fact_5493_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_5494_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_5495_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_5496_Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ B2 ) )
      = ( minus_3314409938677909166at_nat @ ( minus_3314409938677909166at_nat @ A2 @ B2 ) @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ).

% Diff_insert
thf(fact_5497_Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ B2 ) )
      = ( minus_1356011639430497352at_nat @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ).

% Diff_insert
thf(fact_5498_Diff__insert,axiom,
    ! [A2: set_real,A: real,B2: set_real] :
      ( ( minus_minus_set_real @ A2 @ ( insert_real @ A @ B2 ) )
      = ( minus_minus_set_real @ ( minus_minus_set_real @ A2 @ B2 ) @ ( insert_real @ A @ bot_bot_set_real ) ) ) ).

% Diff_insert
thf(fact_5499_Diff__insert,axiom,
    ! [A2: set_int,A: int,B2: set_int] :
      ( ( minus_minus_set_int @ A2 @ ( insert_int @ A @ B2 ) )
      = ( minus_minus_set_int @ ( minus_minus_set_int @ A2 @ B2 ) @ ( insert_int @ A @ bot_bot_set_int ) ) ) ).

% Diff_insert
thf(fact_5500_Diff__insert,axiom,
    ! [A2: set_nat,A: nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ B2 ) )
      = ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ).

% Diff_insert
thf(fact_5501_subset__Diff__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,B2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,C4: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ ( insert9069300056098147895at_nat @ X3 @ C4 ) ) )
      = ( ( ord_le1268244103169919719at_nat @ A2 @ ( minus_3314409938677909166at_nat @ B2 @ C4 ) )
        & ~ ( member8757157785044589968at_nat @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5502_subset__Diff__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,C4: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ ( insert8211810215607154385at_nat @ X3 @ C4 ) ) )
      = ( ( ord_le3146513528884898305at_nat @ A2 @ ( minus_1356011639430497352at_nat @ B2 @ C4 ) )
        & ~ ( member8440522571783428010at_nat @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5503_subset__Diff__insert,axiom,
    ! [A2: set_real,B2: set_real,X3: real,C4: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ ( insert_real @ X3 @ C4 ) ) )
      = ( ( ord_less_eq_set_real @ A2 @ ( minus_minus_set_real @ B2 @ C4 ) )
        & ~ ( member_real @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5504_subset__Diff__insert,axiom,
    ! [A2: set_set_nat,B2: set_set_nat,X3: set_nat,C4: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ ( insert_set_nat @ X3 @ C4 ) ) )
      = ( ( ord_le6893508408891458716et_nat @ A2 @ ( minus_2163939370556025621et_nat @ B2 @ C4 ) )
        & ~ ( member_set_nat @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5505_subset__Diff__insert,axiom,
    ! [A2: set_nat,B2: set_nat,X3: nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ ( insert_nat @ X3 @ C4 ) ) )
      = ( ( ord_less_eq_set_nat @ A2 @ ( minus_minus_set_nat @ B2 @ C4 ) )
        & ~ ( member_nat @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5506_subset__Diff__insert,axiom,
    ! [A2: set_int,B2: set_int,X3: int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ ( insert_int @ X3 @ C4 ) ) )
      = ( ( ord_less_eq_set_int @ A2 @ ( minus_minus_set_int @ B2 @ C4 ) )
        & ~ ( member_int @ X3 @ A2 ) ) ) ).

% subset_Diff_insert
thf(fact_5507_real__0__less__add__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ Y3 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) ) ).

% real_0_less_add_iff
thf(fact_5508_real__add__less__0__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X3 @ Y3 ) @ zero_zero_real )
      = ( ord_less_real @ Y3 @ ( uminus_uminus_real @ X3 ) ) ) ).

% real_add_less_0_iff
thf(fact_5509_zminus1__lemma,axiom,
    ! [A: int,B: int,Q4: int,R2: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R2 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R2 = zero_zero_int ) @ ( uminus_uminus_int @ Q4 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q4 ) @ one_one_int ) ) @ ( if_int @ ( R2 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R2 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_5510_int__le__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I @ K )
     => ( ( P @ K )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_le_induct
thf(fact_5511_int__less__induct,axiom,
    ! [I: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I3: int] :
              ( ( ord_less_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_less_induct
thf(fact_5512_pos__zmult__eq__1__iff__lemma,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
     => ( ( M = one_one_int )
        | ( M
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% pos_zmult_eq_1_iff_lemma
thf(fact_5513_zmult__eq__1__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ( times_times_int @ M @ N2 )
        = one_one_int )
      = ( ( ( M = one_one_int )
          & ( N2 = one_one_int ) )
        | ( ( M
            = ( uminus_uminus_int @ one_one_int ) )
          & ( N2
            = ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zmult_eq_1_iff
thf(fact_5514_signed__take__bit__eq__take__bit__shift,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ ( plus_plus_int @ K3 @ ( 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_5515_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_5516_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_5517_atLeastatMost__psubset__iff,axiom,
    ! [A: set_int,B: set_int,C: set_int,D: set_int] :
      ( ( ord_less_set_set_int @ ( set_or370866239135849197et_int @ A @ B ) @ ( set_or370866239135849197et_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_set_int @ A @ B )
          | ( ( ord_less_eq_set_int @ C @ A )
            & ( ord_less_eq_set_int @ B @ D )
            & ( ( ord_less_set_int @ C @ A )
              | ( ord_less_set_int @ B @ D ) ) ) )
        & ( ord_less_eq_set_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5518_atLeastatMost__psubset__iff,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] :
      ( ( ord_less_set_rat @ ( set_or633870826150836451st_rat @ A @ B ) @ ( set_or633870826150836451st_rat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_rat @ A @ B )
          | ( ( ord_less_eq_rat @ C @ A )
            & ( ord_less_eq_rat @ B @ D )
            & ( ( ord_less_rat @ C @ A )
              | ( ord_less_rat @ B @ D ) ) ) )
        & ( ord_less_eq_rat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5519_atLeastatMost__psubset__iff,axiom,
    ! [A: num,B: num,C: num,D: num] :
      ( ( ord_less_set_num @ ( set_or7049704709247886629st_num @ A @ B ) @ ( set_or7049704709247886629st_num @ C @ D ) )
      = ( ( ~ ( ord_less_eq_num @ A @ B )
          | ( ( ord_less_eq_num @ C @ A )
            & ( ord_less_eq_num @ B @ D )
            & ( ( ord_less_num @ C @ A )
              | ( ord_less_num @ B @ D ) ) ) )
        & ( ord_less_eq_num @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5520_atLeastatMost__psubset__iff,axiom,
    ! [A: nat,B: nat,C: nat,D: nat] :
      ( ( ord_less_set_nat @ ( set_or1269000886237332187st_nat @ A @ B ) @ ( set_or1269000886237332187st_nat @ C @ D ) )
      = ( ( ~ ( ord_less_eq_nat @ A @ B )
          | ( ( ord_less_eq_nat @ C @ A )
            & ( ord_less_eq_nat @ B @ D )
            & ( ( ord_less_nat @ C @ A )
              | ( ord_less_nat @ B @ D ) ) ) )
        & ( ord_less_eq_nat @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5521_atLeastatMost__psubset__iff,axiom,
    ! [A: int,B: int,C: int,D: int] :
      ( ( ord_less_set_int @ ( set_or1266510415728281911st_int @ A @ B ) @ ( set_or1266510415728281911st_int @ C @ D ) )
      = ( ( ~ ( ord_less_eq_int @ A @ B )
          | ( ( ord_less_eq_int @ C @ A )
            & ( ord_less_eq_int @ B @ D )
            & ( ( ord_less_int @ C @ A )
              | ( ord_less_int @ B @ D ) ) ) )
        & ( ord_less_eq_int @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5522_atLeastatMost__psubset__iff,axiom,
    ! [A: real,B: real,C: real,D: real] :
      ( ( ord_less_set_real @ ( set_or1222579329274155063t_real @ A @ B ) @ ( set_or1222579329274155063t_real @ C @ D ) )
      = ( ( ~ ( ord_less_eq_real @ A @ B )
          | ( ( ord_less_eq_real @ C @ A )
            & ( ord_less_eq_real @ B @ D )
            & ( ( ord_less_real @ C @ A )
              | ( ord_less_real @ B @ D ) ) ) )
        & ( ord_less_eq_real @ C @ D ) ) ) ).

% atLeastatMost_psubset_iff
thf(fact_5523_take__bit__signed__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri631733984087533419it_int @ N2 @ A ) )
        = ( bit_se2923211474154528505it_int @ M @ A ) ) ) ).

% take_bit_signed_take_bit
thf(fact_5524_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_5525_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5526_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5527_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_5528_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_5529_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5530_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_le_zero
thf(fact_5531_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_5532_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_5533_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_5534_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) @ zero_zero_rat ) ).

% neg_numeral_less_zero
thf(fact_5535_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5536_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_5537_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_5538_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5539_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5540_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_5541_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_5542_le__minus__one__simps_I1_J,axiom,
    ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% le_minus_one_simps(1)
thf(fact_5543_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_5544_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_5545_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_5546_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% le_minus_one_simps(3)
thf(fact_5547_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_5548_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_5549_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_5550_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% less_minus_one_simps(3)
thf(fact_5551_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_5552_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_5553_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_5554_less__minus__one__simps_I1_J,axiom,
    ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ zero_zero_rat ).

% less_minus_one_simps(1)
thf(fact_5555_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_5556_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_le_one
thf(fact_5557_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5558_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_le_one
thf(fact_5559_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_le_one
thf(fact_5560_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_le_numeral
thf(fact_5561_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5562_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_le_numeral
thf(fact_5563_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_le_numeral
thf(fact_5564_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% neg_numeral_le_neg_one
thf(fact_5565_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% neg_numeral_le_neg_one
thf(fact_5566_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% neg_numeral_le_neg_one
thf(fact_5567_neg__numeral__le__neg__one,axiom,
    ! [M: num] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% neg_numeral_le_neg_one
thf(fact_5568_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_le_neg_one
thf(fact_5569_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5570_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_le_neg_one
thf(fact_5571_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_le_neg_one
thf(fact_5572_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5573_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5574_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5575_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5576_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5577_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5578_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5579_not__neg__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_neg_one_less_neg_numeral
thf(fact_5580_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5581_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5582_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5583_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5584_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) ) ).

% not_numeral_less_neg_one
thf(fact_5585_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) ) ).

% not_numeral_less_neg_one
thf(fact_5586_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) ) ).

% not_numeral_less_neg_one
thf(fact_5587_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5588_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ M ) ) ).

% neg_one_less_numeral
thf(fact_5589_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ M ) ) ).

% neg_one_less_numeral
thf(fact_5590_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ M ) ) ).

% neg_one_less_numeral
thf(fact_5591_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5592_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) ).

% neg_numeral_less_one
thf(fact_5593_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real ) ).

% neg_numeral_less_one
thf(fact_5594_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat ) ).

% neg_numeral_less_one
thf(fact_5595_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5596_eq__minus__divide__eq,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ A @ C )
            = ( uminus1482373934393186551omplex @ B ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5597_eq__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( A
        = ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ A @ C )
            = ( uminus_uminus_real @ B ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5598_eq__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( A
        = ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ A @ C )
            = ( uminus_uminus_rat @ B ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_minus_divide_eq
thf(fact_5599_minus__divide__eq__eq,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ B )
            = ( times_times_complex @ A @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5600_minus__divide__eq__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_real )
         => ( ( uminus_uminus_real @ B )
            = ( times_times_real @ A @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5601_minus__divide__eq__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) )
        = A )
      = ( ( ( C != zero_zero_rat )
         => ( ( uminus_uminus_rat @ B )
            = ( times_times_rat @ A @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% minus_divide_eq_eq
thf(fact_5602_nonzero__neg__divide__eq__eq,axiom,
    ! [B: complex,A: complex,C: complex] :
      ( ( B != zero_zero_complex )
     => ( ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
          = C )
        = ( ( uminus1482373934393186551omplex @ A )
          = ( times_times_complex @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5603_nonzero__neg__divide__eq__eq,axiom,
    ! [B: real,A: real,C: real] :
      ( ( B != zero_zero_real )
     => ( ( ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) )
          = C )
        = ( ( uminus_uminus_real @ A )
          = ( times_times_real @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5604_nonzero__neg__divide__eq__eq,axiom,
    ! [B: rat,A: rat,C: rat] :
      ( ( B != zero_zero_rat )
     => ( ( ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) )
          = C )
        = ( ( uminus_uminus_rat @ A )
          = ( times_times_rat @ C @ B ) ) ) ) ).

% nonzero_neg_divide_eq_eq
thf(fact_5605_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: complex,C: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( C
          = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ( times_times_complex @ C @ B )
          = ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5606_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: real,C: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( C
          = ( uminus_uminus_real @ ( divide_divide_real @ A @ B ) ) )
        = ( ( times_times_real @ C @ B )
          = ( uminus_uminus_real @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5607_nonzero__neg__divide__eq__eq2,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( C
          = ( uminus_uminus_rat @ ( divide_divide_rat @ A @ B ) ) )
        = ( ( times_times_rat @ C @ B )
          = ( uminus_uminus_rat @ A ) ) ) ) ).

% nonzero_neg_divide_eq_eq2
thf(fact_5608_divide__eq__minus__1__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( B != zero_zero_complex )
        & ( A
          = ( uminus1482373934393186551omplex @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5609_divide__eq__minus__1__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( divide_divide_real @ A @ B )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ( B != zero_zero_real )
        & ( A
          = ( uminus_uminus_real @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5610_divide__eq__minus__1__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( divide_divide_rat @ A @ B )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ( B != zero_zero_rat )
        & ( A
          = ( uminus_uminus_rat @ B ) ) ) ) ).

% divide_eq_minus_1_iff
thf(fact_5611_finite__empty__induct,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: produc3843707927480180839at_nat,A7: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A7 )
             => ( ( member8757157785044589968at_nat @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ A4 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
         => ( P @ bot_bo228742789529271731at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_5612_finite__empty__induct,axiom,
    ! [A2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: set_nat,A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( member_set_nat @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ A4 @ bot_bot_set_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_5613_finite__empty__induct,axiom,
    ! [A2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: complex,A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( member_complex @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ A4 @ bot_bot_set_complex ) ) ) ) ) )
         => ( P @ bot_bot_set_complex ) ) ) ) ).

% finite_empty_induct
thf(fact_5614_finite__empty__induct,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: product_prod_nat_nat,A7: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A7 )
             => ( ( member8440522571783428010at_nat @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ A4 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
         => ( P @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_5615_finite__empty__induct,axiom,
    ! [A2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: real,A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( member_real @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ A4 @ bot_bot_set_real ) ) ) ) ) )
         => ( P @ bot_bot_set_real ) ) ) ) ).

% finite_empty_induct
thf(fact_5616_finite__empty__induct,axiom,
    ! [A2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: int,A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( member_int @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ A4 @ bot_bot_set_int ) ) ) ) ) )
         => ( P @ bot_bot_set_int ) ) ) ) ).

% finite_empty_induct
thf(fact_5617_finite__empty__induct,axiom,
    ! [A2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( P @ A2 )
       => ( ! [A4: nat,A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( member_nat @ A4 @ A7 )
               => ( ( P @ A7 )
                 => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ A4 @ bot_bot_set_nat ) ) ) ) ) )
         => ( P @ bot_bot_set_nat ) ) ) ) ).

% finite_empty_induct
thf(fact_5618_infinite__coinduct,axiom,
    ! [X7: set_Pr4329608150637261639at_nat > $o,A2: set_Pr4329608150637261639at_nat] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_Pr4329608150637261639at_nat] :
            ( ( X7 @ A7 )
           => ? [X2: produc3843707927480180839at_nat] :
                ( ( member8757157785044589968at_nat @ X2 @ A7 )
                & ( ( X7 @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) )
                  | ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) ) ) )
       => ~ ( finite4343798906461161616at_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5619_infinite__coinduct,axiom,
    ! [X7: set_complex > $o,A2: set_complex] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_complex] :
            ( ( X7 @ A7 )
           => ? [X2: complex] :
                ( ( member_complex @ X2 @ A7 )
                & ( ( X7 @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) )
                  | ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) ) ) )
       => ~ ( finite3207457112153483333omplex @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5620_infinite__coinduct,axiom,
    ! [X7: set_Pr1261947904930325089at_nat > $o,A2: set_Pr1261947904930325089at_nat] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_Pr1261947904930325089at_nat] :
            ( ( X7 @ A7 )
           => ? [X2: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X2 @ A7 )
                & ( ( X7 @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) )
                  | ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) ) ) )
       => ~ ( finite6177210948735845034at_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5621_infinite__coinduct,axiom,
    ! [X7: set_real > $o,A2: set_real] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_real] :
            ( ( X7 @ A7 )
           => ? [X2: real] :
                ( ( member_real @ X2 @ A7 )
                & ( ( X7 @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) )
                  | ~ ( finite_finite_real @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) ) ) )
       => ~ ( finite_finite_real @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5622_infinite__coinduct,axiom,
    ! [X7: set_int > $o,A2: set_int] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_int] :
            ( ( X7 @ A7 )
           => ? [X2: int] :
                ( ( member_int @ X2 @ A7 )
                & ( ( X7 @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) )
                  | ~ ( finite_finite_int @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) ) ) )
       => ~ ( finite_finite_int @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5623_infinite__coinduct,axiom,
    ! [X7: set_nat > $o,A2: set_nat] :
      ( ( X7 @ A2 )
     => ( ! [A7: set_nat] :
            ( ( X7 @ A7 )
           => ? [X2: nat] :
                ( ( member_nat @ X2 @ A7 )
                & ( ( X7 @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) )
                  | ~ ( finite_finite_nat @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) ) ) )
       => ~ ( finite_finite_nat @ A2 ) ) ) ).

% infinite_coinduct
thf(fact_5624_infinite__remove,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,A: produc3843707927480180839at_nat] :
      ( ~ ( finite4343798906461161616at_nat @ S3 )
     => ~ ( finite4343798906461161616at_nat @ ( minus_3314409938677909166at_nat @ S3 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ) ) ).

% infinite_remove
thf(fact_5625_infinite__remove,axiom,
    ! [S3: set_complex,A: complex] :
      ( ~ ( finite3207457112153483333omplex @ S3 )
     => ~ ( finite3207457112153483333omplex @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ).

% infinite_remove
thf(fact_5626_infinite__remove,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,A: product_prod_nat_nat] :
      ( ~ ( finite6177210948735845034at_nat @ S3 )
     => ~ ( finite6177210948735845034at_nat @ ( minus_1356011639430497352at_nat @ S3 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ) ) ).

% infinite_remove
thf(fact_5627_infinite__remove,axiom,
    ! [S3: set_real,A: real] :
      ( ~ ( finite_finite_real @ S3 )
     => ~ ( finite_finite_real @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ).

% infinite_remove
thf(fact_5628_infinite__remove,axiom,
    ! [S3: set_int,A: int] :
      ( ~ ( finite_finite_int @ S3 )
     => ~ ( finite_finite_int @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ).

% infinite_remove
thf(fact_5629_infinite__remove,axiom,
    ! [S3: set_nat,A: nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ~ ( finite_finite_nat @ ( minus_minus_set_nat @ S3 @ ( insert_nat @ A @ bot_bot_set_nat ) ) ) ) ).

% infinite_remove
thf(fact_5630_Diff__single__insert,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B2 )
     => ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_5631_Diff__single__insert,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B2 )
     => ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_5632_Diff__single__insert,axiom,
    ! [A2: set_real,X3: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B2 )
     => ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_5633_Diff__single__insert,axiom,
    ! [A2: set_nat,X3: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 )
     => ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_5634_Diff__single__insert,axiom,
    ! [A2: set_int,X3: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 )
     => ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) ) ) ).

% Diff_single_insert
thf(fact_5635_subset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le1268244103169919719at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
      = ( ( ( member8757157785044589968at_nat @ X3 @ A2 )
         => ( ord_le1268244103169919719at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
        & ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
         => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5636_subset__insert__iff,axiom,
    ! [A2: set_set_nat,X3: set_nat,B2: set_set_nat] :
      ( ( ord_le6893508408891458716et_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
      = ( ( ( member_set_nat @ X3 @ A2 )
         => ( ord_le6893508408891458716et_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) @ B2 ) )
        & ( ~ ( member_set_nat @ X3 @ A2 )
         => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5637_subset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
      = ( ( ( member8440522571783428010at_nat @ X3 @ A2 )
         => ( ord_le3146513528884898305at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
        & ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
         => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5638_subset__insert__iff,axiom,
    ! [A2: set_real,X3: real,B2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
      = ( ( ( member_real @ X3 @ A2 )
         => ( ord_less_eq_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B2 ) )
        & ( ~ ( member_real @ X3 @ A2 )
         => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5639_subset__insert__iff,axiom,
    ! [A2: set_nat,X3: nat,B2: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
      = ( ( ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
        & ( ~ ( member_nat @ X3 @ A2 )
         => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5640_subset__insert__iff,axiom,
    ! [A2: set_int,X3: int,B2: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
      = ( ( ( member_int @ X3 @ A2 )
         => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 ) )
        & ( ~ ( member_int @ X3 @ A2 )
         => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ).

% subset_insert_iff
thf(fact_5641_take__bit__Suc__minus__1__eq,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( suc @ N2 ) ) @ one_one_Code_integer ) ) ).

% take_bit_Suc_minus_1_eq
thf(fact_5642_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_5643_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_5644_mod__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( modulo_modulo_int @ K @ L )
          = ( modulo_modulo_int @ ( minus_minus_int @ K @ L ) @ L ) ) ) ) ).

% mod_pos_geq
thf(fact_5645_Icc__eq__insert__lb__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( set_or1269000886237332187st_nat @ M @ N2 )
        = ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ).

% Icc_eq_insert_lb_nat
thf(fact_5646_atLeastAtMostSuc__conv,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) )
        = ( insert_nat @ ( suc @ N2 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ).

% atLeastAtMostSuc_conv
thf(fact_5647_atLeastAtMost__insertL,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( insert_nat @ M @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% atLeastAtMost_insertL
thf(fact_5648_plusinfinity,axiom,
    ! [D: int,P5: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int,K2: int] :
            ( ( P5 @ X4 )
            = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( ( ord_less_int @ Z4 @ X4 )
             => ( ( P @ X4 )
                = ( P5 @ X4 ) ) )
         => ( ? [X_12: int] : ( P5 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% plusinfinity
thf(fact_5649_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int,K2: int] :
            ( ( P1 @ X4 )
            = ( P1 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z4: int] :
            ! [X4: int] :
              ( ( ord_less_int @ X4 @ Z4 )
             => ( ( P @ X4 )
                = ( P1 @ X4 ) ) )
         => ( ? [X_12: int] : ( P1 @ X_12 )
           => ? [X_1: int] : ( P @ X_1 ) ) ) ) ) ).

% minusinfinity
thf(fact_5650_int__induct,axiom,
    ! [P: int > $o,K: int,I: int] :
      ( ( P @ K )
     => ( ! [I3: int] :
            ( ( ord_less_eq_int @ K @ I3 )
           => ( ( P @ I3 )
             => ( P @ ( plus_plus_int @ I3 @ one_one_int ) ) ) )
       => ( ! [I3: int] :
              ( ( ord_less_eq_int @ I3 @ K )
             => ( ( P @ I3 )
               => ( P @ ( minus_minus_int @ I3 @ one_one_int ) ) ) )
         => ( P @ I ) ) ) ) ).

% int_induct
thf(fact_5651_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_5652_take__bit__int__less__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_int_less_exp
thf(fact_5653_pos__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_5654_pos__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_less_eq
thf(fact_5655_pos__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_5656_pos__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_less_minus_divide_eq
thf(fact_5657_neg__minus__divide__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_5658_neg__minus__divide__less__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_less_eq
thf(fact_5659_neg__less__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_5660_neg__less__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_less_minus_divide_eq
thf(fact_5661_minus__divide__less__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_5662_minus__divide__less__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_less_eq
thf(fact_5663_less__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_5664_less__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% less_minus_divide_eq
thf(fact_5665_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W2: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5666_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5667_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5668_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W2 ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5669_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5670_eq__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5671_minus__divide__add__eq__iff,axiom,
    ! [Z2: complex,X3: complex,Y3: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5672_minus__divide__add__eq__iff,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z2 ) ) @ Y3 )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5673_minus__divide__add__eq__iff,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z2 ) ) @ Y3 )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5674_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5675_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5676_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = B ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5677_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z2 ) ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5678_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z2 ) ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5679_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z2 ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5680_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: complex,A: complex,B: complex] :
      ( ( ( Z2 = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z2 != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z2 ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5681_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: real,A: real,B: real] :
      ( ( ( Z2 = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z2 != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z2 ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5682_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z2: rat,A: rat,B: rat] :
      ( ( ( Z2 = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z2 != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z2 ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z2 ) ) @ Z2 ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5683_minus__divide__diff__eq__iff,axiom,
    ! [Z2: complex,X3: complex,Y3: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X3 @ Z2 ) ) @ Y3 )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ ( times_times_complex @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5684_minus__divide__diff__eq__iff,axiom,
    ! [Z2: real,X3: real,Y3: real] :
      ( ( Z2 != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X3 @ Z2 ) ) @ Y3 )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ ( times_times_real @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5685_minus__divide__diff__eq__iff,axiom,
    ! [Z2: rat,X3: rat,Y3: rat] :
      ( ( Z2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X3 @ Z2 ) ) @ Y3 )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X3 ) @ ( times_times_rat @ Y3 @ Z2 ) ) @ Z2 ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5686_remove__induct,axiom,
    ! [P: set_Pr4329608150637261639at_nat > $o,B2: set_Pr4329608150637261639at_nat] :
      ( ( P @ bot_bo228742789529271731at_nat )
     => ( ( ~ ( finite4343798906461161616at_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A7 )
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A7 @ B2 )
                 => ( ! [X2: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X2 @ A7 )
                       => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5687_remove__induct,axiom,
    ! [P: set_set_nat > $o,B2: set_set_nat] :
      ( ( P @ bot_bot_set_set_nat )
     => ( ( ~ ( finite1152437895449049373et_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( A7 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A7 @ B2 )
                 => ( ! [X2: set_nat] :
                        ( ( member_set_nat @ X2 @ A7 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5688_remove__induct,axiom,
    ! [P: set_complex > $o,B2: set_complex] :
      ( ( P @ bot_bot_set_complex )
     => ( ( ~ ( finite3207457112153483333omplex @ B2 )
         => ( P @ B2 ) )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B2 )
                 => ( ! [X2: complex] :
                        ( ( member_complex @ X2 @ A7 )
                       => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5689_remove__induct,axiom,
    ! [P: set_Pr1261947904930325089at_nat > $o,B2: set_Pr1261947904930325089at_nat] :
      ( ( P @ bot_bo2099793752762293965at_nat )
     => ( ( ~ ( finite6177210948735845034at_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A7 )
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A7 @ B2 )
                 => ( ! [X2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X2 @ A7 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5690_remove__induct,axiom,
    ! [P: set_real > $o,B2: set_real] :
      ( ( P @ bot_bot_set_real )
     => ( ( ~ ( finite_finite_real @ B2 )
         => ( P @ B2 ) )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B2 )
                 => ( ! [X2: real] :
                        ( ( member_real @ X2 @ A7 )
                       => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5691_remove__induct,axiom,
    ! [P: set_nat > $o,B2: set_nat] :
      ( ( P @ bot_bot_set_nat )
     => ( ( ~ ( finite_finite_nat @ B2 )
         => ( P @ B2 ) )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B2 )
                 => ( ! [X2: nat] :
                        ( ( member_nat @ X2 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5692_remove__induct,axiom,
    ! [P: set_int > $o,B2: set_int] :
      ( ( P @ bot_bot_set_int )
     => ( ( ~ ( finite_finite_int @ B2 )
         => ( P @ B2 ) )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B2 )
                 => ( ! [X2: int] :
                        ( ( member_int @ X2 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% remove_induct
thf(fact_5693_finite__remove__induct,axiom,
    ! [B2: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ B2 )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [A7: set_Pr4329608150637261639at_nat] :
              ( ( finite4343798906461161616at_nat @ A7 )
             => ( ( A7 != bot_bo228742789529271731at_nat )
               => ( ( ord_le1268244103169919719at_nat @ A7 @ B2 )
                 => ( ! [X2: produc3843707927480180839at_nat] :
                        ( ( member8757157785044589968at_nat @ X2 @ A7 )
                       => ( P @ ( minus_3314409938677909166at_nat @ A7 @ ( insert9069300056098147895at_nat @ X2 @ bot_bo228742789529271731at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5694_finite__remove__induct,axiom,
    ! [B2: set_set_nat,P: set_set_nat > $o] :
      ( ( finite1152437895449049373et_nat @ B2 )
     => ( ( P @ bot_bot_set_set_nat )
       => ( ! [A7: set_set_nat] :
              ( ( finite1152437895449049373et_nat @ A7 )
             => ( ( A7 != bot_bot_set_set_nat )
               => ( ( ord_le6893508408891458716et_nat @ A7 @ B2 )
                 => ( ! [X2: set_nat] :
                        ( ( member_set_nat @ X2 @ A7 )
                       => ( P @ ( minus_2163939370556025621et_nat @ A7 @ ( insert_set_nat @ X2 @ bot_bot_set_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5695_finite__remove__induct,axiom,
    ! [B2: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [A7: set_complex] :
              ( ( finite3207457112153483333omplex @ A7 )
             => ( ( A7 != bot_bot_set_complex )
               => ( ( ord_le211207098394363844omplex @ A7 @ B2 )
                 => ( ! [X2: complex] :
                        ( ( member_complex @ X2 @ A7 )
                       => ( P @ ( minus_811609699411566653omplex @ A7 @ ( insert_complex @ X2 @ bot_bot_set_complex ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5696_finite__remove__induct,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [A7: set_Pr1261947904930325089at_nat] :
              ( ( finite6177210948735845034at_nat @ A7 )
             => ( ( A7 != bot_bo2099793752762293965at_nat )
               => ( ( ord_le3146513528884898305at_nat @ A7 @ B2 )
                 => ( ! [X2: product_prod_nat_nat] :
                        ( ( member8440522571783428010at_nat @ X2 @ A7 )
                       => ( P @ ( minus_1356011639430497352at_nat @ A7 @ ( insert8211810215607154385at_nat @ X2 @ bot_bo2099793752762293965at_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5697_finite__remove__induct,axiom,
    ! [B2: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ B2 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [A7: set_real] :
              ( ( finite_finite_real @ A7 )
             => ( ( A7 != bot_bot_set_real )
               => ( ( ord_less_eq_set_real @ A7 @ B2 )
                 => ( ! [X2: real] :
                        ( ( member_real @ X2 @ A7 )
                       => ( P @ ( minus_minus_set_real @ A7 @ ( insert_real @ X2 @ bot_bot_set_real ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5698_finite__remove__induct,axiom,
    ! [B2: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ B2 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [A7: set_nat] :
              ( ( finite_finite_nat @ A7 )
             => ( ( A7 != bot_bot_set_nat )
               => ( ( ord_less_eq_set_nat @ A7 @ B2 )
                 => ( ! [X2: nat] :
                        ( ( member_nat @ X2 @ A7 )
                       => ( P @ ( minus_minus_set_nat @ A7 @ ( insert_nat @ X2 @ bot_bot_set_nat ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5699_finite__remove__induct,axiom,
    ! [B2: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ B2 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [A7: set_int] :
              ( ( finite_finite_int @ A7 )
             => ( ( A7 != bot_bot_set_int )
               => ( ( ord_less_eq_set_int @ A7 @ B2 )
                 => ( ! [X2: int] :
                        ( ( member_int @ X2 @ A7 )
                       => ( P @ ( minus_minus_set_int @ A7 @ ( insert_int @ X2 @ bot_bot_set_int ) ) ) )
                   => ( P @ A7 ) ) ) ) )
         => ( P @ B2 ) ) ) ) ).

% finite_remove_induct
thf(fact_5700_signed__take__bit__int__greater__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_greater_self_iff
thf(fact_5701_finite__induct__select,axiom,
    ! [S3: set_Pr4329608150637261639at_nat,P: set_Pr4329608150637261639at_nat > $o] :
      ( ( finite4343798906461161616at_nat @ S3 )
     => ( ( P @ bot_bo228742789529271731at_nat )
       => ( ! [T4: set_Pr4329608150637261639at_nat] :
              ( ( ord_le2604355607129572851at_nat @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X2: produc3843707927480180839at_nat] :
                    ( ( member8757157785044589968at_nat @ X2 @ ( minus_3314409938677909166at_nat @ S3 @ T4 ) )
                    & ( P @ ( insert9069300056098147895at_nat @ X2 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5702_finite__induct__select,axiom,
    ! [S3: set_complex,P: set_complex > $o] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( P @ bot_bot_set_complex )
       => ( ! [T4: set_complex] :
              ( ( ord_less_set_complex @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X2: complex] :
                    ( ( member_complex @ X2 @ ( minus_811609699411566653omplex @ S3 @ T4 ) )
                    & ( P @ ( insert_complex @ X2 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5703_finite__induct__select,axiom,
    ! [S3: set_Pr1261947904930325089at_nat,P: set_Pr1261947904930325089at_nat > $o] :
      ( ( finite6177210948735845034at_nat @ S3 )
     => ( ( P @ bot_bo2099793752762293965at_nat )
       => ( ! [T4: set_Pr1261947904930325089at_nat] :
              ( ( ord_le7866589430770878221at_nat @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X2: product_prod_nat_nat] :
                    ( ( member8440522571783428010at_nat @ X2 @ ( minus_1356011639430497352at_nat @ S3 @ T4 ) )
                    & ( P @ ( insert8211810215607154385at_nat @ X2 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5704_finite__induct__select,axiom,
    ! [S3: set_real,P: set_real > $o] :
      ( ( finite_finite_real @ S3 )
     => ( ( P @ bot_bot_set_real )
       => ( ! [T4: set_real] :
              ( ( ord_less_set_real @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X2: real] :
                    ( ( member_real @ X2 @ ( minus_minus_set_real @ S3 @ T4 ) )
                    & ( P @ ( insert_real @ X2 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5705_finite__induct__select,axiom,
    ! [S3: set_int,P: set_int > $o] :
      ( ( finite_finite_int @ S3 )
     => ( ( P @ bot_bot_set_int )
       => ( ! [T4: set_int] :
              ( ( ord_less_set_int @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X2: int] :
                    ( ( member_int @ X2 @ ( minus_minus_set_int @ S3 @ T4 ) )
                    & ( P @ ( insert_int @ X2 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5706_finite__induct__select,axiom,
    ! [S3: set_nat,P: set_nat > $o] :
      ( ( finite_finite_nat @ S3 )
     => ( ( P @ bot_bot_set_nat )
       => ( ! [T4: set_nat] :
              ( ( ord_less_set_nat @ T4 @ S3 )
             => ( ( P @ T4 )
               => ? [X2: nat] :
                    ( ( member_nat @ X2 @ ( minus_minus_set_nat @ S3 @ T4 ) )
                    & ( P @ ( insert_nat @ X2 @ T4 ) ) ) ) )
         => ( P @ S3 ) ) ) ) ).

% finite_induct_select
thf(fact_5707_psubset__insert__iff,axiom,
    ! [A2: set_Pr4329608150637261639at_nat,X3: produc3843707927480180839at_nat,B2: set_Pr4329608150637261639at_nat] :
      ( ( ord_le2604355607129572851at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ B2 ) )
      = ( ( ( member8757157785044589968at_nat @ X3 @ B2 )
         => ( ord_le2604355607129572851at_nat @ A2 @ B2 ) )
        & ( ~ ( member8757157785044589968at_nat @ X3 @ B2 )
         => ( ( ( member8757157785044589968at_nat @ X3 @ A2 )
             => ( ord_le2604355607129572851at_nat @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) ) @ B2 ) )
            & ( ~ ( member8757157785044589968at_nat @ X3 @ A2 )
             => ( ord_le1268244103169919719at_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5708_psubset__insert__iff,axiom,
    ! [A2: set_set_nat,X3: set_nat,B2: set_set_nat] :
      ( ( ord_less_set_set_nat @ A2 @ ( insert_set_nat @ X3 @ B2 ) )
      = ( ( ( member_set_nat @ X3 @ B2 )
         => ( ord_less_set_set_nat @ A2 @ B2 ) )
        & ( ~ ( member_set_nat @ X3 @ B2 )
         => ( ( ( member_set_nat @ X3 @ A2 )
             => ( ord_less_set_set_nat @ ( minus_2163939370556025621et_nat @ A2 @ ( insert_set_nat @ X3 @ bot_bot_set_set_nat ) ) @ B2 ) )
            & ( ~ ( member_set_nat @ X3 @ A2 )
             => ( ord_le6893508408891458716et_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5709_psubset__insert__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ B2 ) )
      = ( ( ( member8440522571783428010at_nat @ X3 @ B2 )
         => ( ord_le7866589430770878221at_nat @ A2 @ B2 ) )
        & ( ~ ( member8440522571783428010at_nat @ X3 @ B2 )
         => ( ( ( member8440522571783428010at_nat @ X3 @ A2 )
             => ( ord_le7866589430770878221at_nat @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) ) @ B2 ) )
            & ( ~ ( member8440522571783428010at_nat @ X3 @ A2 )
             => ( ord_le3146513528884898305at_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5710_psubset__insert__iff,axiom,
    ! [A2: set_real,X3: real,B2: set_real] :
      ( ( ord_less_set_real @ A2 @ ( insert_real @ X3 @ B2 ) )
      = ( ( ( member_real @ X3 @ B2 )
         => ( ord_less_set_real @ A2 @ B2 ) )
        & ( ~ ( member_real @ X3 @ B2 )
         => ( ( ( member_real @ X3 @ A2 )
             => ( ord_less_set_real @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) @ B2 ) )
            & ( ~ ( member_real @ X3 @ A2 )
             => ( ord_less_eq_set_real @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5711_psubset__insert__iff,axiom,
    ! [A2: set_nat,X3: nat,B2: set_nat] :
      ( ( ord_less_set_nat @ A2 @ ( insert_nat @ X3 @ B2 ) )
      = ( ( ( member_nat @ X3 @ B2 )
         => ( ord_less_set_nat @ A2 @ B2 ) )
        & ( ~ ( member_nat @ X3 @ B2 )
         => ( ( ( member_nat @ X3 @ A2 )
             => ( ord_less_set_nat @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ X3 @ bot_bot_set_nat ) ) @ B2 ) )
            & ( ~ ( member_nat @ X3 @ A2 )
             => ( ord_less_eq_set_nat @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5712_psubset__insert__iff,axiom,
    ! [A2: set_int,X3: int,B2: set_int] :
      ( ( ord_less_set_int @ A2 @ ( insert_int @ X3 @ B2 ) )
      = ( ( ( member_int @ X3 @ B2 )
         => ( ord_less_set_int @ A2 @ B2 ) )
        & ( ~ ( member_int @ X3 @ B2 )
         => ( ( ( member_int @ X3 @ A2 )
             => ( ord_less_set_int @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) @ B2 ) )
            & ( ~ ( member_int @ X3 @ A2 )
             => ( ord_less_eq_set_int @ A2 @ B2 ) ) ) ) ) ) ).

% psubset_insert_iff
thf(fact_5713_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int] :
            ( ( P @ X4 )
           => ( P @ ( minus_minus_int @ X4 @ D ) ) )
       => ( ( ord_less_eq_int @ zero_zero_int @ K )
         => ! [X2: int] :
              ( ( P @ X2 )
             => ( P @ ( minus_minus_int @ X2 @ ( times_times_int @ K @ D ) ) ) ) ) ) ) ).

% decr_mult_lemma
thf(fact_5714_verit__less__mono__div__int2,axiom,
    ! [A2: int,B2: int,N2: int] :
      ( ( ord_less_eq_int @ A2 @ B2 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B2 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_5715_div__eq__minus1,axiom,
    ! [B: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ B )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% div_eq_minus1
thf(fact_5716_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_5717_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_5718_take__bit__nat__eq__self,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M ) ) ).

% take_bit_nat_eq_self
thf(fact_5719_take__bit__nat__less__exp,axiom,
    ! [N2: nat,M: nat] : ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% take_bit_nat_less_exp
thf(fact_5720_take__bit__nat__eq__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( bit_se2925701944663578781it_nat @ N2 @ M )
        = M )
      = ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_nat_eq_self_iff
thf(fact_5721_take__bit__int__less__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ K ) ) ).

% take_bit_int_less_self_iff
thf(fact_5722_take__bit__int__greater__eq__self__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ K @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
      = ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_int_greater_eq_self_iff
thf(fact_5723_le__minus__divide__eq,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ A @ zero_zero_real ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_5724_le__minus__divide__eq,axiom,
    ! [A: rat,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ) ) ) ) ).

% le_minus_divide_eq
thf(fact_5725_minus__divide__le__eq,axiom,
    ! [B: real,C: real,A: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_5726_minus__divide__le__eq,axiom,
    ! [B: rat,C: rat,A: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ) ) ).

% minus_divide_le_eq
thf(fact_5727_neg__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_5728_neg__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% neg_le_minus_divide_eq
thf(fact_5729_neg__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_5730_neg__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ C @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% neg_minus_divide_le_eq
thf(fact_5731_pos__le__minus__divide__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) )
        = ( ord_less_eq_real @ ( times_times_real @ A @ C ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_5732_pos__le__minus__divide__eq,axiom,
    ! [C: rat,A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ A @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) )
        = ( ord_less_eq_rat @ ( times_times_rat @ A @ C ) @ ( uminus_uminus_rat @ B ) ) ) ) ).

% pos_le_minus_divide_eq
thf(fact_5733_pos__minus__divide__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ B @ C ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( times_times_real @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_5734_pos__minus__divide__le__eq,axiom,
    ! [C: rat,B: rat,A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ C )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ B @ C ) ) @ A )
        = ( ord_less_eq_rat @ ( uminus_uminus_rat @ B ) @ ( times_times_rat @ A @ C ) ) ) ) ).

% pos_minus_divide_le_eq
thf(fact_5735_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5736_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5737_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5738_less__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5739_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_5740_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_5741_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_5742_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_5743_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5744_neg__one__power__add__eq__neg__one__power__diff,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( plus_plus_nat @ N2 @ K ) )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( minus_minus_nat @ N2 @ K ) ) ) ) ).

% neg_one_power_add_eq_neg_one_power_diff
thf(fact_5745_signed__take__bit__int__eq__self,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_ri631733984087533419it_int @ N2 @ K )
          = K ) ) ) ).

% signed_take_bit_int_eq_self
thf(fact_5746_signed__take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_ri631733984087533419it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% signed_take_bit_int_eq_self_iff
thf(fact_5747_take__bit__eq__0__iff,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( ( bit_se1745604003318907178nteger @ N2 @ A )
        = zero_z3403309356797280102nteger )
      = ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ A ) ) ).

% take_bit_eq_0_iff
thf(fact_5748_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_5749_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_5750_take__bit__nat__less__self__iff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) @ M )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M ) ) ).

% take_bit_nat_less_self_iff
thf(fact_5751_take__bit__int__eq__self,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( bit_se2923211474154528505it_int @ N2 @ K )
          = K ) ) ) ).

% take_bit_int_eq_self
thf(fact_5752_take__bit__int__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = K )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ).

% take_bit_int_eq_self_iff
thf(fact_5753_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W2: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5754_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W2: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5755_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W2 ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5756_le__divide__eq__numeral_I2_J,axiom,
    ! [W2: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) @ B ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W2 ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5757_square__le__1,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_5758_square__le__1,axiom,
    ! [X3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X3 )
     => ( ( ord_le3102999989581377725nteger @ X3 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_5759_square__le__1,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_5760_square__le__1,axiom,
    ! [X3: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X3 )
     => ( ( ord_less_eq_int @ X3 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_5761_div__pos__geq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ L )
     => ( ( ord_less_eq_int @ L @ K )
       => ( ( divide_divide_int @ K @ L )
          = ( plus_plus_int @ ( divide_divide_int @ ( minus_minus_int @ K @ L ) @ L ) @ one_one_int ) ) ) ) ).

% div_pos_geq
thf(fact_5762_div__pos__neg__trivial,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_eq_int @ ( plus_plus_int @ K @ L ) @ zero_zero_int )
       => ( ( divide_divide_int @ K @ L )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% div_pos_neg_trivial
thf(fact_5763_take__bit__int__greater__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_int @ K @ zero_zero_int )
     => ( ord_less_eq_int @ ( plus_plus_int @ K @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_int_greater_eq
thf(fact_5764_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_5765_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_5766_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_5767_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus_uminus_rat @ one_one_rat ) ) ).

% power_minus1_odd
thf(fact_5768_power__minus1__odd,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% power_minus1_odd
thf(fact_5769_take__bit__Suc,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se1745604003318907178nteger @ ( suc @ N2 ) @ A )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( bit_se1745604003318907178nteger @ N2 @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ).

% take_bit_Suc
thf(fact_5770_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_5771_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_5772_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_5773_stable__imp__take__bit__eq,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N2 @ A )
            = zero_z3403309356797280102nteger ) )
        & ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
         => ( ( bit_se1745604003318907178nteger @ N2 @ A )
            = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ one_one_Code_integer ) ) ) ) ) ).

% stable_imp_take_bit_eq
thf(fact_5774_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_5775_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_5776_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_5777_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_5778_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A4: real,B4: real,C2: real] :
            ( ( P @ A4 @ B4 )
           => ( ( P @ B4 @ C2 )
             => ( ( ord_less_eq_real @ A4 @ B4 )
               => ( ( ord_less_eq_real @ B4 @ C2 )
                 => ( P @ A4 @ C2 ) ) ) ) )
       => ( ! [X4: real] :
              ( ( ord_less_eq_real @ A @ X4 )
             => ( ( ord_less_eq_real @ X4 @ B )
               => ? [D3: real] :
                    ( ( ord_less_real @ zero_zero_real @ D3 )
                    & ! [A4: real,B4: real] :
                        ( ( ( ord_less_eq_real @ A4 @ X4 )
                          & ( ord_less_eq_real @ X4 @ B4 )
                          & ( ord_less_real @ ( minus_minus_real @ B4 @ A4 ) @ D3 ) )
                       => ( P @ A4 @ B4 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_5779_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_5780_and__int_Oelims,axiom,
    ! [X3: int,Xa2: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( ( member_int @ X3 @ ( 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 ) ) ) )
         => ( Y3
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
        & ( ~ ( ( member_int @ X3 @ ( 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 ) ) ) )
         => ( Y3
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_5781_and__int_Osimps,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( member_int @ K3 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ 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 ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) ) )
          @ ( plus_plus_int
            @ ( zero_n2684676970156552555ol_int
              @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 )
                & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L2 ) ) )
            @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_int.simps
thf(fact_5782_Diff__idemp,axiom,
    ! [A2: set_nat,B2: set_nat] :
      ( ( minus_minus_set_nat @ ( minus_minus_set_nat @ A2 @ B2 ) @ B2 )
      = ( minus_minus_set_nat @ A2 @ B2 ) ) ).

% Diff_idemp
thf(fact_5783_Diff__iff,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
      = ( ( member8440522571783428010at_nat @ C @ A2 )
        & ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_5784_Diff__iff,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
      = ( ( member_real @ C @ A2 )
        & ~ ( member_real @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_5785_Diff__iff,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
      = ( ( member_set_nat @ C @ A2 )
        & ~ ( member_set_nat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_5786_Diff__iff,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
      = ( ( member_int @ C @ A2 )
        & ~ ( member_int @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_5787_Diff__iff,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
      = ( ( member_nat @ C @ A2 )
        & ~ ( member_nat @ C @ B2 ) ) ) ).

% Diff_iff
thf(fact_5788_DiffI,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ A2 )
     => ( ~ ( member8440522571783428010at_nat @ C @ B2 )
       => ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_5789_DiffI,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ A2 )
     => ( ~ ( member_real @ C @ B2 )
       => ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_5790_DiffI,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ A2 )
     => ( ~ ( member_set_nat @ C @ B2 )
       => ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_5791_DiffI,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ A2 )
     => ( ~ ( member_int @ C @ B2 )
       => ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_5792_DiffI,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ A2 )
     => ( ~ ( member_nat @ C @ B2 )
       => ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) ) ) ) ).

% DiffI
thf(fact_5793_ComplI,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ~ ( member8440522571783428010at_nat @ C @ A2 )
     => ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) ) ) ).

% ComplI
thf(fact_5794_ComplI,axiom,
    ! [C: real,A2: set_real] :
      ( ~ ( member_real @ C @ A2 )
     => ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) ) ) ).

% ComplI
thf(fact_5795_ComplI,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ~ ( member_set_nat @ C @ A2 )
     => ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_5796_ComplI,axiom,
    ! [C: nat,A2: set_nat] :
      ( ~ ( member_nat @ C @ A2 )
     => ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) ) ) ).

% ComplI
thf(fact_5797_ComplI,axiom,
    ! [C: int,A2: set_int] :
      ( ~ ( member_int @ C @ A2 )
     => ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% ComplI
thf(fact_5798_Compl__iff,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
      = ( ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_5799_Compl__iff,axiom,
    ! [C: real,A2: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
      = ( ~ ( member_real @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_5800_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_5801_Compl__iff,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
      = ( ~ ( member_nat @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_5802_Compl__iff,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
      = ( ~ ( member_int @ C @ A2 ) ) ) ).

% Compl_iff
thf(fact_5803_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_5804_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_5805_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_5806_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_5807_bit_Oconj__zero__left,axiom,
    ! [X3: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X3 )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_5808_bit_Oconj__zero__right,axiom,
    ! [X3: int] :
      ( ( bit_se725231765392027082nd_int @ X3 @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_5809_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_5810_semiring__norm_I80_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(80)
thf(fact_5811_and__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        & ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% and_negative_int_iff
thf(fact_5812_semiring__norm_I81_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(81)
thf(fact_5813_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_5814_concat__bit__negative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_int @ ( bit_concat_bit @ N2 @ K @ L ) @ zero_zero_int )
      = ( ord_less_int @ L @ zero_zero_int ) ) ).

% concat_bit_negative_iff
thf(fact_5815_semiring__norm_I79_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(79)
thf(fact_5816_semiring__norm_I74_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% semiring_norm(74)
thf(fact_5817_and__numerals_I5_J,axiom,
    ! [X3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_5818_and__numerals_I5_J,axiom,
    ! [X3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_5819_and__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_5820_and__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_5821_div__Suc__eq__div__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( divide_divide_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% div_Suc_eq_div_add3
thf(fact_5822_Suc__div__eq__add3__div__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_div_eq_add3_div_numeral
thf(fact_5823_mod__Suc__eq__mod__add3,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ ( suc @ ( suc @ N2 ) ) ) )
      = ( modulo_modulo_nat @ M @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ N2 ) ) ) ).

% mod_Suc_eq_mod_add3
thf(fact_5824_Suc__mod__eq__add3__mod__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ ( numeral_numeral_nat @ V ) )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ ( numeral_numeral_nat @ V ) ) ) ).

% Suc_mod_eq_add3_mod_numeral
thf(fact_5825_and__numerals_I7_J,axiom,
    ! [X3: num,Y3: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% and_numerals(7)
thf(fact_5826_and__numerals_I7_J,axiom,
    ! [X3: num,Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% and_numerals(7)
thf(fact_5827_DiffD2,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
     => ~ ( member8440522571783428010at_nat @ C @ B2 ) ) ).

% DiffD2
thf(fact_5828_DiffD2,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
     => ~ ( member_real @ C @ B2 ) ) ).

% DiffD2
thf(fact_5829_DiffD2,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
     => ~ ( member_set_nat @ C @ B2 ) ) ).

% DiffD2
thf(fact_5830_DiffD2,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
     => ~ ( member_int @ C @ B2 ) ) ).

% DiffD2
thf(fact_5831_DiffD2,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
     => ~ ( member_nat @ C @ B2 ) ) ).

% DiffD2
thf(fact_5832_DiffD1,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
     => ( member8440522571783428010at_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_5833_DiffD1,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
     => ( member_real @ C @ A2 ) ) ).

% DiffD1
thf(fact_5834_DiffD1,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
     => ( member_set_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_5835_DiffD1,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
     => ( member_int @ C @ A2 ) ) ).

% DiffD1
thf(fact_5836_DiffD1,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
     => ( member_nat @ C @ A2 ) ) ).

% DiffD1
thf(fact_5837_DiffE,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,B2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
     => ~ ( ( member8440522571783428010at_nat @ C @ A2 )
         => ( member8440522571783428010at_nat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_5838_DiffE,axiom,
    ! [C: real,A2: set_real,B2: set_real] :
      ( ( member_real @ C @ ( minus_minus_set_real @ A2 @ B2 ) )
     => ~ ( ( member_real @ C @ A2 )
         => ( member_real @ C @ B2 ) ) ) ).

% DiffE
thf(fact_5839_DiffE,axiom,
    ! [C: set_nat,A2: set_set_nat,B2: set_set_nat] :
      ( ( member_set_nat @ C @ ( minus_2163939370556025621et_nat @ A2 @ B2 ) )
     => ~ ( ( member_set_nat @ C @ A2 )
         => ( member_set_nat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_5840_DiffE,axiom,
    ! [C: int,A2: set_int,B2: set_int] :
      ( ( member_int @ C @ ( minus_minus_set_int @ A2 @ B2 ) )
     => ~ ( ( member_int @ C @ A2 )
         => ( member_int @ C @ B2 ) ) ) ).

% DiffE
thf(fact_5841_DiffE,axiom,
    ! [C: nat,A2: set_nat,B2: set_nat] :
      ( ( member_nat @ C @ ( minus_minus_set_nat @ A2 @ B2 ) )
     => ~ ( ( member_nat @ C @ A2 )
         => ( member_nat @ C @ B2 ) ) ) ).

% DiffE
thf(fact_5842_ComplD,axiom,
    ! [C: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ C @ ( uminus6524753893492686040at_nat @ A2 ) )
     => ~ ( member8440522571783428010at_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_5843_ComplD,axiom,
    ! [C: real,A2: set_real] :
      ( ( member_real @ C @ ( uminus612125837232591019t_real @ A2 ) )
     => ~ ( member_real @ C @ A2 ) ) ).

% ComplD
thf(fact_5844_ComplD,axiom,
    ! [C: set_nat,A2: set_set_nat] :
      ( ( member_set_nat @ C @ ( uminus613421341184616069et_nat @ A2 ) )
     => ~ ( member_set_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_5845_ComplD,axiom,
    ! [C: nat,A2: set_nat] :
      ( ( member_nat @ C @ ( uminus5710092332889474511et_nat @ A2 ) )
     => ~ ( member_nat @ C @ A2 ) ) ).

% ComplD
thf(fact_5846_ComplD,axiom,
    ! [C: int,A2: set_int] :
      ( ( member_int @ C @ ( uminus1532241313380277803et_int @ A2 ) )
     => ~ ( member_int @ C @ A2 ) ) ).

% ComplD
thf(fact_5847_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L: int,R2: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R2 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R2 ) ) ).

% concat_bit_assoc
thf(fact_5848_xor__num_Ocases,axiom,
    ! [X3: product_prod_num_num] :
      ( ( X3
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N3: num] :
            ( X3
           != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
       => ( ! [N3: num] :
              ( X3
             != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
         => ( ! [M3: num] :
                ( X3
               != ( product_Pair_num_num @ ( bit0 @ M3 ) @ one ) )
           => ( ! [M3: num,N3: num] :
                  ( X3
                 != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit0 @ N3 ) ) )
             => ( ! [M3: num,N3: num] :
                    ( X3
                   != ( product_Pair_num_num @ ( bit0 @ M3 ) @ ( bit1 @ N3 ) ) )
               => ( ! [M3: num] :
                      ( X3
                     != ( product_Pair_num_num @ ( bit1 @ M3 ) @ one ) )
                 => ( ! [M3: num,N3: num] :
                        ( X3
                       != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit0 @ N3 ) ) )
                   => ~ ! [M3: num,N3: num] :
                          ( X3
                         != ( product_Pair_num_num @ ( bit1 @ M3 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_5849_and__less__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_int @ L @ zero_zero_int )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ K ) ) ).

% and_less_eq
thf(fact_5850_AND__upper1_H_H,axiom,
    ! [Y3: int,Z2: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z2 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y3 @ Ya ) @ Z2 ) ) ) ).

% AND_upper1''
thf(fact_5851_AND__upper2_H_H,axiom,
    ! [Y3: int,Z2: int,X3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_int @ Y3 @ Z2 )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X3 @ Y3 ) @ Z2 ) ) ) ).

% AND_upper2''
thf(fact_5852_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_5853_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_5854_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_5855_numeral__Bit1,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_Bit1
thf(fact_5856_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_5857_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_5858_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I2 ) @ bot_bot_set_int @ ( insert_int @ I2 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_5859_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_5860_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_5861_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_5862_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q4: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_5863_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_5864_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_5865_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_5866_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_5867_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q4 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_5868_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q4 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_5869_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q4: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_5870_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q4 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5871_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q4 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5872_cong__exp__iff__simps_I7_J,axiom,
    ! [Q4: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q4 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q4 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5873_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X4: int,K2: int] :
            ( ( P @ X4 )
            = ( P @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X5: int] : ( P @ X5 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_5874_bset_I3_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X2 = T )
             => ( ( minus_minus_int @ X2 @ D6 )
                = T ) ) ) ) ) ).

% bset(3)
thf(fact_5875_bset_I4_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ( member_int @ T @ B2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X2 != T )
             => ( ( minus_minus_int @ X2 @ D6 )
               != T ) ) ) ) ) ).

% bset(4)
thf(fact_5876_bset_I5_J,axiom,
    ! [D6: int,B2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B2 )
                 => ( X2
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ X2 @ T )
           => ( ord_less_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).

% bset(5)
thf(fact_5877_bset_I7_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ( member_int @ T @ B2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ T @ X2 )
             => ( ord_less_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).

% bset(7)
thf(fact_5878_aset_I3_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X2 = T )
             => ( ( plus_plus_int @ X2 @ D6 )
                = T ) ) ) ) ) ).

% aset(3)
thf(fact_5879_aset_I4_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ( member_int @ T @ A2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( X2 != T )
             => ( ( plus_plus_int @ X2 @ D6 )
               != T ) ) ) ) ) ).

% aset(4)
thf(fact_5880_aset_I5_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ( member_int @ T @ A2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_int @ X2 @ T )
             => ( ord_less_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).

% aset(5)
thf(fact_5881_aset_I7_J,axiom,
    ! [D6: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X2
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_int @ T @ X2 )
           => ( ord_less_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).

% aset(7)
thf(fact_5882_Suc__div__eq__add3__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( divide_divide_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( divide_divide_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_div_eq_add3_div
thf(fact_5883_Suc__mod__eq__add3__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( suc @ ( suc @ ( suc @ M ) ) ) @ N2 )
      = ( modulo_modulo_nat @ ( plus_plus_nat @ ( numeral_numeral_nat @ ( bit1 @ one ) ) @ M ) @ N2 ) ) ).

% Suc_mod_eq_add3_mod
thf(fact_5884_aset_I8_J,axiom,
    ! [D6: int,A2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X2
                   != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ T @ X2 )
           => ( ord_less_eq_int @ T @ ( plus_plus_int @ X2 @ D6 ) ) ) ) ) ).

% aset(8)
thf(fact_5885_aset_I6_J,axiom,
    ! [D6: int,T: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ( member_int @ ( plus_plus_int @ T @ one_one_int ) @ A2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ X2 @ T )
             => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D6 ) @ T ) ) ) ) ) ).

% aset(6)
thf(fact_5886_bset_I8_J,axiom,
    ! [D6: int,T: int,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ( member_int @ ( minus_minus_int @ T @ one_one_int ) @ B2 )
       => ! [X2: int] :
            ( ! [Xa3: int] :
                ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B2 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
           => ( ( ord_less_eq_int @ T @ X2 )
             => ( ord_less_eq_int @ T @ ( minus_minus_int @ X2 @ D6 ) ) ) ) ) ) ).

% bset(8)
thf(fact_5887_bset_I6_J,axiom,
    ! [D6: int,B2: set_int,T: int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ! [X2: int] :
          ( ! [Xa3: int] :
              ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B2 )
                 => ( X2
                   != ( plus_plus_int @ Xb2 @ Xa3 ) ) ) )
         => ( ( ord_less_eq_int @ X2 @ T )
           => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D6 ) @ T ) ) ) ) ).

% bset(6)
thf(fact_5888_cppi,axiom,
    ! [D6: int,P: int > $o,P5: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ Z4 @ X4 )
           => ( ( P @ X4 )
              = ( P5 @ X4 ) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ A2 )
                     => ( X4
                       != ( minus_minus_int @ Xb3 @ Xa ) ) ) )
             => ( ( P @ X4 )
               => ( P @ ( plus_plus_int @ X4 @ D6 ) ) ) )
         => ( ! [X4: int,K2: int] :
                ( ( P5 @ X4 )
                = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
           => ( ( ? [X5: int] : ( P @ X5 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
                    & ( P5 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ A2 )
                        & ( P @ ( minus_minus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_5889_cpmi,axiom,
    ! [D6: int,P: int > $o,P5: int > $o,B2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D6 )
     => ( ? [Z4: int] :
          ! [X4: int] :
            ( ( ord_less_int @ X4 @ Z4 )
           => ( ( P @ X4 )
              = ( P5 @ X4 ) ) )
       => ( ! [X4: int] :
              ( ! [Xa: int] :
                  ( ( member_int @ Xa @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ B2 )
                     => ( X4
                       != ( plus_plus_int @ Xb3 @ Xa ) ) ) )
             => ( ( P @ X4 )
               => ( P @ ( minus_minus_int @ X4 @ D6 ) ) ) )
         => ( ! [X4: int,K2: int] :
                ( ( P5 @ X4 )
                = ( P5 @ ( minus_minus_int @ X4 @ ( times_times_int @ K2 @ D6 ) ) ) )
           => ( ( ? [X5: int] : ( P @ X5 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
                    & ( P5 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D6 ) )
                    & ? [Y: int] :
                        ( ( member_int @ Y @ B2 )
                        & ( P @ ( plus_plus_int @ Y @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_5890_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_5891_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_5892_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_5893_mod__exhaust__less__4,axiom,
    ! [M: nat] :
      ( ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = zero_zero_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = one_one_nat )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      | ( ( modulo_modulo_nat @ M @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
        = ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) ).

% mod_exhaust_less_4
thf(fact_5894_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_5895_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_5896_divmod__algorithm__code_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) ) ) )
      & ( ~ ( ord_less_eq_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(7)
thf(fact_5897_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5026877609467782581ep_nat @ ( bit1 @ N2 ) @ ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_5898_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique5024387138958732305ep_int @ ( bit1 @ N2 ) @ ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_5899_divmod__algorithm__code_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) ) ) )
      & ( ~ ( ord_less_num @ M @ N2 )
       => ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
          = ( unique4921790084139445826nteger @ ( bit1 @ N2 ) @ ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ ( bit1 @ N2 ) ) ) ) ) ) ) ).

% divmod_algorithm_code(8)
thf(fact_5900_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_5901_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_5902_and__int_Opelims,axiom,
    ! [X3: int,Xa2: int,Y3: int] :
      ( ( ( bit_se725231765392027082nd_int @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
       => ~ ( ( ( ( ( member_int @ X3 @ ( 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 ) ) ) )
               => ( Y3
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) ) ) ) )
              & ( ~ ( ( member_int @ X3 @ ( 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 ) ) ) )
               => ( Y3
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X3 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa2 ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X3 @ ( 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 @ X3 @ Xa2 ) ) ) ) ) ).

% and_int.pelims
thf(fact_5903_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_5904_and__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_5905_and__nat__numerals_I3_J,axiom,
    ! [X3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_5906_and__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_5907_and__nat__numerals_I4_J,axiom,
    ! [X3: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_5908_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ M @ one )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ M ) @ zero_zero_int ) ) ).

% divmod_algorithm_code(2)
thf(fact_5909_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique5055182867167087721od_nat @ M @ one )
      = ( product_Pair_nat_nat @ ( numeral_numeral_nat @ M ) @ zero_zero_nat ) ) ).

% divmod_algorithm_code(2)
thf(fact_5910_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_5911_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5912_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5913_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5914_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5915_add__neg__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ) ).

% add_neg_numeral_special(6)
thf(fact_5916_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_5917_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_5918_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_5919_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_5920_add__neg__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).

% add_neg_numeral_special(5)
thf(fact_5921_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_5922_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_5923_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_5924_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_5925_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_5926_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_5927_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_5928_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_5929_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numera6690914467698888265omplex @ ( inc @ X3 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X3 ) @ one_one_complex ) ) ).

% numeral_inc
thf(fact_5930_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numera1916890842035813515d_enat @ ( inc @ X3 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X3 ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_inc
thf(fact_5931_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numeral_numeral_real @ ( inc @ X3 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X3 ) @ one_one_real ) ) ).

% numeral_inc
thf(fact_5932_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numeral_numeral_rat @ ( inc @ X3 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X3 ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_5933_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numeral_numeral_nat @ ( inc @ X3 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X3 ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_5934_numeral__inc,axiom,
    ! [X3: num] :
      ( ( numeral_numeral_int @ ( inc @ X3 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X3 ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_5935_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M2: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_int_def
thf(fact_5936_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M2: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M2 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5937_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M2: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5938_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M2: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5939_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M2: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M2 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_5940_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M2: nat,N: nat] :
          ( if_nat
          @ ( ( M2 = zero_zero_nat )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M2 @ ( 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 @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_5941_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M2: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_5942_and__int_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [K2: int,L4: int] :
            ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ K2 @ L4 ) )
           => ( ( ~ ( ( member_int @ K2 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ L4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( P @ ( divide_divide_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
             => ( P @ K2 @ L4 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% and_int.pinduct
thf(fact_5943_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M2: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M2 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M2 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5944_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M2: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M2 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M2 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5945_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M2: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M2 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M2 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M2 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5946_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I3: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I3 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I3 @ J2 )
               => ( P @ ( plus_plus_int @ I3 @ one_one_int ) @ J2 ) )
             => ( P @ I3 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_5947_signed__take__bit__eq__take__bit__minus,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ K3 ) @ ( times_times_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( suc @ N ) ) @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_take_bit_minus
thf(fact_5948_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_5949_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_5950_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5951_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5952_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5953_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_5954_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5955_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5956_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5957_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_5958_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5959_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5960_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5961_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_5962_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5963_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5964_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5965_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_5966_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ring_1_of_int_real @ W2 )
        = ( ring_1_of_int_real @ Z2 ) )
      = ( W2 = Z2 ) ) ).

% of_int_eq_iff
thf(fact_5967_of__int__eq__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ring_1_of_int_rat @ W2 )
        = ( ring_1_of_int_rat @ Z2 ) )
      = ( W2 = Z2 ) ) ).

% of_int_eq_iff
thf(fact_5968_bit__0__eq,axiom,
    ( ( bit_se1146084159140164899it_int @ zero_zero_int )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_5969_bit__0__eq,axiom,
    ( ( bit_se1148574629649215175it_nat @ zero_zero_nat )
    = bot_bot_nat_o ) ).

% bit_0_eq
thf(fact_5970_of__int__0,axiom,
    ( ( ring_17405671764205052669omplex @ zero_zero_int )
    = zero_zero_complex ) ).

% of_int_0
thf(fact_5971_of__int__0,axiom,
    ( ( ring_1_of_int_int @ zero_zero_int )
    = zero_zero_int ) ).

% of_int_0
thf(fact_5972_of__int__0,axiom,
    ( ( ring_1_of_int_real @ zero_zero_int )
    = zero_zero_real ) ).

% of_int_0
thf(fact_5973_of__int__0,axiom,
    ( ( ring_1_of_int_rat @ zero_zero_int )
    = zero_zero_rat ) ).

% of_int_0
thf(fact_5974_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_complex
        = ( ring_17405671764205052669omplex @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5975_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_int
        = ( ring_1_of_int_int @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5976_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_real
        = ( ring_1_of_int_real @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5977_of__int__0__eq__iff,axiom,
    ! [Z2: int] :
      ( ( zero_zero_rat
        = ( ring_1_of_int_rat @ Z2 ) )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_0_eq_iff
thf(fact_5978_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_17405671764205052669omplex @ Z2 )
        = zero_zero_complex )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5979_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = zero_zero_int )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5980_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_real @ Z2 )
        = zero_zero_real )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5981_of__int__eq__0__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = zero_zero_rat )
      = ( Z2 = zero_zero_int ) ) ).

% of_int_eq_0_iff
thf(fact_5982_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% of_int_le_iff
thf(fact_5983_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% of_int_le_iff
thf(fact_5984_of__int__le__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ W2 @ Z2 ) ) ).

% of_int_le_iff
thf(fact_5985_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ( ring_1_of_int_real @ Z2 )
        = ( numeral_numeral_real @ N2 ) )
      = ( Z2
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5986_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = ( numeral_numeral_rat @ N2 ) )
      = ( Z2
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5987_of__int__eq__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = ( numeral_numeral_int @ N2 ) )
      = ( Z2
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_5988_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_5989_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_5990_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_5991_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% of_int_less_iff
thf(fact_5992_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% of_int_less_iff
thf(fact_5993_of__int__less__iff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ W2 @ Z2 ) ) ).

% of_int_less_iff
thf(fact_5994_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_17405671764205052669omplex @ Z2 )
        = one_one_complex )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5995_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_int @ Z2 )
        = one_one_int )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5996_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_real @ Z2 )
        = one_one_real )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5997_of__int__eq__1__iff,axiom,
    ! [Z2: int] :
      ( ( ( ring_1_of_int_rat @ Z2 )
        = one_one_rat )
      = ( Z2 = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_5998_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_5999_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_6000_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_6001_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_6002_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_17405671764205052669omplex @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_complex @ ( ring_17405671764205052669omplex @ W2 ) @ ( ring_17405671764205052669omplex @ Z2 ) ) ) ).

% of_int_mult
thf(fact_6003_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_real @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_mult
thf(fact_6004_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_mult
thf(fact_6005_of__int__mult,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( times_times_int @ W2 @ Z2 ) )
      = ( times_times_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_mult
thf(fact_6006_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_add
thf(fact_6007_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_add
thf(fact_6008_of__int__add,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W2 @ Z2 ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_add
thf(fact_6009_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_1_of_int_int @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus_uminus_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_minus
thf(fact_6010_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_1_of_int_real @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus_uminus_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_minus
thf(fact_6011_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus_uminus_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_minus
thf(fact_6012_of__int__minus,axiom,
    ! [Z2: int] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ Z2 ) )
      = ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ Z2 ) ) ) ).

% of_int_minus
thf(fact_6013_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_real @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_minus_real @ ( ring_1_of_int_real @ W2 ) @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_diff
thf(fact_6014_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_rat @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_minus_rat @ ( ring_1_of_int_rat @ W2 ) @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_diff
thf(fact_6015_of__int__diff,axiom,
    ! [W2: int,Z2: int] :
      ( ( ring_1_of_int_int @ ( minus_minus_int @ W2 @ Z2 ) )
      = ( minus_minus_int @ ( ring_1_of_int_int @ W2 ) @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_diff
thf(fact_6016_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_6017_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_6018_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral_nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_6019_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ( ring_1_of_int_rat @ X3 )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
      = ( X3
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6020_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ( ring_1_of_int_real @ X3 )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
      = ( X3
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6021_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ( ring_1_of_int_int @ X3 )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( X3
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6022_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ( ring_17405671764205052669omplex @ X3 )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 ) )
      = ( X3
        = ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_6023_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 )
        = ( ring_1_of_int_rat @ X3 ) )
      = ( ( power_power_int @ B @ W2 )
        = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6024_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 )
        = ( ring_1_of_int_real @ X3 ) )
      = ( ( power_power_int @ B @ W2 )
        = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6025_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 )
        = ( ring_1_of_int_int @ X3 ) )
      = ( ( power_power_int @ B @ W2 )
        = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6026_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W2 )
        = ( ring_17405671764205052669omplex @ X3 ) )
      = ( ( power_power_int @ B @ W2 )
        = X3 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_6027_of__int__power,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z2 @ N2 ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z2 ) @ N2 ) ) ).

% of_int_power
thf(fact_6028_of__int__power,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z2 @ N2 ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z2 ) @ N2 ) ) ).

% of_int_power
thf(fact_6029_of__int__power,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z2 @ N2 ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z2 ) @ N2 ) ) ).

% of_int_power
thf(fact_6030_of__int__power,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z2 @ N2 ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z2 ) @ N2 ) ) ).

% of_int_power
thf(fact_6031_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_6032_of__int__of__bool,axiom,
    ! [P: $o] :
      ( ( ring_1_of_int_rat @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% of_int_of_bool
thf(fact_6033_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_6034_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_6035_bit__numeral__Bit0__Suc__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( suc @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% bit_numeral_Bit0_Suc_iff
thf(fact_6036_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_6037_bit__numeral__Bit1__Suc__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( suc @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% bit_numeral_Bit1_Suc_iff
thf(fact_6038_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_6039_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_6040_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_6041_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_6042_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_6043_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_6044_signed__take__bit__negative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ zero_zero_int )
      = ( bit_se1146084159140164899it_int @ K @ N2 ) ) ).

% signed_take_bit_negative_iff
thf(fact_6045_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_6046_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_6047_of__int__0__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_le_iff
thf(fact_6048_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_6049_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_6050_of__int__le__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z2 @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_6051_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ zero_zero_real )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_6052_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ zero_zero_rat )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_6053_of__int__less__0__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ zero_zero_int )
      = ( ord_less_int @ Z2 @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_6054_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_6055_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_6056_of__int__0__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% of_int_0_less_iff
thf(fact_6057_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_6058_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_6059_of__int__le__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_6060_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_6061_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_6062_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_le_iff
thf(fact_6063_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_6064_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_6065_of__int__less__numeral__iff,axiom,
    ! [Z2: int,N2: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_int @ Z2 @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_6066_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_6067_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_6068_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z2: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z2 ) ) ).

% of_int_numeral_less_iff
thf(fact_6069_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_6070_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_6071_of__int__le__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
      = ( ord_less_eq_int @ Z2 @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_6072_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_6073_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_6074_of__int__1__le__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_eq_int @ one_one_int @ Z2 ) ) ).

% of_int_1_le_iff
thf(fact_6075_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_6076_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_6077_of__int__less__1__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z2 ) @ one_one_int )
      = ( ord_less_int @ Z2 @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_6078_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_6079_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_6080_of__int__1__less__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% of_int_1_less_iff
thf(fact_6081_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6082_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6083_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_6084_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6085_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6086_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( ord_less_eq_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_6087_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y3 )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6088_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y3 )
        = ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6089_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y3 )
        = ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6090_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y3 )
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_6091_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X3 ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6092_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 )
        = ( ring_1_of_int_real @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6093_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 )
        = ( ring_1_of_int_rat @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6094_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
        = ( ring_1_of_int_int @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
        = Y3 ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_6095_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) @ ( ring_1_of_int_real @ X3 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6096_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) @ ( ring_1_of_int_rat @ X3 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6097_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W2: nat,X3: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) @ ( ring_1_of_int_int @ X3 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W2 ) @ X3 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_6098_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X3 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W2 ) )
      = ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6099_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X3 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W2 ) )
      = ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6100_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X3: int,B: int,W2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X3 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W2 ) )
      = ( ord_less_int @ X3 @ ( power_power_int @ B @ W2 ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_6101_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_6102_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_6103_bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_6104_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_6105_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_6106_bit__mod__2__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( bit_se9216721137139052372nteger @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N2 )
      = ( ( N2 = zero_zero_nat )
        & ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_mod_2_iff
thf(fact_6107_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_6108_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_6109_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6110_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6111_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_6112_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6113_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6114_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_6115_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6116_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6117_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_6118_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X3 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6119_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X3 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6120_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_6121_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y3 )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6122_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y3 )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6123_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y3 )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6124_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y3 )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6125_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y3 )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_6126_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X3 ) ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6127_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
        = ( ring_1_of_int_int @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6128_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X3 ) ) @ N2 )
        = ( ring_1_of_int_real @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6129_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X3 ) ) @ N2 )
        = ( ring_1_of_int_rat @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6130_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X3 ) ) @ N2 )
        = ( ring_18347121197199848620nteger @ Y3 ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X3 ) ) @ N2 )
        = Y3 ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_6131_bit__disjunctive__add__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( bit_se1146084159140164899it_int @ A @ N3 )
          | ~ ( bit_se1146084159140164899it_int @ B @ N3 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ B ) @ N2 )
        = ( ( bit_se1146084159140164899it_int @ A @ N2 )
          | ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_6132_bit__disjunctive__add__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( bit_se1148574629649215175it_nat @ A @ N3 )
          | ~ ( bit_se1148574629649215175it_nat @ B @ N3 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
        = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
          | ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ) ).

% bit_disjunctive_add_iff
thf(fact_6133_ex__le__of__int,axiom,
    ! [X3: real] :
    ? [Z3: int] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_le_of_int
thf(fact_6134_ex__le__of__int,axiom,
    ! [X3: rat] :
    ? [Z3: int] : ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_le_of_int
thf(fact_6135_ex__less__of__int,axiom,
    ! [X3: real] :
    ? [Z3: int] : ( ord_less_real @ X3 @ ( ring_1_of_int_real @ Z3 ) ) ).

% ex_less_of_int
thf(fact_6136_ex__less__of__int,axiom,
    ! [X3: rat] :
    ? [Z3: int] : ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ Z3 ) ) ).

% ex_less_of_int
thf(fact_6137_ex__of__int__less,axiom,
    ! [X3: real] :
    ? [Z3: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z3 ) @ X3 ) ).

% ex_of_int_less
thf(fact_6138_ex__of__int__less,axiom,
    ! [X3: rat] :
    ? [Z3: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 ) ).

% ex_of_int_less
thf(fact_6139_mult__of__int__commute,axiom,
    ! [X3: int,Y3: complex] :
      ( ( times_times_complex @ ( ring_17405671764205052669omplex @ X3 ) @ Y3 )
      = ( times_times_complex @ Y3 @ ( ring_17405671764205052669omplex @ X3 ) ) ) ).

% mult_of_int_commute
thf(fact_6140_mult__of__int__commute,axiom,
    ! [X3: int,Y3: real] :
      ( ( times_times_real @ ( ring_1_of_int_real @ X3 ) @ Y3 )
      = ( times_times_real @ Y3 @ ( ring_1_of_int_real @ X3 ) ) ) ).

% mult_of_int_commute
thf(fact_6141_mult__of__int__commute,axiom,
    ! [X3: int,Y3: rat] :
      ( ( times_times_rat @ ( ring_1_of_int_rat @ X3 ) @ Y3 )
      = ( times_times_rat @ Y3 @ ( ring_1_of_int_rat @ X3 ) ) ) ).

% mult_of_int_commute
thf(fact_6142_mult__of__int__commute,axiom,
    ! [X3: int,Y3: int] :
      ( ( times_times_int @ ( ring_1_of_int_int @ X3 ) @ Y3 )
      = ( times_times_int @ Y3 @ ( ring_1_of_int_int @ X3 ) ) ) ).

% mult_of_int_commute
thf(fact_6143_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_6144_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_6145_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1146084159140164899it_int @ one_one_int @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_6146_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_6147_bit__take__bit__iff,axiom,
    ! [M: nat,A: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se2925701944663578781it_nat @ M @ A ) @ N2 )
      = ( ( ord_less_nat @ N2 @ M )
        & ( bit_se1148574629649215175it_nat @ A @ N2 ) ) ) ).

% bit_take_bit_iff
thf(fact_6148_bit__take__bit__iff,axiom,
    ! [M: nat,A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ M @ A ) @ N2 )
      = ( ( ord_less_nat @ N2 @ M )
        & ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).

% bit_take_bit_iff
thf(fact_6149_bit__of__bool__iff,axiom,
    ! [B: $o,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( zero_n2684676970156552555ol_int @ B ) @ N2 )
      = ( B
        & ( N2 = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_6150_bit__of__bool__iff,axiom,
    ! [B: $o,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( zero_n2687167440665602831ol_nat @ B ) @ N2 )
      = ( B
        & ( N2 = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_6151_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_6152_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_6153_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_6154_of__int__nonneg,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_nonneg
thf(fact_6155_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% of_int_pos
thf(fact_6156_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% of_int_pos
thf(fact_6157_of__int__pos,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z2 ) ) ) ).

% of_int_pos
thf(fact_6158_floor__exists,axiom,
    ! [X3: real] :
    ? [Z3: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z3 ) @ X3 )
      & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_6159_floor__exists,axiom,
    ! [X3: rat] :
    ? [Z3: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z3 ) @ X3 )
      & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z3 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_6160_floor__exists1,axiom,
    ! [X3: real] :
    ? [X4: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ X3 )
      & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ X4 @ one_one_int ) ) )
      & ! [Y6: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y6 ) @ X3 )
            & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
         => ( Y6 = X4 ) ) ) ).

% floor_exists1
thf(fact_6161_floor__exists1,axiom,
    ! [X3: rat] :
    ? [X4: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ X3 )
      & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X4 @ one_one_int ) ) )
      & ! [Y6: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y6 ) @ X3 )
            & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y6 @ one_one_int ) ) ) )
         => ( Y6 = X4 ) ) ) ).

% floor_exists1
thf(fact_6162_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_6163_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_6164_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_6165_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_6166_bit__imp__take__bit__positive,axiom,
    ! [N2: nat,M: nat,K: int] :
      ( ( ord_less_nat @ N2 @ M )
     => ( ( bit_se1146084159140164899it_int @ K @ N2 )
       => ( ord_less_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ M @ K ) ) ) ) ).

% bit_imp_take_bit_positive
thf(fact_6167_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N: int,M2: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M2 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_6168_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N: int,M2: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M2 ) ) ) ) ).

% int_less_real_le
thf(fact_6169_bit__concat__bit__iff,axiom,
    ! [M: nat,K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_concat_bit @ M @ K @ L ) @ N2 )
      = ( ( ( ord_less_nat @ N2 @ M )
          & ( bit_se1146084159140164899it_int @ K @ N2 ) )
        | ( ( ord_less_eq_nat @ M @ N2 )
          & ( bit_se1146084159140164899it_int @ L @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% bit_concat_bit_iff
thf(fact_6170_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_6171_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_6172_bit__Suc,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( bit_se9216721137139052372nteger @ A @ ( suc @ N2 ) )
      = ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ N2 ) ) ).

% bit_Suc
thf(fact_6173_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_6174_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_6175_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M6: nat] :
              ( ( ord_less_eq_nat @ N3 @ M6 )
             => ( ( bit_se1146084159140164899it_int @ K @ M6 )
                = ( 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_6176_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_6177_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_6178_even__bit__succ__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ A ) @ N2 )
        = ( ( bit_se9216721137139052372nteger @ A @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% even_bit_succ_iff
thf(fact_6179_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_6180_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_6181_odd__bit__iff__bit__pred,axiom,
    ! [A: code_integer,N2: nat] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( bit_se9216721137139052372nteger @ A @ N2 )
        = ( ( bit_se9216721137139052372nteger @ ( minus_8373710615458151222nteger @ A @ one_one_Code_integer ) @ N2 )
          | ( N2 = zero_zero_nat ) ) ) ) ).

% odd_bit_iff_bit_pred
thf(fact_6182_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_6183_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_6184_bit__sum__mult__2__cases,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se9216721137139052372nteger @ A @ ( suc @ J2 ) )
     => ( ( bit_se9216721137139052372nteger @ ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) @ N2 )
        = ( ( ( N2 = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
          & ( ( N2 != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_6185_bit__sum__mult__2__cases,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1146084159140164899it_int @ A @ ( suc @ J2 ) )
     => ( ( bit_se1146084159140164899it_int @ ( plus_plus_int @ A @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ 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 ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_6186_bit__sum__mult__2__cases,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ! [J2: nat] :
          ~ ( bit_se1148574629649215175it_nat @ A @ ( suc @ J2 ) )
     => ( ( bit_se1148574629649215175it_nat @ ( plus_plus_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) @ 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 ) ) @ B ) @ N2 ) ) ) ) ) ).

% bit_sum_mult_2_cases
thf(fact_6187_bit__rec,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A3: code_integer,N: nat] :
          ( ( ( N = zero_zero_nat )
           => ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 ) )
          & ( ( N != zero_zero_nat )
           => ( bit_se9216721137139052372nteger @ ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( minus_minus_nat @ N @ one_one_nat ) ) ) ) ) ) ).

% bit_rec
thf(fact_6188_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_6189_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_6190_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_6191_round__unique,axiom,
    ! [X3: real,Y3: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y3 ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y3 ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X3 )
          = Y3 ) ) ) ).

% round_unique
thf(fact_6192_round__unique,axiom,
    ! [X3: rat,Y3: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y3 ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y3 ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X3 )
          = Y3 ) ) ) ).

% round_unique
thf(fact_6193_of__int__round__gt,axiom,
    ! [X3: real] : ( ord_less_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) ) ).

% of_int_round_gt
thf(fact_6194_of__int__round__gt,axiom,
    ! [X3: rat] : ( ord_less_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) ) ).

% of_int_round_gt
thf(fact_6195_of__int__round__ge,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( minus_minus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) ) ).

% of_int_round_ge
thf(fact_6196_of__int__round__ge,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) ) ).

% of_int_round_ge
thf(fact_6197_of__int__round__le,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ ( plus_plus_real @ X3 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_6198_of__int__round__le,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ ( plus_plus_rat @ X3 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_6199_divmod__BitM__2__eq,axiom,
    ! [M: num] :
      ( ( unique5052692396658037445od_int @ ( bitM @ M ) @ ( bit0 @ one ) )
      = ( product_Pair_int_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ one_one_int ) ) ).

% divmod_BitM_2_eq
thf(fact_6200_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_6201_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_6202_neg__numeral__le__ceiling,axiom,
    ! [V: num,X3: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X3 ) ) ).

% neg_numeral_le_ceiling
thf(fact_6203_neg__numeral__le__ceiling,axiom,
    ! [V: num,X3: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X3 ) ) ).

% neg_numeral_le_ceiling
thf(fact_6204_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_6205_ceiling__zero,axiom,
    ( ( archim2889992004027027881ng_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_6206_ceiling__zero,axiom,
    ( ( archim7802044766580827645g_real @ zero_zero_real )
    = zero_zero_int ) ).

% ceiling_zero
thf(fact_6207_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2002935070580805687sk_nat @ N2 )
        = zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6208_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2000444600071755411sk_int @ N2 )
        = zero_zero_int )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6209_mask__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% mask_0
thf(fact_6210_mask__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
    = zero_zero_int ) ).

% mask_0
thf(fact_6211_round__0,axiom,
    ( ( archim8280529875227126926d_real @ zero_zero_real )
    = zero_zero_int ) ).

% round_0
thf(fact_6212_round__0,axiom,
    ( ( archim7778729529865785530nd_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% round_0
thf(fact_6213_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_6214_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_6215_ceiling__add__of__int,axiom,
    ! [X3: rat,Z2: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 ) ) ).

% ceiling_add_of_int
thf(fact_6216_ceiling__add__of__int,axiom,
    ! [X3: real,Z2: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 ) ) ).

% ceiling_add_of_int
thf(fact_6217_ceiling__le__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_6218_ceiling__le__zero,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X3 @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_6219_zero__less__ceiling,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ zero_zero_rat @ X3 ) ) ).

% zero_less_ceiling
thf(fact_6220_zero__less__ceiling,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% zero_less_ceiling
thf(fact_6221_ceiling__le__numeral,axiom,
    ! [X3: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X3 @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_6222_ceiling__le__numeral,axiom,
    ! [X3: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X3 @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_6223_ceiling__less__one,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int )
      = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_6224_ceiling__less__one,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int )
      = ( ord_less_eq_rat @ X3 @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_6225_one__le__ceiling,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ zero_zero_rat @ X3 ) ) ).

% one_le_ceiling
thf(fact_6226_one__le__ceiling,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% one_le_ceiling
thf(fact_6227_numeral__less__ceiling,axiom,
    ! [V: num,X3: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X3 ) ) ).

% numeral_less_ceiling
thf(fact_6228_numeral__less__ceiling,axiom,
    ! [V: num,X3: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X3 ) ) ).

% numeral_less_ceiling
thf(fact_6229_ceiling__le__one,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int )
      = ( ord_less_eq_real @ X3 @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_6230_ceiling__le__one,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int )
      = ( ord_less_eq_rat @ X3 @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_6231_one__less__ceiling,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ one_one_rat @ X3 ) ) ).

% one_less_ceiling
thf(fact_6232_one__less__ceiling,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ one_one_real @ X3 ) ) ).

% one_less_ceiling
thf(fact_6233_ceiling__add__numeral,axiom,
    ! [X3: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ V ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_6234_ceiling__add__numeral,axiom,
    ! [X3: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_6235_ceiling__add__one,axiom,
    ! [X3: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_6236_ceiling__add__one,axiom,
    ! [X3: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_6237_ceiling__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X3 @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_6238_ceiling__less__zero,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X3 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_6239_zero__le__ceiling,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 ) ) ).

% zero_le_ceiling
thf(fact_6240_zero__le__ceiling,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X3 ) ) ).

% zero_le_ceiling
thf(fact_6241_ceiling__less__numeral,axiom,
    ! [X3: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_6242_ceiling__less__numeral,axiom,
    ! [X3: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_6243_numeral__le__ceiling,axiom,
    ! [V: num,X3: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X3 ) ) ).

% numeral_le_ceiling
thf(fact_6244_numeral__le__ceiling,axiom,
    ! [V: num,X3: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X3 ) ) ).

% numeral_le_ceiling
thf(fact_6245_ceiling__le__neg__numeral,axiom,
    ! [X3: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_6246_ceiling__le__neg__numeral,axiom,
    ! [X3: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_6247_neg__numeral__less__ceiling,axiom,
    ! [V: num,X3: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X3 ) ) ).

% neg_numeral_less_ceiling
thf(fact_6248_neg__numeral__less__ceiling,axiom,
    ! [V: num,X3: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X3 ) ) ).

% neg_numeral_less_ceiling
thf(fact_6249_ceiling__less__neg__numeral,axiom,
    ! [X3: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_6250_ceiling__less__neg__numeral,axiom,
    ! [X3: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_6251_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_6252_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_6253_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_6254_ceiling__mono,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ Y3 @ X3 )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y3 ) @ ( archim7802044766580827645g_real @ X3 ) ) ) ).

% ceiling_mono
thf(fact_6255_ceiling__mono,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_eq_rat @ Y3 @ X3 )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y3 ) @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ).

% ceiling_mono
thf(fact_6256_le__of__int__ceiling,axiom,
    ! [X3: real] : ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) ) ).

% le_of_int_ceiling
thf(fact_6257_le__of__int__ceiling,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ).

% le_of_int_ceiling
thf(fact_6258_ceiling__less__cancel,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim2889992004027027881ng_rat @ Y3 ) )
     => ( ord_less_rat @ X3 @ Y3 ) ) ).

% ceiling_less_cancel
thf(fact_6259_ceiling__less__cancel,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ ( archim7802044766580827645g_real @ Y3 ) )
     => ( ord_less_real @ X3 @ Y3 ) ) ).

% ceiling_less_cancel
thf(fact_6260_not__mask__negative__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_6261_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_6262_ceiling__le__iff,axiom,
    ! [X3: real,Z2: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 )
      = ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% ceiling_le_iff
thf(fact_6263_ceiling__le__iff,axiom,
    ! [X3: rat,Z2: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 )
      = ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% ceiling_le_iff
thf(fact_6264_ceiling__le,axiom,
    ! [X3: real,A: int] :
      ( ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ A ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X3 ) @ A ) ) ).

% ceiling_le
thf(fact_6265_ceiling__le,axiom,
    ! [X3: rat,A: int] :
      ( ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ A ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X3 ) @ A ) ) ).

% ceiling_le
thf(fact_6266_less__ceiling__iff,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_int @ Z2 @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ).

% less_ceiling_iff
thf(fact_6267_less__ceiling__iff,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_int @ Z2 @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ).

% less_ceiling_iff
thf(fact_6268_ceiling__add__le,axiom,
    ! [X3: rat,Y3: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X3 @ Y3 ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X3 ) @ ( archim2889992004027027881ng_rat @ Y3 ) ) ) ).

% ceiling_add_le
thf(fact_6269_ceiling__add__le,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X3 ) @ ( archim7802044766580827645g_real @ Y3 ) ) ) ).

% ceiling_add_le
thf(fact_6270_round__mono,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X3 ) @ ( archim7778729529865785530nd_rat @ Y3 ) ) ) ).

% round_mono
thf(fact_6271_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_6272_eval__nat__numeral_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_nat @ ( bit0 @ N2 ) )
      = ( suc @ ( numeral_numeral_nat @ ( bitM @ N2 ) ) ) ) ).

% eval_nat_numeral(2)
thf(fact_6273_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_6274_of__int__ceiling__le__add__one,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ ( plus_plus_rat @ R2 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_6275_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_6276_of__int__ceiling__diff__one__le,axiom,
    ! [R2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R2 ) ) @ one_one_rat ) @ R2 ) ).

% of_int_ceiling_diff_one_le
thf(fact_6277_ceiling__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) @ T )
              & ( ord_less_eq_real @ T @ ( ring_1_of_int_real @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_6278_ceiling__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) @ T )
              & ( ord_less_eq_rat @ T @ ( ring_1_of_int_rat @ I2 ) ) )
           => ( P @ I2 ) ) ) ) ).

% ceiling_split
thf(fact_6279_ceiling__eq__iff,axiom,
    ! [X3: real,A: int] :
      ( ( ( archim7802044766580827645g_real @ X3 )
        = A )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X3 )
        & ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_6280_ceiling__eq__iff,axiom,
    ! [X3: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X3 )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X3 )
        & ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_6281_ceiling__unique,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ Z2 ) )
       => ( ( archim7802044766580827645g_real @ X3 )
          = Z2 ) ) ) ).

% ceiling_unique
thf(fact_6282_ceiling__unique,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) )
       => ( ( archim2889992004027027881ng_rat @ X3 )
          = Z2 ) ) ) ).

% ceiling_unique
thf(fact_6283_ceiling__correct,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) @ one_one_real ) @ X3 )
      & ( ord_less_eq_real @ X3 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X3 ) ) ) ) ).

% ceiling_correct
thf(fact_6284_ceiling__correct,axiom,
    ! [X3: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) @ one_one_rat ) @ X3 )
      & ( ord_less_eq_rat @ X3 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X3 ) ) ) ) ).

% ceiling_correct
thf(fact_6285_mult__ceiling__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_6286_mult__ceiling__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ).

% mult_ceiling_le
thf(fact_6287_ceiling__less__iff,axiom,
    ! [X3: real,Z2: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X3 ) @ Z2 )
      = ( ord_less_eq_real @ X3 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_6288_ceiling__less__iff,axiom,
    ! [X3: rat,Z2: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X3 ) @ Z2 )
      = ( ord_less_eq_rat @ X3 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_6289_le__ceiling__iff,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_eq_int @ Z2 @ ( archim2889992004027027881ng_rat @ X3 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 ) ) ).

% le_ceiling_iff
thf(fact_6290_le__ceiling__iff,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_eq_int @ Z2 @ ( archim7802044766580827645g_real @ X3 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 ) ) ).

% le_ceiling_iff
thf(fact_6291_ceiling__divide__upper,axiom,
    ! [Q4: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q4 )
     => ( ord_less_eq_real @ P4 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).

% ceiling_divide_upper
thf(fact_6292_ceiling__divide__upper,axiom,
    ! [Q4: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_eq_rat @ P4 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) ) ) ).

% ceiling_divide_upper
thf(fact_6293_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_6294_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_6295_semiring__bit__operations__class_Oeven__mask__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se2119862282449309892nteger @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% semiring_bit_operations_class.even_mask_iff
thf(fact_6296_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_6297_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_6298_ceiling__divide__lower,axiom,
    ! [Q4: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q4 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_6299_ceiling__divide__lower,axiom,
    ! [Q4: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) @ P4 ) ) ).

% ceiling_divide_lower
thf(fact_6300_ceiling__eq,axiom,
    ! [N2: int,X3: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X3 )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_6301_ceiling__eq,axiom,
    ! [N2: int,X3: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N2 ) @ X3 )
     => ( ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N2 ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X3 )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_6302_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_6303_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_6304_horner__sum__of__bool__2__less,axiom,
    ! [Bs: list_o] : ( ord_less_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( size_size_list_o @ Bs ) ) ) ).

% horner_sum_of_bool_2_less
thf(fact_6305_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_6306_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_6307_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_6308_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_6309_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_6310_the__elem__eq,axiom,
    ! [X3: produc3843707927480180839at_nat] :
      ( ( the_el221668144340439132at_nat @ ( insert9069300056098147895at_nat @ X3 @ bot_bo228742789529271731at_nat ) )
      = X3 ) ).

% the_elem_eq
thf(fact_6311_the__elem__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( the_el2281957884133575798at_nat @ ( insert8211810215607154385at_nat @ X3 @ bot_bo2099793752762293965at_nat ) )
      = X3 ) ).

% the_elem_eq
thf(fact_6312_the__elem__eq,axiom,
    ! [X3: real] :
      ( ( the_elem_real @ ( insert_real @ X3 @ bot_bot_set_real ) )
      = X3 ) ).

% the_elem_eq
thf(fact_6313_the__elem__eq,axiom,
    ! [X3: nat] :
      ( ( the_elem_nat @ ( insert_nat @ X3 @ bot_bot_set_nat ) )
      = X3 ) ).

% the_elem_eq
thf(fact_6314_the__elem__eq,axiom,
    ! [X3: int] :
      ( ( the_elem_int @ ( insert_int @ X3 @ bot_bot_set_int ) )
      = X3 ) ).

% the_elem_eq
thf(fact_6315_vebt__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
        = Y3 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( Y3
              = ( ~ ( ( ( 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] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
           => Y3 )
         => ( ( ? [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
             => Y3 )
           => ( ( ? [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
               => Y3 )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                   => ( Y3
                      = ( ~ ( ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(1)
thf(fact_6316_set__vebt_H__def,axiom,
    ( vEBT_VEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_vebt_member @ T2 ) ) ) ) ).

% set_vebt'_def
thf(fact_6317_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
          @ ^ [X: real] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6318_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
          @ ^ [X: list_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6319_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
          @ ^ [X: set_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6320_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
          @ ^ [X: nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6321_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
          @ ^ [X: int] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6322_finite__Collect__conjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        | ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6323_finite__Collect__conjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        | ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) )
     => ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              & ( Q @ X ) ) ) ) ) ).

% finite_Collect_conjI
thf(fact_6324_finite__Collect__disjI,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [X: real] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_real @ ( collect_real @ P ) )
        & ( finite_finite_real @ ( collect_real @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6325_finite__Collect__disjI,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
        & ( finite8100373058378681591st_nat @ ( collect_list_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6326_finite__Collect__disjI,axiom,
    ! [P: set_nat > $o,Q: set_nat > $o] :
      ( ( finite1152437895449049373et_nat
        @ ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
        & ( finite1152437895449049373et_nat @ ( collect_set_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6327_finite__Collect__disjI,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [X: nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_nat @ ( collect_nat @ P ) )
        & ( finite_finite_nat @ ( collect_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6328_finite__Collect__disjI,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [X: int] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite_finite_int @ ( collect_int @ P ) )
        & ( finite_finite_int @ ( collect_int @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6329_finite__Collect__disjI,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [X: complex] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
        & ( finite3207457112153483333omplex @ ( collect_complex @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6330_finite__Collect__disjI,axiom,
    ! [P: product_prod_nat_nat > $o,Q: product_prod_nat_nat > $o] :
      ( ( finite6177210948735845034at_nat
        @ ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( P @ X )
              | ( Q @ X ) ) ) )
      = ( ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
        & ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ Q ) ) ) ) ).

% finite_Collect_disjI
thf(fact_6331_bit_Oxor__self,axiom,
    ! [X3: int] :
      ( ( bit_se6526347334894502574or_int @ X3 @ X3 )
      = zero_zero_int ) ).

% bit.xor_self
thf(fact_6332_xor__self__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ A )
      = zero_zero_nat ) ).

% xor_self_eq
thf(fact_6333_xor__self__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ A )
      = zero_zero_int ) ).

% xor_self_eq
thf(fact_6334_xor_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
      = A ) ).

% xor.left_neutral
thf(fact_6335_xor_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
      = A ) ).

% xor.left_neutral
thf(fact_6336_xor_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
      = A ) ).

% xor.right_neutral
thf(fact_6337_xor_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
      = A ) ).

% xor.right_neutral
thf(fact_6338_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: real > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
              ? [Y: real] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: real] :
              ( ( P @ Y )
             => ( finite_finite_real
                @ ( collect_real
                  @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6339_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: nat > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: real] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: real] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6340_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: int > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
              ? [Y: real] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: real] :
              ( ( P @ Y )
             => ( finite_finite_int
                @ ( collect_int
                  @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6341_finite__Collect__bounded__ex,axiom,
    ! [P: real > $o,Q: complex > real > $o] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X: complex] :
              ? [Y: real] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: real] :
              ( ( P @ Y )
             => ( finite3207457112153483333omplex
                @ ( collect_complex
                  @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6342_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: real > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_real
                @ ( collect_real
                  @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6343_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: nat > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6344_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: int > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [X: int] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite_finite_int
                @ ( collect_int
                  @ ^ [X: int] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6345_finite__Collect__bounded__ex,axiom,
    ! [P: nat > $o,Q: complex > nat > $o] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [X: complex] :
              ? [Y: nat] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: nat] :
              ( ( P @ Y )
             => ( finite3207457112153483333omplex
                @ ( collect_complex
                  @ ^ [X: complex] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6346_finite__Collect__bounded__ex,axiom,
    ! [P: int > $o,Q: real > int > $o] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [X: real] :
              ? [Y: int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: int] :
              ( ( P @ Y )
             => ( finite_finite_real
                @ ( collect_real
                  @ ^ [X: real] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6347_finite__Collect__bounded__ex,axiom,
    ! [P: int > $o,Q: nat > int > $o] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [X: nat] :
              ? [Y: int] :
                ( ( P @ Y )
                & ( Q @ X @ Y ) ) ) )
        = ( ! [Y: int] :
              ( ( P @ Y )
             => ( finite_finite_nat
                @ ( collect_nat
                  @ ^ [X: nat] : ( Q @ X @ Y ) ) ) ) ) ) ) ).

% finite_Collect_bounded_ex
thf(fact_6348_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_6349_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_6350_finite__Collect__subsets,axiom,
    ! [A2: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite9047747110432174090at_nat
        @ ( collec5514110066124741708at_nat
          @ ^ [B5: set_Pr1261947904930325089at_nat] : ( ord_le3146513528884898305at_nat @ B5 @ A2 ) ) ) ) ).

% finite_Collect_subsets
thf(fact_6351_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_6352_singleton__conv2,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ( ^ [Y4: produc3843707927480180839at_nat,Z: produc3843707927480180839at_nat] : Y4 = Z
          @ A ) )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv2
thf(fact_6353_singleton__conv2,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ( ^ [Y4: list_nat,Z: list_nat] : Y4 = Z
          @ A ) )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv2
thf(fact_6354_singleton__conv2,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ( ^ [Y4: set_nat,Z: set_nat] : Y4 = Z
          @ A ) )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv2
thf(fact_6355_singleton__conv2,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ( ^ [Y4: product_prod_nat_nat,Z: product_prod_nat_nat] : Y4 = Z
          @ A ) )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv2
thf(fact_6356_singleton__conv2,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ( ^ [Y4: real,Z: real] : Y4 = Z
          @ A ) )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv2
thf(fact_6357_singleton__conv2,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ( ^ [Y4: nat,Z: nat] : Y4 = Z
          @ A ) )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv2
thf(fact_6358_singleton__conv2,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ( ^ [Y4: int,Z: int] : Y4 = Z
          @ A ) )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv2
thf(fact_6359_singleton__conv,axiom,
    ! [A: produc3843707927480180839at_nat] :
      ( ( collec6321179662152712658at_nat
        @ ^ [X: produc3843707927480180839at_nat] : X = A )
      = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) ).

% singleton_conv
thf(fact_6360_singleton__conv,axiom,
    ! [A: list_nat] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] : X = A )
      = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) ).

% singleton_conv
thf(fact_6361_singleton__conv,axiom,
    ! [A: set_nat] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] : X = A )
      = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) ).

% singleton_conv
thf(fact_6362_singleton__conv,axiom,
    ! [A: product_prod_nat_nat] :
      ( ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] : X = A )
      = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) ).

% singleton_conv
thf(fact_6363_singleton__conv,axiom,
    ! [A: real] :
      ( ( collect_real
        @ ^ [X: real] : X = A )
      = ( insert_real @ A @ bot_bot_set_real ) ) ).

% singleton_conv
thf(fact_6364_singleton__conv,axiom,
    ! [A: nat] :
      ( ( collect_nat
        @ ^ [X: nat] : X = A )
      = ( insert_nat @ A @ bot_bot_set_nat ) ) ).

% singleton_conv
thf(fact_6365_singleton__conv,axiom,
    ! [A: int] :
      ( ( collect_int
        @ ^ [X: int] : X = A )
      = ( insert_int @ A @ bot_bot_set_int ) ) ).

% singleton_conv
thf(fact_6366_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_6367_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_6368_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A @ I2 )
            & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_6369_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_6370_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_eq_int @ A @ I2 )
            & ( ord_less_int @ I2 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_6371_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I2: int] :
            ( ( ord_less_int @ A @ I2 )
            & ( ord_less_eq_int @ I2 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_6372_xor__nat__numerals_I4_J,axiom,
    ! [X3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X3 ) ) ) ).

% xor_nat_numerals(4)
thf(fact_6373_xor__nat__numerals_I3_J,axiom,
    ! [X3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).

% xor_nat_numerals(3)
thf(fact_6374_xor__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) ) ).

% xor_nat_numerals(2)
thf(fact_6375_xor__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% xor_nat_numerals(1)
thf(fact_6376_xor__numerals_I6_J,axiom,
    ! [X3: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_6377_xor__numerals_I6_J,axiom,
    ! [X3: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X3 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_6378_xor__numerals_I4_J,axiom,
    ! [X3: num,Y3: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X3 ) @ ( numeral_numeral_nat @ Y3 ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_6379_xor__numerals_I4_J,axiom,
    ! [X3: num,Y3: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X3 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y3 ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X3 ) @ ( numeral_numeral_int @ Y3 ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_6380_empty__def,axiom,
    ( bot_bot_set_list_nat
    = ( collect_list_nat
      @ ^ [X: list_nat] : $false ) ) ).

% empty_def
thf(fact_6381_empty__def,axiom,
    ( bot_bot_set_set_nat
    = ( collect_set_nat
      @ ^ [X: set_nat] : $false ) ) ).

% empty_def
thf(fact_6382_empty__def,axiom,
    ( bot_bo2099793752762293965at_nat
    = ( collec3392354462482085612at_nat
      @ ^ [X: product_prod_nat_nat] : $false ) ) ).

% empty_def
thf(fact_6383_empty__def,axiom,
    ( bot_bot_set_real
    = ( collect_real
      @ ^ [X: real] : $false ) ) ).

% empty_def
thf(fact_6384_empty__def,axiom,
    ( bot_bot_set_nat
    = ( collect_nat
      @ ^ [X: nat] : $false ) ) ).

% empty_def
thf(fact_6385_empty__def,axiom,
    ( bot_bot_set_int
    = ( collect_int
      @ ^ [X: int] : $false ) ) ).

% empty_def
thf(fact_6386_Collect__conv__if2,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X: produc3843707927480180839at_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X: produc3843707927480180839at_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6387_Collect__conv__if2,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6388_Collect__conv__if2,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6389_Collect__conv__if2,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6390_Collect__conv__if2,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if2
thf(fact_6391_Collect__conv__if2,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if2
thf(fact_6392_Collect__conv__if2,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A = X )
                & ( P @ X ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( A = X )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if2
thf(fact_6393_Collect__conv__if,axiom,
    ! [P: produc3843707927480180839at_nat > $o,A: produc3843707927480180839at_nat] :
      ( ( ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X: produc3843707927480180839at_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec6321179662152712658at_nat
            @ ^ [X: produc3843707927480180839at_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo228742789529271731at_nat ) ) ) ).

% Collect_conv_if
thf(fact_6394_Collect__conv__if,axiom,
    ! [P: list_nat > $o,A: list_nat] :
      ( ( ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_list_nat @ A @ bot_bot_set_list_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_list_nat ) ) ) ).

% Collect_conv_if
thf(fact_6395_Collect__conv__if,axiom,
    ! [P: set_nat > $o,A: set_nat] :
      ( ( ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_set_nat @ A @ bot_bot_set_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_6396_Collect__conv__if,axiom,
    ! [P: product_prod_nat_nat > $o,A: product_prod_nat_nat] :
      ( ( ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bo2099793752762293965at_nat ) ) ) ).

% Collect_conv_if
thf(fact_6397_Collect__conv__if,axiom,
    ! [P: real > $o,A: real] :
      ( ( ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_real @ A @ bot_bot_set_real ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_real
            @ ^ [X: real] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_real ) ) ) ).

% Collect_conv_if
thf(fact_6398_Collect__conv__if,axiom,
    ! [P: nat > $o,A: nat] :
      ( ( ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_nat @ A @ bot_bot_set_nat ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_nat
            @ ^ [X: nat] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_nat ) ) ) ).

% Collect_conv_if
thf(fact_6399_Collect__conv__if,axiom,
    ! [P: int > $o,A: int] :
      ( ( ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A )
                & ( P @ X ) ) )
          = ( insert_int @ A @ bot_bot_set_int ) ) )
      & ( ~ ( P @ A )
       => ( ( collect_int
            @ ^ [X: int] :
                ( ( X = A )
                & ( P @ X ) ) )
          = bot_bot_set_int ) ) ) ).

% Collect_conv_if
thf(fact_6400_pred__subset__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le2646555220125990790_nat_o
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R )
        @ ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6401_pred__subset__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ord_le6741204236512500942_int_o
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R )
        @ ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) )
      = ( ord_le2843351958646193337nt_int @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6402_pred__subset__eq2,axiom,
    ! [R: set_Pr8056137968301705908nteger,S3: set_Pr8056137968301705908nteger] :
      ( ( ord_le3636971675376928563eger_o
        @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R )
        @ ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S3 ) )
      = ( ord_le3216752416896350996nteger @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6403_pred__subset__eq2,axiom,
    ! [R: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
      ( ( ord_le4340812435750786203eger_o
        @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R )
        @ ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S3 ) )
      = ( ord_le653643898420964396nteger @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6404_pred__subset__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
      ( ( ord_le5643404153117327598_int_o
        @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R )
        @ ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S3 ) )
      = ( ord_le8725513860283290265nt_int @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6405_pred__subset__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
      ( ( ord_le2124322318746777828_int_o
        @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R )
        @ ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S3 ) )
      = ( ord_le135402666524580259nt_int @ R @ S3 ) ) ).

% pred_subset_eq2
thf(fact_6406_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
      = ( ( dvd_dvd_real @ A @ B )
        & ~ ( dvd_dvd_real @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6407_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6408_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6409_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( ( dvd_dvd_Code_integer @ A @ B )
        & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_6410_pred__equals__eq2,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ R ) )
        = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6411_pred__equals__eq2,axiom,
    ! [R: set_Pr958786334691620121nt_int,S3: set_Pr958786334691620121nt_int] :
      ( ( ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ R ) )
        = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6412_pred__equals__eq2,axiom,
    ! [R: set_Pr8056137968301705908nteger,S3: set_Pr8056137968301705908nteger] :
      ( ( ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ R ) )
        = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6413_pred__equals__eq2,axiom,
    ! [R: set_Pr1281608226676607948nteger,S3: set_Pr1281608226676607948nteger] :
      ( ( ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ R ) )
        = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6414_pred__equals__eq2,axiom,
    ! [R: set_Pr9222295170931077689nt_int,S3: set_Pr9222295170931077689nt_int] :
      ( ( ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ R ) )
        = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6415_pred__equals__eq2,axiom,
    ! [R: set_Pr1872883991513573699nt_int,S3: set_Pr1872883991513573699nt_int] :
      ( ( ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ R ) )
        = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ S3 ) ) )
      = ( R = S3 ) ) ).

% pred_equals_eq2
thf(fact_6416_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ A ) )
        @ ( collect_real
          @ ^ [C3: real] : ( dvd_dvd_real @ C3 @ B ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6417_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ A ) )
        @ ( collect_nat
          @ ^ [C3: nat] : ( dvd_dvd_nat @ C3 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6418_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C3: code_integer] : ( dvd_dvd_Code_integer @ C3 @ B ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6419_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ A ) )
        @ ( collect_int
          @ ^ [C3: int] : ( dvd_dvd_int @ C3 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_6420_less__set__def,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( ord_le549003669493604880_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_6421_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( ord_less_real_o
          @ ^ [X: real] : ( member_real @ X @ A5 )
          @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_6422_less__set__def,axiom,
    ( ord_less_set_set_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( ord_less_set_nat_o
          @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
          @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_6423_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A5 )
          @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_6424_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ord_less_int_o
          @ ^ [X: int] : ( member_int @ X @ A5 )
          @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ).

% less_set_def
thf(fact_6425_Collect__subset,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6426_Collect__subset,axiom,
    ! [A2: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6427_Collect__subset,axiom,
    ! [A2: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( member_list_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6428_Collect__subset,axiom,
    ! [A2: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6429_Collect__subset,axiom,
    ! [A2: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6430_Collect__subset,axiom,
    ! [A2: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_6431_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_6432_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X: real] : ( member_real @ X @ A5 )
          @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_6433_less__eq__set__def,axiom,
    ( ord_le6893508408891458716et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( ord_le3964352015994296041_nat_o
          @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
          @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_6434_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A5 )
          @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_6435_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X: int] : ( member_int @ X @ A5 )
          @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ).

% less_eq_set_def
thf(fact_6436_prop__restrict,axiom,
    ! [X3: product_prod_nat_nat,Z5: set_Pr1261947904930325089at_nat,X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ( member8440522571783428010at_nat @ X3 @ Z5 )
     => ( ( ord_le3146513528884898305at_nat @ Z5
          @ ( collec3392354462482085612at_nat
            @ ^ [X: product_prod_nat_nat] :
                ( ( member8440522571783428010at_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_6437_prop__restrict,axiom,
    ! [X3: real,Z5: set_real,X7: set_real,P: real > $o] :
      ( ( member_real @ X3 @ Z5 )
     => ( ( ord_less_eq_set_real @ Z5
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_6438_prop__restrict,axiom,
    ! [X3: list_nat,Z5: set_list_nat,X7: set_list_nat,P: list_nat > $o] :
      ( ( member_list_nat @ X3 @ Z5 )
     => ( ( ord_le6045566169113846134st_nat @ Z5
          @ ( collect_list_nat
            @ ^ [X: list_nat] :
                ( ( member_list_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_6439_prop__restrict,axiom,
    ! [X3: set_nat,Z5: set_set_nat,X7: set_set_nat,P: set_nat > $o] :
      ( ( member_set_nat @ X3 @ Z5 )
     => ( ( ord_le6893508408891458716et_nat @ Z5
          @ ( collect_set_nat
            @ ^ [X: set_nat] :
                ( ( member_set_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_6440_prop__restrict,axiom,
    ! [X3: nat,Z5: set_nat,X7: set_nat,P: nat > $o] :
      ( ( member_nat @ X3 @ Z5 )
     => ( ( ord_less_eq_set_nat @ Z5
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_6441_prop__restrict,axiom,
    ! [X3: int,Z5: set_int,X7: set_int,P: int > $o] :
      ( ( member_int @ X3 @ Z5 )
     => ( ( ord_less_eq_set_int @ Z5
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ X7 )
                & ( P @ X ) ) ) )
       => ( P @ X3 ) ) ) ).

% prop_restrict
thf(fact_6442_Collect__restrict,axiom,
    ! [X7: set_Pr1261947904930325089at_nat,P: product_prod_nat_nat > $o] :
      ( ord_le3146513528884898305at_nat
      @ ( collec3392354462482085612at_nat
        @ ^ [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_6443_Collect__restrict,axiom,
    ! [X7: set_real,P: real > $o] :
      ( ord_less_eq_set_real
      @ ( collect_real
        @ ^ [X: real] :
            ( ( member_real @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_6444_Collect__restrict,axiom,
    ! [X7: set_list_nat,P: list_nat > $o] :
      ( ord_le6045566169113846134st_nat
      @ ( collect_list_nat
        @ ^ [X: list_nat] :
            ( ( member_list_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_6445_Collect__restrict,axiom,
    ! [X7: set_set_nat,P: set_nat > $o] :
      ( ord_le6893508408891458716et_nat
      @ ( collect_set_nat
        @ ^ [X: set_nat] :
            ( ( member_set_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_6446_Collect__restrict,axiom,
    ! [X7: set_nat,P: nat > $o] :
      ( ord_less_eq_set_nat
      @ ( collect_nat
        @ ^ [X: nat] :
            ( ( member_nat @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_6447_Collect__restrict,axiom,
    ! [X7: set_int,P: int > $o] :
      ( ord_less_eq_set_int
      @ ( collect_int
        @ ^ [X: int] :
            ( ( member_int @ X @ X7 )
            & ( P @ X ) ) )
      @ X7 ) ).

% Collect_restrict
thf(fact_6448_pred__subset__eq,axiom,
    ! [R: set_Pr1261947904930325089at_nat,S3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le704812498762024988_nat_o
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ R )
        @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ S3 ) )
      = ( ord_le3146513528884898305at_nat @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_6449_pred__subset__eq,axiom,
    ! [R: set_real,S3: set_real] :
      ( ( ord_less_eq_real_o
        @ ^ [X: real] : ( member_real @ X @ R )
        @ ^ [X: real] : ( member_real @ X @ S3 ) )
      = ( ord_less_eq_set_real @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_6450_pred__subset__eq,axiom,
    ! [R: set_set_nat,S3: set_set_nat] :
      ( ( ord_le3964352015994296041_nat_o
        @ ^ [X: set_nat] : ( member_set_nat @ X @ R )
        @ ^ [X: set_nat] : ( member_set_nat @ X @ S3 ) )
      = ( ord_le6893508408891458716et_nat @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_6451_pred__subset__eq,axiom,
    ! [R: set_nat,S3: set_nat] :
      ( ( ord_less_eq_nat_o
        @ ^ [X: nat] : ( member_nat @ X @ R )
        @ ^ [X: nat] : ( member_nat @ X @ S3 ) )
      = ( ord_less_eq_set_nat @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_6452_pred__subset__eq,axiom,
    ! [R: set_int,S3: set_int] :
      ( ( ord_less_eq_int_o
        @ ^ [X: int] : ( member_int @ X @ R )
        @ ^ [X: int] : ( member_int @ X @ S3 ) )
      = ( ord_less_eq_set_int @ R @ S3 ) ) ).

% pred_subset_eq
thf(fact_6453_lambda__one,axiom,
    ( ( ^ [X: complex] : X )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_6454_lambda__one,axiom,
    ( ( ^ [X: real] : X )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_6455_lambda__one,axiom,
    ( ( ^ [X: rat] : X )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_6456_lambda__one,axiom,
    ( ( ^ [X: nat] : X )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_6457_lambda__one,axiom,
    ( ( ^ [X: int] : X )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_6458_lambda__zero,axiom,
    ( ( ^ [H: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_6459_lambda__zero,axiom,
    ( ( ^ [H: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_6460_lambda__zero,axiom,
    ( ( ^ [H: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_6461_lambda__zero,axiom,
    ( ( ^ [H: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_6462_lambda__zero,axiom,
    ( ( ^ [H: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_6463_uminus__set__def,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( uminus8676089048583255045_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6464_uminus__set__def,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A5: set_real] :
          ( collect_real
          @ ( uminus_uminus_real_o
            @ ^ [X: real] : ( member_real @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6465_uminus__set__def,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A5: set_list_nat] :
          ( collect_list_nat
          @ ( uminus5770388063884162150_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6466_uminus__set__def,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A5: set_set_nat] :
          ( collect_set_nat
          @ ( uminus6401447641752708672_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6467_uminus__set__def,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A5: set_nat] :
          ( collect_nat
          @ ( uminus_uminus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6468_uminus__set__def,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A5: set_int] :
          ( collect_int
          @ ( uminus_uminus_int_o
            @ ^ [X: int] : ( member_int @ X @ A5 ) ) ) ) ) ).

% uminus_set_def
thf(fact_6469_Collect__neg__eq,axiom,
    ! [P: real > $o] :
      ( ( collect_real
        @ ^ [X: real] :
            ~ ( P @ X ) )
      = ( uminus612125837232591019t_real @ ( collect_real @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6470_Collect__neg__eq,axiom,
    ! [P: list_nat > $o] :
      ( ( collect_list_nat
        @ ^ [X: list_nat] :
            ~ ( P @ X ) )
      = ( uminus3195874150345416415st_nat @ ( collect_list_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6471_Collect__neg__eq,axiom,
    ! [P: set_nat > $o] :
      ( ( collect_set_nat
        @ ^ [X: set_nat] :
            ~ ( P @ X ) )
      = ( uminus613421341184616069et_nat @ ( collect_set_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6472_Collect__neg__eq,axiom,
    ! [P: nat > $o] :
      ( ( collect_nat
        @ ^ [X: nat] :
            ~ ( P @ X ) )
      = ( uminus5710092332889474511et_nat @ ( collect_nat @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6473_Collect__neg__eq,axiom,
    ! [P: int > $o] :
      ( ( collect_int
        @ ^ [X: int] :
            ~ ( P @ X ) )
      = ( uminus1532241313380277803et_int @ ( collect_int @ P ) ) ) ).

% Collect_neg_eq
thf(fact_6474_Compl__eq,axiom,
    ( uminus6524753893492686040at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ~ ( member8440522571783428010at_nat @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6475_Compl__eq,axiom,
    ( uminus612125837232591019t_real
    = ( ^ [A5: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ~ ( member_real @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6476_Compl__eq,axiom,
    ( uminus3195874150345416415st_nat
    = ( ^ [A5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ~ ( member_list_nat @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6477_Compl__eq,axiom,
    ( uminus613421341184616069et_nat
    = ( ^ [A5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ~ ( member_set_nat @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6478_Compl__eq,axiom,
    ( uminus5710092332889474511et_nat
    = ( ^ [A5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ~ ( member_nat @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6479_Compl__eq,axiom,
    ( uminus1532241313380277803et_int
    = ( ^ [A5: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ~ ( member_int @ X @ A5 ) ) ) ) ).

% Compl_eq
thf(fact_6480_set__diff__eq,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A5 )
              & ~ ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6481_set__diff__eq,axiom,
    ( minus_minus_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( member_real @ X @ A5 )
              & ~ ( member_real @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6482_set__diff__eq,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( member_list_nat @ X @ A5 )
              & ~ ( member_list_nat @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6483_set__diff__eq,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( member_set_nat @ X @ A5 )
              & ~ ( member_set_nat @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6484_set__diff__eq,axiom,
    ( minus_minus_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( member_int @ X @ A5 )
              & ~ ( member_int @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6485_set__diff__eq,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( member_nat @ X @ A5 )
              & ~ ( member_nat @ X @ B5 ) ) ) ) ) ).

% set_diff_eq
thf(fact_6486_insert__compr,axiom,
    ( insert9069300056098147895at_nat
    = ( ^ [A3: produc3843707927480180839at_nat,B5: set_Pr4329608150637261639at_nat] :
          ( collec6321179662152712658at_nat
          @ ^ [X: produc3843707927480180839at_nat] :
              ( ( X = A3 )
              | ( member8757157785044589968at_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_6487_insert__compr,axiom,
    ( insert8211810215607154385at_nat
    = ( ^ [A3: product_prod_nat_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ^ [X: product_prod_nat_nat] :
              ( ( X = A3 )
              | ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_6488_insert__compr,axiom,
    ( insert_real
    = ( ^ [A3: real,B5: set_real] :
          ( collect_real
          @ ^ [X: real] :
              ( ( X = A3 )
              | ( member_real @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_6489_insert__compr,axiom,
    ( insert_list_nat
    = ( ^ [A3: list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ^ [X: list_nat] :
              ( ( X = A3 )
              | ( member_list_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_6490_insert__compr,axiom,
    ( insert_set_nat
    = ( ^ [A3: set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] :
              ( ( X = A3 )
              | ( member_set_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_6491_insert__compr,axiom,
    ( insert_nat
    = ( ^ [A3: nat,B5: set_nat] :
          ( collect_nat
          @ ^ [X: nat] :
              ( ( X = A3 )
              | ( member_nat @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_6492_insert__compr,axiom,
    ( insert_int
    = ( ^ [A3: int,B5: set_int] :
          ( collect_int
          @ ^ [X: int] :
              ( ( X = A3 )
              | ( member_int @ X @ B5 ) ) ) ) ) ).

% insert_compr
thf(fact_6493_minus__set__def,axiom,
    ( minus_1356011639430497352at_nat
    = ( ^ [A5: set_Pr1261947904930325089at_nat,B5: set_Pr1261947904930325089at_nat] :
          ( collec3392354462482085612at_nat
          @ ( minus_2270307095948843157_nat_o
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A5 )
            @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_6494_minus__set__def,axiom,
    ( minus_minus_set_real
    = ( ^ [A5: set_real,B5: set_real] :
          ( collect_real
          @ ( minus_minus_real_o
            @ ^ [X: real] : ( member_real @ X @ A5 )
            @ ^ [X: real] : ( member_real @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_6495_minus__set__def,axiom,
    ( minus_7954133019191499631st_nat
    = ( ^ [A5: set_list_nat,B5: set_list_nat] :
          ( collect_list_nat
          @ ( minus_1139252259498527702_nat_o
            @ ^ [X: list_nat] : ( member_list_nat @ X @ A5 )
            @ ^ [X: list_nat] : ( member_list_nat @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_6496_minus__set__def,axiom,
    ( minus_2163939370556025621et_nat
    = ( ^ [A5: set_set_nat,B5: set_set_nat] :
          ( collect_set_nat
          @ ( minus_6910147592129066416_nat_o
            @ ^ [X: set_nat] : ( member_set_nat @ X @ A5 )
            @ ^ [X: set_nat] : ( member_set_nat @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_6497_minus__set__def,axiom,
    ( minus_minus_set_int
    = ( ^ [A5: set_int,B5: set_int] :
          ( collect_int
          @ ( minus_minus_int_o
            @ ^ [X: int] : ( member_int @ X @ A5 )
            @ ^ [X: int] : ( member_int @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_6498_minus__set__def,axiom,
    ( minus_minus_set_nat
    = ( ^ [A5: set_nat,B5: set_nat] :
          ( collect_nat
          @ ( minus_minus_nat_o
            @ ^ [X: nat] : ( member_nat @ X @ A5 )
            @ ^ [X: nat] : ( member_nat @ X @ B5 ) ) ) ) ) ).

% minus_set_def
thf(fact_6499_insert__Collect,axiom,
    ! [A: product_prod_nat_nat,P: product_prod_nat_nat > $o] :
      ( ( insert8211810215607154385at_nat @ A @ ( collec3392354462482085612at_nat @ P ) )
      = ( collec3392354462482085612at_nat
        @ ^ [U2: product_prod_nat_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6500_insert__Collect,axiom,
    ! [A: produc3843707927480180839at_nat,P: produc3843707927480180839at_nat > $o] :
      ( ( insert9069300056098147895at_nat @ A @ ( collec6321179662152712658at_nat @ P ) )
      = ( collec6321179662152712658at_nat
        @ ^ [U2: produc3843707927480180839at_nat] :
            ( ( U2 != A )
           => ( P @ U2 ) ) ) ) ).

% insert_Collect
thf(fact_6501_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_6502_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_6503_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_6504_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_6505_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_6506_not__finite__existsD,axiom,
    ! [P: real > $o] :
      ( ~ ( finite_finite_real @ ( collect_real @ P ) )
     => ? [X_1: real] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6507_not__finite__existsD,axiom,
    ! [P: list_nat > $o] :
      ( ~ ( finite8100373058378681591st_nat @ ( collect_list_nat @ P ) )
     => ? [X_1: list_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6508_not__finite__existsD,axiom,
    ! [P: set_nat > $o] :
      ( ~ ( finite1152437895449049373et_nat @ ( collect_set_nat @ P ) )
     => ? [X_1: set_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6509_not__finite__existsD,axiom,
    ! [P: nat > $o] :
      ( ~ ( finite_finite_nat @ ( collect_nat @ P ) )
     => ? [X_1: nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6510_not__finite__existsD,axiom,
    ! [P: int > $o] :
      ( ~ ( finite_finite_int @ ( collect_int @ P ) )
     => ? [X_1: int] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6511_not__finite__existsD,axiom,
    ! [P: complex > $o] :
      ( ~ ( finite3207457112153483333omplex @ ( collect_complex @ P ) )
     => ? [X_1: complex] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6512_not__finite__existsD,axiom,
    ! [P: product_prod_nat_nat > $o] :
      ( ~ ( finite6177210948735845034at_nat @ ( collec3392354462482085612at_nat @ P ) )
     => ? [X_1: product_prod_nat_nat] : ( P @ X_1 ) ) ).

% not_finite_existsD
thf(fact_6513_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_nat,R: real > nat > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B2 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6514_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_int,R: real > int > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: int] :
              ( ( member_int @ X4 @ B2 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6515_pigeonhole__infinite__rel,axiom,
    ! [A2: set_real,B2: set_complex,R: real > complex > $o] :
      ( ~ ( finite_finite_real @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: complex] :
              ( ( member_complex @ X4 @ B2 )
              & ~ ( finite_finite_real
                  @ ( collect_real
                    @ ^ [A3: real] :
                        ( ( member_real @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6516_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_nat,R: nat > nat > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6517_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_int,R: nat > int > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: int] :
              ( ( member_int @ X4 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6518_pigeonhole__infinite__rel,axiom,
    ! [A2: set_nat,B2: set_complex,R: nat > complex > $o] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: complex] :
              ( ( member_complex @ X4 @ B2 )
              & ~ ( finite_finite_nat
                  @ ( collect_nat
                    @ ^ [A3: nat] :
                        ( ( member_nat @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6519_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_nat,R: int > nat > $o] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B2 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A3: int] :
                        ( ( member_int @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6520_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_int,R: int > int > $o] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( finite_finite_int @ B2 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ? [Xa: int] :
                  ( ( member_int @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: int] :
              ( ( member_int @ X4 @ B2 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A3: int] :
                        ( ( member_int @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6521_pigeonhole__infinite__rel,axiom,
    ! [A2: set_int,B2: set_complex,R: int > complex > $o] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ? [Xa: complex] :
                  ( ( member_complex @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: complex] :
              ( ( member_complex @ X4 @ B2 )
              & ~ ( finite_finite_int
                  @ ( collect_int
                    @ ^ [A3: int] :
                        ( ( member_int @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6522_pigeonhole__infinite__rel,axiom,
    ! [A2: set_complex,B2: set_nat,R: complex > nat > $o] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
             => ? [Xa: nat] :
                  ( ( member_nat @ Xa @ B2 )
                  & ( R @ X4 @ Xa ) ) )
         => ? [X4: nat] :
              ( ( member_nat @ X4 @ B2 )
              & ~ ( finite3207457112153483333omplex
                  @ ( collect_complex
                    @ ^ [A3: complex] :
                        ( ( member_complex @ A3 @ A2 )
                        & ( R @ A3 @ X4 ) ) ) ) ) ) ) ) ).

% pigeonhole_infinite_rel
thf(fact_6523_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X: real,Y: real] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6524_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu3: nat] :
              ? [X: real,Y: real] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6525_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > int] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [Uu3: int] :
              ? [X: real,Y: real] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6526_finite__image__set2,axiom,
    ! [P: real > $o,Q: real > $o,F: real > real > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_real @ ( collect_real @ Q ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Uu3: complex] :
              ? [X: real,Y: real] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6527_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X: real,Y: nat] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6528_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu3: nat] :
              ? [X: real,Y: nat] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6529_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > int] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [Uu3: int] :
              ? [X: real,Y: nat] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6530_finite__image__set2,axiom,
    ! [P: real > $o,Q: nat > $o,F: real > nat > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_nat @ ( collect_nat @ Q ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [Uu3: complex] :
              ? [X: real,Y: nat] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6531_finite__image__set2,axiom,
    ! [P: real > $o,Q: int > $o,F: real > int > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_int @ ( collect_int @ Q ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [Uu3: real] :
              ? [X: real,Y: int] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6532_finite__image__set2,axiom,
    ! [P: real > $o,Q: int > $o,F: real > int > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( ( finite_finite_int @ ( collect_int @ Q ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [Uu3: nat] :
              ? [X: real,Y: int] :
                ( ( Uu3
                  = ( F @ X @ Y ) )
                & ( P @ X )
                & ( Q @ Y ) ) ) ) ) ) ).

% finite_image_set2
thf(fact_6533_finite__image__set,axiom,
    ! [P: real > $o,F: real > real] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Uu3: real] :
            ? [X: real] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6534_finite__image__set,axiom,
    ! [P: real > $o,F: real > nat] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu3: nat] :
            ? [X: real] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6535_finite__image__set,axiom,
    ! [P: real > $o,F: real > int] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [Uu3: int] :
            ? [X: real] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6536_finite__image__set,axiom,
    ! [P: real > $o,F: real > complex] :
      ( ( finite_finite_real @ ( collect_real @ P ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Uu3: complex] :
            ? [X: real] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6537_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Uu3: real] :
            ? [X: nat] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6538_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu3: nat] :
            ? [X: nat] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6539_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [Uu3: int] :
            ? [X: nat] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6540_finite__image__set,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Uu3: complex] :
            ? [X: nat] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6541_finite__image__set,axiom,
    ! [P: int > $o,F: int > real] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( finite_finite_real
        @ ( collect_real
          @ ^ [Uu3: real] :
            ? [X: int] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6542_finite__image__set,axiom,
    ! [P: int > $o,F: int > nat] :
      ( ( finite_finite_int @ ( collect_int @ P ) )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [Uu3: nat] :
            ? [X: int] :
              ( ( Uu3
                = ( F @ X ) )
              & ( P @ X ) ) ) ) ) ).

% finite_image_set
thf(fact_6543_bot__empty__eq2,axiom,
    ( bot_bot_int_int_o
    = ( ^ [X: int,Y: int] : ( member5262025264175285858nt_int @ ( product_Pair_int_int @ X @ Y ) @ bot_bo1796632182523588997nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_6544_bot__empty__eq2,axiom,
    ( bot_bo5358457235160185703eger_o
    = ( ^ [X: code_integer > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member3068662437193594005nteger @ ( produc6137756002093451184nteger @ X @ Y ) @ bot_bo3145834390647256904nteger ) ) ) ).

% bot_empty_eq2
thf(fact_6545_bot__empty__eq2,axiom,
    ( bot_bo3000040243691356879eger_o
    = ( ^ [X: produc6241069584506657477e_term > option6357759511663192854e_term,Y: produc8923325533196201883nteger] : ( member4164122664394876845nteger @ ( produc8603105652947943368nteger @ X @ Y ) @ bot_bo5443222936135328352nteger ) ) ) ).

% bot_empty_eq2
thf(fact_6546_bot__empty__eq2,axiom,
    ( bot_bo8662317086119403298_int_o
    = ( ^ [X: produc8551481072490612790e_term > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7618704894036264090nt_int @ ( produc5700946648718959541nt_int @ X @ Y ) @ bot_bo572930865798478029nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_6547_bot__empty__eq2,axiom,
    ( bot_bo1403522918969695512_int_o
    = ( ^ [X: int > option6357759511663192854e_term,Y: product_prod_int_int] : ( member7034335876925520548nt_int @ ( produc4305682042979456191nt_int @ X @ Y ) @ bot_bo4508923176915781079nt_int ) ) ) ).

% bot_empty_eq2
thf(fact_6548_bot__empty__eq2,axiom,
    ( bot_bot_nat_nat_o
    = ( ^ [X: nat,Y: nat] : ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ X @ Y ) @ bot_bo2099793752762293965at_nat ) ) ) ).

% bot_empty_eq2
thf(fact_6549_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_6550_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_6551_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_6552_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_6553_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit0 @ N2 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_6554_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_6555_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_6556_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T2: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T2 ) ) ) ) ).

% set_vebt_def
thf(fact_6557_finite__divisors__int,axiom,
    ! [I: int] :
      ( ( I != zero_zero_int )
     => ( finite_finite_int
        @ ( collect_int
          @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ I ) ) ) ) ).

% finite_divisors_int
thf(fact_6558_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_6559_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_6560_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_6561_numeral__code_I3_J,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bit1 @ N2 ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ N2 ) @ ( numeral_numeral_rat @ N2 ) ) @ one_one_rat ) ) ).

% numeral_code(3)
thf(fact_6562_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_6563_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_6564_finite__divisors__nat,axiom,
    ! [M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( finite_finite_nat
        @ ( collect_nat
          @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M ) ) ) ) ).

% finite_divisors_nat
thf(fact_6565_set__conv__nth,axiom,
    ( set_real2
    = ( ^ [Xs2: list_real] :
          ( collect_real
          @ ^ [Uu3: real] :
            ? [I2: nat] :
              ( ( Uu3
                = ( nth_real @ Xs2 @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_size_list_real @ Xs2 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_6566_set__conv__nth,axiom,
    ( set_list_nat2
    = ( ^ [Xs2: list_list_nat] :
          ( collect_list_nat
          @ ^ [Uu3: list_nat] :
            ? [I2: nat] :
              ( ( Uu3
                = ( nth_list_nat @ Xs2 @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_s3023201423986296836st_nat @ Xs2 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_6567_set__conv__nth,axiom,
    ( set_set_nat2
    = ( ^ [Xs2: list_set_nat] :
          ( collect_set_nat
          @ ^ [Uu3: set_nat] :
            ? [I2: nat] :
              ( ( Uu3
                = ( nth_set_nat @ Xs2 @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_s3254054031482475050et_nat @ Xs2 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_6568_set__conv__nth,axiom,
    ( set_VEBT_VEBT2
    = ( ^ [Xs2: list_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [Uu3: vEBT_VEBT] :
            ? [I2: nat] :
              ( ( Uu3
                = ( nth_VEBT_VEBT @ Xs2 @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_s6755466524823107622T_VEBT @ Xs2 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_6569_set__conv__nth,axiom,
    ( set_o2
    = ( ^ [Xs2: list_o] :
          ( collect_o
          @ ^ [Uu3: $o] :
            ? [I2: nat] :
              ( ( Uu3
                = ( nth_o @ Xs2 @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_size_list_o @ Xs2 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_6570_set__conv__nth,axiom,
    ( set_nat2
    = ( ^ [Xs2: list_nat] :
          ( collect_nat
          @ ^ [Uu3: nat] :
            ? [I2: nat] :
              ( ( Uu3
                = ( nth_nat @ Xs2 @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_size_list_nat @ Xs2 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_6571_set__conv__nth,axiom,
    ( set_int2
    = ( ^ [Xs2: list_int] :
          ( collect_int
          @ ^ [Uu3: int] :
            ? [I2: nat] :
              ( ( Uu3
                = ( nth_int @ Xs2 @ I2 ) )
              & ( ord_less_nat @ I2 @ ( size_size_list_int @ Xs2 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_6572_finite__lists__length__eq,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ( size_s3451745648224563538omplex @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6573_finite__lists__length__eq,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs2: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
              & ( ( size_s5460976970255530739at_nat @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6574_finite__lists__length__eq,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ( size_s6755466524823107622T_VEBT @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6575_finite__lists__length__eq,axiom,
    ! [A2: set_o,N2: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_o @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6576_finite__lists__length__eq,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_nat @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6577_finite__lists__length__eq,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ( size_size_list_int @ Xs2 )
                = N2 ) ) ) ) ) ).

% finite_lists_length_eq
thf(fact_6578_finite__lists__length__le,axiom,
    ! [A2: set_complex,N2: nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( finite8712137658972009173omplex
        @ ( collect_list_complex
          @ ^ [Xs2: list_complex] :
              ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s3451745648224563538omplex @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6579_finite__lists__length__le,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,N2: nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( finite500796754983035824at_nat
        @ ( collec3343600615725829874at_nat
          @ ^ [Xs2: list_P6011104703257516679at_nat] :
              ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s5460976970255530739at_nat @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6580_finite__lists__length__le,axiom,
    ! [A2: set_VEBT_VEBT,N2: nat] :
      ( ( finite5795047828879050333T_VEBT @ A2 )
     => ( finite3004134309566078307T_VEBT
        @ ( collec5608196760682091941T_VEBT
          @ ^ [Xs2: list_VEBT_VEBT] :
              ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_s6755466524823107622T_VEBT @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6581_finite__lists__length__le,axiom,
    ! [A2: set_o,N2: nat] :
      ( ( finite_finite_o @ A2 )
     => ( finite_finite_list_o
        @ ( collect_list_o
          @ ^ [Xs2: list_o] :
              ( ( ord_less_eq_set_o @ ( set_o2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_o @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6582_finite__lists__length__le,axiom,
    ! [A2: set_nat,N2: nat] :
      ( ( finite_finite_nat @ A2 )
     => ( finite8100373058378681591st_nat
        @ ( collect_list_nat
          @ ^ [Xs2: list_nat] :
              ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_nat @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6583_finite__lists__length__le,axiom,
    ! [A2: set_int,N2: nat] :
      ( ( finite_finite_int @ A2 )
     => ( finite3922522038869484883st_int
        @ ( collect_list_int
          @ ^ [Xs2: list_int] :
              ( ( ord_less_eq_set_int @ ( set_int2 @ Xs2 ) @ A2 )
              & ( ord_less_eq_nat @ ( size_size_list_int @ Xs2 ) @ N2 ) ) ) ) ) ).

% finite_lists_length_le
thf(fact_6584_vebt__buildup_Osimps_I3_J,axiom,
    ! [Va: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( 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 ) ) )
       => ( ( vEBT_vebt_buildup @ ( suc @ ( suc @ Va ) ) )
          = ( 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.simps(3)
thf(fact_6585_vebt__buildup_Oelims,axiom,
    ! [X3: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X3 )
        = Y3 )
     => ( ( ( X3 = zero_zero_nat )
         => ( Y3
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X3
              = ( suc @ zero_zero_nat ) )
           => ( Y3
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va2: nat] :
                ( ( X3
                  = ( suc @ ( suc @ Va2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y3
                        = ( 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 ) ) )
                     => ( Y3
                        = ( 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.elims
thf(fact_6586_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X3: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X3 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_6587_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_6588_dbl__dec__def,axiom,
    ( neg_nu6075765906172075777c_real
    = ( ^ [X: real] : ( minus_minus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_dec_def
thf(fact_6589_dbl__dec__def,axiom,
    ( neg_nu3179335615603231917ec_rat
    = ( ^ [X: rat] : ( minus_minus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_dec_def
thf(fact_6590_dbl__dec__def,axiom,
    ( neg_nu3811975205180677377ec_int
    = ( ^ [X: int] : ( minus_minus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_dec_def
thf(fact_6591_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ X3 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_6592_vebt__member_Osimps_I5_J,axiom,
    ! [Mi: nat,Ma: nat,Va: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,X3: nat] :
      ( ( vEBT_vebt_member @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ ( suc @ Va ) ) @ TreeList2 @ Summary ) @ X3 )
      = ( ( X3 != Mi )
       => ( ( X3 != Ma )
         => ( ~ ( ord_less_nat @ X3 @ Mi )
            & ( ~ ( ord_less_nat @ X3 @ Mi )
             => ( ~ ( ord_less_nat @ Ma @ X3 )
                & ( ~ ( ord_less_nat @ Ma @ X3 )
                 => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
                     => ( vEBT_vebt_member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ ( suc @ Va ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.simps(5)
thf(fact_6593_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc: vEBT_VEBT,X3: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc ) @ X3 )
      = ( ( X3 = Mi )
        | ( X3 = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X3 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_6594_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
        = Y3 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X3
              = ( vEBT_Leaf @ A4 @ B4 ) )
           => ( Y3
              = ( ~ ( ( ( 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] :
                ( X3
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y3 )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( Y3
                  = ( ~ ( ( ( 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_6595_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X3
              = ( 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] :
              ( ? [S2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
             => ~ ( ( ( 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_6596_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X3
              = ( 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] :
              ( X3
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( ( ( 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_6597_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ! [Mi2: nat,Ma2: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X3
                = ( 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] :
                  ( X3
                  = ( 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] :
                    ( X3
                    = ( 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_6598_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ 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 @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_6599_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M2: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_6600_one__xor__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ one_one_Code_integer @ A )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_6601_one__xor__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ A )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_6602_one__xor__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ A )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% one_xor_eq
thf(fact_6603_xor__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3222712562003087583nteger @ A @ one_one_Code_integer )
      = ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ ( zero_n356916108424825756nteger @ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n356916108424825756nteger
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_6604_xor__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ one_one_nat )
      = ( minus_minus_nat @ ( plus_plus_nat @ A @ ( zero_n2687167440665602831ol_nat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2687167440665602831ol_nat
          @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_6605_xor__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ one_one_int )
      = ( minus_minus_int @ ( plus_plus_int @ A @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
        @ ( zero_n2684676970156552555ol_int
          @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% xor_one_eq
thf(fact_6606_vebt__member_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X3 @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X3
              = ( 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,Va2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Summary2: vEBT_VEBT] :
                  ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                              & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(2)
thf(fact_6607_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X3
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X3
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X3
                    = ( 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] :
                      ( X3
                      = ( 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] :
                        ( X3
                        = ( 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_6608_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y3 )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X3
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y3 )
         => ( ! [Mi2: nat,Ma2: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y3
                  = ( ~ ( ( Xa2 = Mi2 )
                        | ( Xa2 = Ma2 ) ) ) ) )
           => ( ! [Mi2: nat,Ma2: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc2: vEBT_VEBT] :
                      ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                 => ( Y3
                    = ( ~ ( ( 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] :
                        ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                   => ( Y3
                      = ( ~ ( ( ( 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_6609_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( groups9119017779487936845_o_nat @ zero_n2687167440665602831ol_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Bs ) @ N2 )
      = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N2 ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_6610_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( groups9116527308978886569_o_int @ zero_n2684676970156552555ol_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Bs ) @ N2 )
      = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N2 ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_6611_vebt__member_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
     => ( ! [A4: $o,B4: $o] :
            ( ( X3
              = ( 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] :
              ( X3
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
         => ( ! [V2: product_prod_nat_nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                ( X3
               != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
           => ( ! [V2: product_prod_nat_nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( X3
                 != ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
             => ~ ! [Mi2: nat,Ma2: nat,Va2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Summary2: vEBT_VEBT] :
                        ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.elims(3)
thf(fact_6612_of__int__code__if,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] :
          ( if_complex @ ( K3 = zero_zero_int ) @ zero_zero_complex
          @ ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_complex
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( ring_17405671764205052669omplex @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_complex ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6613_of__int__code__if,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] :
          ( if_int @ ( K3 = zero_zero_int ) @ zero_zero_int
          @ ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( ring_1_of_int_int @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_int
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_int ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6614_of__int__code__if,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] :
          ( if_real @ ( K3 = zero_zero_int ) @ zero_zero_real
          @ ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( ring_1_of_int_real @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_real
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_real ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6615_of__int__code__if,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] :
          ( if_rat @ ( K3 = zero_zero_int ) @ zero_zero_rat
          @ ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( ring_1_of_int_rat @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_rat
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( ring_1_of_int_rat @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_rat ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6616_of__int__code__if,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
          @ ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( ring_18347121197199848620nteger @ ( uminus_uminus_int @ K3 ) ) )
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% of_int_code_if
thf(fact_6617_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_6618_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_6619_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_6620_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > complex,Y3: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6621_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6622_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > complex,Y3: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6623_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( times_times_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_complex ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6624_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > real,Y3: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6625_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6626_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > real,Y3: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6627_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( times_times_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_real ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6628_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > rat,Y3: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( times_times_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6629_prod_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != one_one_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != one_one_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( times_times_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != one_one_rat ) ) ) ) ) ) ).

% prod.finite_Collect_op
thf(fact_6630_vebt__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_vebt_member @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( Y3
                  = ( ( ( 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] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ~ Y3
                 => ~ ( 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] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ zero_zero_nat @ Uy2 @ Uz2 ) )
                 => ( ~ Y3
                   => ~ ( 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] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ V2 ) @ ( suc @ zero_zero_nat ) @ Vb2 @ Vc2 ) )
                   => ( ~ Y3
                     => ~ ( 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,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( Y3
                          = ( ( 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( 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 @ Va2 ) ) @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(1)
thf(fact_6631_vebt__member_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_vebt_member @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X3
                = ( 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] :
                ( ( X3
                  = ( 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] :
                  ( ( X3
                    = ( 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] :
                    ( ( X3
                      = ( 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,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
                     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(3)
thf(fact_6632_xor__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
       != ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% xor_negative_int_iff
thf(fact_6633_XOR__upper,axiom,
    ! [X3: int,N2: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% XOR_upper
thf(fact_6634_vebt__member_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_vebt_member @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X3
                = ( 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,Va2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_vebt_member_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ ( suc @ Va2 ) ) @ 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 @ Va2 ) ) @ ( 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 @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa2 @ ( divide_divide_nat @ ( suc @ ( suc @ Va2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% vebt_member.pelims(2)
thf(fact_6635_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > complex,Y3: real > complex] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6636_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > complex,Y3: nat > complex] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6637_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > complex,Y3: int > complex] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_complex ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_complex ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6638_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > complex,Y3: complex > complex] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_complex ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_complex ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( plus_plus_complex @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_complex ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6639_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > real,Y3: real > real] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6640_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > real,Y3: nat > real] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6641_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_int,X3: int > real,Y3: int > real] :
      ( ( finite_finite_int
        @ ( collect_int
          @ ^ [I2: int] :
              ( ( member_int @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite_finite_int
          @ ( collect_int
            @ ^ [I2: int] :
                ( ( member_int @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6642_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_complex,X3: complex > real,Y3: complex > real] :
      ( ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [I2: complex] :
              ( ( member_complex @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_real ) ) ) )
     => ( ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_real ) ) ) )
       => ( finite3207457112153483333omplex
          @ ( collect_complex
            @ ^ [I2: complex] :
                ( ( member_complex @ I2 @ I5 )
                & ( ( plus_plus_real @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_real ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6643_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_real,X3: real > rat,Y3: real > rat] :
      ( ( finite_finite_real
        @ ( collect_real
          @ ^ [I2: real] :
              ( ( member_real @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_real
          @ ( collect_real
            @ ^ [I2: real] :
                ( ( member_real @ I2 @ I5 )
                & ( ( plus_plus_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6644_sum_Ofinite__Collect__op,axiom,
    ! [I5: set_nat,X3: nat > rat,Y3: nat > rat] :
      ( ( finite_finite_nat
        @ ( collect_nat
          @ ^ [I2: nat] :
              ( ( member_nat @ I2 @ I5 )
              & ( ( X3 @ I2 )
               != zero_zero_rat ) ) ) )
     => ( ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( Y3 @ I2 )
                 != zero_zero_rat ) ) ) )
       => ( finite_finite_nat
          @ ( collect_nat
            @ ^ [I2: nat] :
                ( ( member_nat @ I2 @ I5 )
                & ( ( plus_plus_rat @ ( X3 @ I2 ) @ ( Y3 @ I2 ) )
                 != zero_zero_rat ) ) ) ) ) ) ).

% sum.finite_Collect_op
thf(fact_6645_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( Y3
                  = ( ( ( 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] :
                ( ( X3
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y3
                 => ~ ( 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,S2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
                 => ( ( Y3
                      = ( ( ( 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 @ S2 ) @ Xa2 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_6646_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X3
                = ( 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,S2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ 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_6647_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [A4: $o,B4: $o] :
              ( ( X3
                = ( 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] :
                ( ( X3
                  = ( 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,S2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S2 ) @ 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_6648_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3
                  = ( 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] :
                  ( ( X3
                    = ( 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] :
                    ( ( X3
                      = ( 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] :
                      ( ( X3
                        = ( 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_6649_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y3
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa2 ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y3
                 => ~ ( 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] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y3
                      = ( ( 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] :
                    ( ( X3
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc2 ) )
                   => ( ( Y3
                        = ( ( 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] :
                      ( ( X3
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                     => ( ( Y3
                          = ( ( ( 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_6650_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_membermima @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Mi2: nat,Ma2: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X3
                = ( 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] :
                ( ( X3
                  = ( 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] :
                  ( ( X3
                    = ( 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_6651_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_6652_arsinh__0,axiom,
    ( ( arsinh_real @ zero_zero_real )
    = zero_zero_real ) ).

% arsinh_0
thf(fact_6653_artanh__0,axiom,
    ( ( artanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% artanh_0
thf(fact_6654_vebt__buildup_Opelims,axiom,
    ! [X3: nat,Y3: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X3 )
        = Y3 )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X3 )
       => ( ( ( X3 = zero_zero_nat )
           => ( ( Y3
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X3
                = ( suc @ zero_zero_nat ) )
             => ( ( Y3
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va2: nat] :
                  ( ( X3
                    = ( suc @ ( suc @ Va2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y3
                          = ( 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 ) ) )
                       => ( Y3
                          = ( 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 ) ) ) ) ) ) ) ) )
                   => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ ( suc @ Va2 ) ) ) ) ) ) ) ) ) ).

% vebt_buildup.pelims
thf(fact_6655_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) @ one_one_int ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6656_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) @ one_one_nat ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6657_divmod__algorithm__code_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) @ one_one_Code_integer ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(6)
thf(fact_6658_artanh__def,axiom,
    ( artanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( ln_ln_real @ ( divide_divide_real @ ( plus_plus_real @ one_one_real @ X ) @ ( minus_minus_real @ one_one_real @ X ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% artanh_def
thf(fact_6659_ln__inj__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ( ln_ln_real @ X3 )
            = ( ln_ln_real @ Y3 ) )
          = ( X3 = Y3 ) ) ) ) ).

% ln_inj_iff
thf(fact_6660_ln__less__cancel__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
          = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_6661_case__prod__conv,axiom,
    ! [F: nat > nat > nat,A: nat,B: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6662_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,A: nat,B: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6663_case__prod__conv,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6664_case__prod__conv,axiom,
    ! [F: int > int > product_prod_int_int,A: int,B: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6665_case__prod__conv,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
      = ( F @ A @ B ) ) ).

% case_prod_conv
thf(fact_6666_ln__le__cancel__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) )
          = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_6667_ln__less__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
        = ( ord_less_real @ X3 @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_6668_ln__gt__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
        = ( ord_less_real @ one_one_real @ X3 ) ) ) ).

% ln_gt_zero_iff
thf(fact_6669_ln__eq__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ( ln_ln_real @ X3 )
          = zero_zero_real )
        = ( X3 = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_6670_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_6671_ln__le__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_6672_ln__ge__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
        = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).

% ln_ge_zero_iff
thf(fact_6673_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5052692396658037445od_int @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [Q5: int,R5: int] : ( product_Pair_int_int @ Q5 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6674_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique5055182867167087721od_nat @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc2626176000494625587at_nat
        @ ^ [Q5: nat,R5: nat] : ( product_Pair_nat_nat @ Q5 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique5055182867167087721od_nat @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6675_divmod__algorithm__code_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( unique3479559517661332726nteger @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( produc6916734918728496179nteger
        @ ^ [Q5: code_integer,R5: code_integer] : ( produc1086072967326762835nteger @ Q5 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ R5 ) )
        @ ( unique3479559517661332726nteger @ M @ N2 ) ) ) ).

% divmod_algorithm_code(5)
thf(fact_6676_prod_Ocase__distrib,axiom,
    ! [H2: nat > nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6677_prod_Ocase__distrib,axiom,
    ! [H2: $o > $o,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6678_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > $o,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4947309494688390418_int_o
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6679_prod_Ocase__distrib,axiom,
    ! [H2: $o > product_prod_int_int,F: int > int > $o,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4947309494688390418_int_o @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6680_prod_Ocase__distrib,axiom,
    ! [H2: nat > product_prod_nat_nat > $o,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6681_prod_Ocase__distrib,axiom,
    ! [H2: ( product_prod_nat_nat > $o ) > nat,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6682_prod_Ocase__distrib,axiom,
    ! [H2: product_prod_int_int > product_prod_int_int,F: int > int > product_prod_int_int,Prod: product_prod_int_int] :
      ( ( H2 @ ( produc4245557441103728435nt_int @ F @ Prod ) )
      = ( produc4245557441103728435nt_int
        @ ^ [X15: int,X24: int] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6683_prod_Ocase__distrib,axiom,
    ! [H2: nat > product_prod_nat_nat > product_prod_nat_nat,F: nat > nat > nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc6842872674320459806at_nat @ F @ Prod ) )
      = ( produc27273713700761075at_nat
        @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6684_prod_Ocase__distrib,axiom,
    ! [H2: ( product_prod_nat_nat > product_prod_nat_nat ) > nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc27273713700761075at_nat @ F @ Prod ) )
      = ( produc6842872674320459806at_nat
        @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6685_prod_Ocase__distrib,axiom,
    ! [H2: ( product_prod_nat_nat > $o ) > product_prod_nat_nat > $o,F: nat > nat > product_prod_nat_nat > $o,Prod: product_prod_nat_nat] :
      ( ( H2 @ ( produc8739625826339149834_nat_o @ F @ Prod ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X15: nat,X24: nat] : ( H2 @ ( F @ X15 @ X24 ) )
        @ Prod ) ) ).

% prod.case_distrib
thf(fact_6686_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > nat,X1: nat,X22: nat] :
      ( ( produc6842872674320459806at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_6687_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,X1: nat,X22: nat] :
      ( ( produc27273713700761075at_nat @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_6688_old_Oprod_Ocase,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,X1: nat,X22: nat] :
      ( ( produc8739625826339149834_nat_o @ F @ ( product_Pair_nat_nat @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_6689_old_Oprod_Ocase,axiom,
    ! [F: int > int > product_prod_int_int,X1: int,X22: int] :
      ( ( produc4245557441103728435nt_int @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_6690_old_Oprod_Ocase,axiom,
    ! [F: int > int > $o,X1: int,X22: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ X1 @ X22 ) )
      = ( F @ X1 @ X22 ) ) ).

% old.prod.case
thf(fact_6691_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: nat > nat > nat,G: nat > nat > nat,P4: product_prod_nat_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
            = Q4 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q4 )
       => ( ( produc6842872674320459806at_nat @ F @ P4 )
          = ( produc6842872674320459806at_nat @ G @ Q4 ) ) ) ) ).

% split_cong
thf(fact_6692_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: nat > nat > product_prod_nat_nat > product_prod_nat_nat,P4: product_prod_nat_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
            = Q4 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q4 )
       => ( ( produc27273713700761075at_nat @ F @ P4 )
          = ( produc27273713700761075at_nat @ G @ Q4 ) ) ) ) ).

% split_cong
thf(fact_6693_split__cong,axiom,
    ! [Q4: product_prod_nat_nat,F: nat > nat > product_prod_nat_nat > $o,G: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( ( product_Pair_nat_nat @ X4 @ Y5 )
            = Q4 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q4 )
       => ( ( produc8739625826339149834_nat_o @ F @ P4 )
          = ( produc8739625826339149834_nat_o @ G @ Q4 ) ) ) ) ).

% split_cong
thf(fact_6694_split__cong,axiom,
    ! [Q4: product_prod_int_int,F: int > int > product_prod_int_int,G: int > int > product_prod_int_int,P4: product_prod_int_int] :
      ( ! [X4: int,Y5: int] :
          ( ( ( product_Pair_int_int @ X4 @ Y5 )
            = Q4 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q4 )
       => ( ( produc4245557441103728435nt_int @ F @ P4 )
          = ( produc4245557441103728435nt_int @ G @ Q4 ) ) ) ) ).

% split_cong
thf(fact_6695_split__cong,axiom,
    ! [Q4: product_prod_int_int,F: int > int > $o,G: int > int > $o,P4: product_prod_int_int] :
      ( ! [X4: int,Y5: int] :
          ( ( ( product_Pair_int_int @ X4 @ Y5 )
            = Q4 )
         => ( ( F @ X4 @ Y5 )
            = ( G @ X4 @ Y5 ) ) )
     => ( ( P4 = Q4 )
       => ( ( produc4947309494688390418_int_o @ F @ P4 )
          = ( produc4947309494688390418_int_o @ G @ Q4 ) ) ) ) ).

% split_cong
thf(fact_6696_cond__case__prod__eta,axiom,
    ! [F: nat > nat > nat,G: product_prod_nat_nat > nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
     => ( ( produc6842872674320459806at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6697_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > product_prod_nat_nat,G: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ! [X4: nat,Y5: nat] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
     => ( ( produc27273713700761075at_nat @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6698_cond__case__prod__eta,axiom,
    ! [F: nat > nat > product_prod_nat_nat > $o,G: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ! [X4: nat,Y5: nat] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) )
     => ( ( produc8739625826339149834_nat_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6699_cond__case__prod__eta,axiom,
    ! [F: int > int > product_prod_int_int,G: product_prod_int_int > product_prod_int_int] :
      ( ! [X4: int,Y5: int] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
     => ( ( produc4245557441103728435nt_int @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6700_cond__case__prod__eta,axiom,
    ! [F: int > int > $o,G: product_prod_int_int > $o] :
      ( ! [X4: int,Y5: int] :
          ( ( F @ X4 @ Y5 )
          = ( G @ ( product_Pair_int_int @ X4 @ Y5 ) ) )
     => ( ( produc4947309494688390418_int_o @ F )
        = G ) ) ).

% cond_case_prod_eta
thf(fact_6701_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > nat] :
      ( ( produc6842872674320459806at_nat
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6702_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > product_prod_nat_nat] :
      ( ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6703_case__prod__eta,axiom,
    ! [F: product_prod_nat_nat > product_prod_nat_nat > $o] :
      ( ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] : ( F @ ( product_Pair_nat_nat @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6704_case__prod__eta,axiom,
    ! [F: product_prod_int_int > product_prod_int_int] :
      ( ( produc4245557441103728435nt_int
        @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6705_case__prod__eta,axiom,
    ! [F: product_prod_int_int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [X: int,Y: int] : ( F @ ( product_Pair_int_int @ X @ Y ) ) )
      = F ) ).

% case_prod_eta
thf(fact_6706_case__prodE2,axiom,
    ! [Q: nat > $o,P: nat > nat > nat,Z2: product_prod_nat_nat] :
      ( ( Q @ ( produc6842872674320459806at_nat @ P @ Z2 ) )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( Z2
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_6707_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > product_prod_nat_nat ) > $o,P: nat > nat > product_prod_nat_nat > product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( Q @ ( produc27273713700761075at_nat @ P @ Z2 ) )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( Z2
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_6708_case__prodE2,axiom,
    ! [Q: ( product_prod_nat_nat > $o ) > $o,P: nat > nat > product_prod_nat_nat > $o,Z2: product_prod_nat_nat] :
      ( ( Q @ ( produc8739625826339149834_nat_o @ P @ Z2 ) )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( Z2
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_6709_case__prodE2,axiom,
    ! [Q: product_prod_int_int > $o,P: int > int > product_prod_int_int,Z2: product_prod_int_int] :
      ( ( Q @ ( produc4245557441103728435nt_int @ P @ Z2 ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( Z2
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_6710_case__prodE2,axiom,
    ! [Q: $o > $o,P: int > int > $o,Z2: product_prod_int_int] :
      ( ( Q @ ( produc4947309494688390418_int_o @ P @ Z2 ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( Z2
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( Q @ ( P @ X4 @ Y5 ) ) ) ) ).

% case_prodE2
thf(fact_6711_ln__less__self,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).

% ln_less_self
thf(fact_6712_ln__bound,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ X3 ) ) ).

% ln_bound
thf(fact_6713_ln__gt__zero__imp__gt__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ord_less_real @ one_one_real @ X3 ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_6714_ln__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X3 ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_6715_ln__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X3 ) ) ) ).

% ln_gt_zero
thf(fact_6716_ln__div,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ln_ln_real @ ( divide_divide_real @ X3 @ Y3 ) )
          = ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).

% ln_div
thf(fact_6717_ln__2__less__1,axiom,
    ord_less_real @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ one_one_real ).

% ln_2_less_1
thf(fact_6718_ln__ge__zero__imp__ge__one,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X3 ) )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_6719_ln__diff__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) @ ( divide_divide_real @ ( minus_minus_real @ X3 @ Y3 ) @ Y3 ) ) ) ) ).

% ln_diff_le
thf(fact_6720_ln__mult,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ( ln_ln_real @ ( times_times_real @ X3 @ Y3 ) )
          = ( plus_plus_real @ ( ln_ln_real @ X3 ) @ ( ln_ln_real @ Y3 ) ) ) ) ) ).

% ln_mult
thf(fact_6721_ln__eq__minus__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ( ln_ln_real @ X3 )
          = ( minus_minus_real @ X3 @ one_one_real ) )
       => ( X3 = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_6722_ln__le__minus__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( minus_minus_real @ X3 @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_6723_ln__add__one__self__le__self2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X3 ) ) @ X3 ) ) ).

% ln_add_one_self_le_self2
thf(fact_6724_ln__one__minus__pos__upper__bound,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X3 ) ) @ ( uminus_uminus_real @ X3 ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_6725_divmod__step__nat__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_nat_def
thf(fact_6726_divmod__step__int__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_int_def
thf(fact_6727_divmod__step__def,axiom,
    ( unique5026877609467782581ep_nat
    = ( ^ [L2: num] :
          ( produc2626176000494625587at_nat
          @ ^ [Q5: nat,R5: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L2 ) @ R5 ) @ ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ one_one_nat ) @ ( minus_minus_nat @ R5 @ ( numeral_numeral_nat @ L2 ) ) ) @ ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_6728_divmod__step__def,axiom,
    ( unique5024387138958732305ep_int
    = ( ^ [L2: num] :
          ( produc4245557441103728435nt_int
          @ ^ [Q5: int,R5: int] : ( if_Pro3027730157355071871nt_int @ ( ord_less_eq_int @ ( numeral_numeral_int @ L2 ) @ R5 ) @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ one_one_int ) @ ( minus_minus_int @ R5 @ ( numeral_numeral_int @ L2 ) ) ) @ ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_6729_divmod__step__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_def
thf(fact_6730_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M2: nat,N: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N = zero_zero_nat )
            | ( ord_less_nat @ M2 @ N ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M2 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M2 @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_6731_tanh__ln__real,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( tanh_real @ ( ln_ln_real @ X3 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_6732_Sum__Icc__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( plus_plus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Icc_nat
thf(fact_6733_arith__series__nat,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I2 @ 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_6734_case__prod__Pair__iden,axiom,
    ! [P4: produc8763457246119570046nteger] :
      ( ( produc3906647086178084059nteger @ produc6137756002093451184nteger @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_6735_case__prod__Pair__iden,axiom,
    ! [P4: produc1908205239877642774nteger] :
      ( ( produc6512950862096126219nteger @ produc8603105652947943368nteger @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_6736_case__prod__Pair__iden,axiom,
    ! [P4: produc2285326912895808259nt_int] :
      ( ( produc8492565224438309093nt_int @ produc5700946648718959541nt_int @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_6737_case__prod__Pair__iden,axiom,
    ! [P4: produc7773217078559923341nt_int] :
      ( ( produc5122537100556696953nt_int @ produc4305682042979456191nt_int @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_6738_case__prod__Pair__iden,axiom,
    ! [P4: product_prod_int_int] :
      ( ( produc4245557441103728435nt_int @ product_Pair_int_int @ P4 )
      = P4 ) ).

% case_prod_Pair_iden
thf(fact_6739_round__unique_H,axiom,
    ! [X3: real,N2: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X3 )
        = N2 ) ) ).

% round_unique'
thf(fact_6740_round__unique_H,axiom,
    ! [X3: rat,N2: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X3 )
        = N2 ) ) ).

% round_unique'
thf(fact_6741_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_6742_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_6743_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_6744_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_6745_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_6746_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_6747_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_6748_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_6749_case__prodI,axiom,
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ A @ B )
     => ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) ) ) ).

% case_prodI
thf(fact_6750_case__prodI,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( F @ A @ B )
     => ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) ) ) ).

% case_prodI
thf(fact_6751_case__prodI,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_6752_case__prodI,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( F @ A @ B )
     => ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_6753_case__prodI,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( F @ A @ B )
     => ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) ) ) ).

% case_prodI
thf(fact_6754_case__prodI2,axiom,
    ! [P4: produc8763457246119570046nteger,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc6137756002093451184nteger @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc127349428274296955eger_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6755_case__prodI2,axiom,
    ! [P4: produc1908205239877642774nteger,C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o] :
      ( ! [A4: produc6241069584506657477e_term > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc8603105652947943368nteger @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc6253627499356882019eger_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6756_case__prodI2,axiom,
    ! [P4: produc2285326912895808259nt_int,C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A4: produc8551481072490612790e_term > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P4
            = ( produc5700946648718959541nt_int @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc1573362020775583542_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6757_case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc2558449545302689196_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6758_case__prodI2,axiom,
    ! [P4: product_prod_int_int,C: int > int > $o] :
      ( ! [A4: int,B4: int] :
          ( ( P4
            = ( product_Pair_int_int @ A4 @ B4 ) )
         => ( C @ A4 @ B4 ) )
     => ( produc4947309494688390418_int_o @ C @ P4 ) ) ).

% case_prodI2
thf(fact_6759_mem__case__prodI,axiom,
    ! [Z2: real,C: int > int > set_real,A: int,B: int] :
      ( ( member_real @ Z2 @ ( C @ A @ B ) )
     => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6760_mem__case__prodI,axiom,
    ! [Z2: nat,C: int > int > set_nat,A: int,B: int] :
      ( ( member_nat @ Z2 @ ( C @ A @ B ) )
     => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6761_mem__case__prodI,axiom,
    ! [Z2: int,C: int > int > set_int,A: int,B: int] :
      ( ( member_int @ Z2 @ ( C @ A @ B ) )
     => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6762_mem__case__prodI,axiom,
    ! [Z2: set_nat,C: int > int > set_set_nat,A: int,B: int] :
      ( ( member_set_nat @ Z2 @ ( C @ A @ B ) )
     => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6763_mem__case__prodI,axiom,
    ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,A: int,B: int] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( C @ A @ B ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6764_mem__case__prodI,axiom,
    ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_real @ Z2 @ ( C @ A @ B ) )
     => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6765_mem__case__prodI,axiom,
    ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_nat @ Z2 @ ( C @ A @ B ) )
     => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6766_mem__case__prodI,axiom,
    ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( member_int @ Z2 @ ( C @ A @ B ) )
     => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ ( produc6137756002093451184nteger @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6767_mem__case__prodI,axiom,
    ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_real @ Z2 @ ( C @ A @ B ) )
     => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6768_mem__case__prodI,axiom,
    ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( member_nat @ Z2 @ ( C @ A @ B ) )
     => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ ( produc4305682042979456191nt_int @ A @ B ) ) ) ) ).

% mem_case_prodI
thf(fact_6769_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z2: real,C: int > int > set_real] :
      ( ! [A4: int,B4: int] :
          ( ( P4
            = ( product_Pair_int_int @ A4 @ B4 ) )
         => ( member_real @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6770_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z2: nat,C: int > int > set_nat] :
      ( ! [A4: int,B4: int] :
          ( ( P4
            = ( product_Pair_int_int @ A4 @ B4 ) )
         => ( member_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6771_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z2: int,C: int > int > set_int] :
      ( ! [A4: int,B4: int] :
          ( ( P4
            = ( product_Pair_int_int @ A4 @ B4 ) )
         => ( member_int @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6772_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z2: set_nat,C: int > int > set_set_nat] :
      ( ! [A4: int,B4: int] :
          ( ( P4
            = ( product_Pair_int_int @ A4 @ B4 ) )
         => ( member_set_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6773_mem__case__prodI2,axiom,
    ! [P4: product_prod_int_int,Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat] :
      ( ! [A4: int,B4: int] :
          ( ( P4
            = ( product_Pair_int_int @ A4 @ B4 ) )
         => ( member8440522571783428010at_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6774_mem__case__prodI2,axiom,
    ! [P4: produc8763457246119570046nteger,Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real] :
      ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc6137756002093451184nteger @ A4 @ B4 ) )
         => ( member_real @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6775_mem__case__prodI2,axiom,
    ! [P4: produc8763457246119570046nteger,Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat] :
      ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc6137756002093451184nteger @ A4 @ B4 ) )
         => ( member_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6776_mem__case__prodI2,axiom,
    ! [P4: produc8763457246119570046nteger,Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int] :
      ( ! [A4: code_integer > option6357759511663192854e_term,B4: produc8923325533196201883nteger] :
          ( ( P4
            = ( produc6137756002093451184nteger @ A4 @ B4 ) )
         => ( member_int @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6777_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
         => ( member_real @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6778_mem__case__prodI2,axiom,
    ! [P4: produc7773217078559923341nt_int,Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat] :
      ( ! [A4: int > option6357759511663192854e_term,B4: product_prod_int_int] :
          ( ( P4
            = ( produc4305682042979456191nt_int @ A4 @ B4 ) )
         => ( member_nat @ Z2 @ ( C @ A4 @ B4 ) ) )
     => ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) ) ) ).

% mem_case_prodI2
thf(fact_6779_case__prodI2_H,axiom,
    ! [P4: product_prod_nat_nat,C: nat > nat > product_prod_nat_nat > $o,X3: product_prod_nat_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( ( product_Pair_nat_nat @ A4 @ B4 )
            = P4 )
         => ( C @ A4 @ B4 @ X3 ) )
     => ( produc8739625826339149834_nat_o @ C @ P4 @ X3 ) ) ).

% case_prodI2'
thf(fact_6780_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_6781_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_6782_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_6783_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_6784_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_6785_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_6786_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_6787_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_6788_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_6789_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_6790_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_6791_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_6792_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_6793_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_6794_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_6795_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_6796_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_6797_abs__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = ( times_3573771949741848930nteger @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_6798_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_6799_abs__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ A ) )
      = ( times_times_rat @ A @ A ) ) ).

% abs_mult_self_eq
thf(fact_6800_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_6801_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_6802_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_6803_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_6804_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_6805_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_6806_abs__add__abs,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) )
      = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_add_abs
thf(fact_6807_abs__add__abs,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) )
      = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_add_abs
thf(fact_6808_abs__add__abs,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) )
      = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_add_abs
thf(fact_6809_abs__add__abs,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) )
      = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_add_abs
thf(fact_6810_abs__divide,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_divide
thf(fact_6811_abs__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_divide
thf(fact_6812_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_6813_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_6814_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_6815_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_6816_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_6817_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_6818_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_6819_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_6820_dvd__abs__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ M @ ( abs_abs_int @ K ) )
      = ( dvd_dvd_int @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_6821_dvd__abs__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ M @ ( abs_abs_real @ K ) )
      = ( dvd_dvd_real @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_6822_dvd__abs__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ M @ ( abs_abs_rat @ K ) )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_6823_dvd__abs__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ M @ ( abs_abs_Code_integer @ K ) )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% dvd_abs_iff
thf(fact_6824_abs__dvd__iff,axiom,
    ! [M: int,K: int] :
      ( ( dvd_dvd_int @ ( abs_abs_int @ M ) @ K )
      = ( dvd_dvd_int @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_6825_abs__dvd__iff,axiom,
    ! [M: real,K: real] :
      ( ( dvd_dvd_real @ ( abs_abs_real @ M ) @ K )
      = ( dvd_dvd_real @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_6826_abs__dvd__iff,axiom,
    ! [M: rat,K: rat] :
      ( ( dvd_dvd_rat @ ( abs_abs_rat @ M ) @ K )
      = ( dvd_dvd_rat @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_6827_abs__dvd__iff,axiom,
    ! [M: code_integer,K: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( abs_abs_Code_integer @ M ) @ K )
      = ( dvd_dvd_Code_integer @ M @ K ) ) ).

% abs_dvd_iff
thf(fact_6828_of__int__abs,axiom,
    ! [X3: int] :
      ( ( ring_1_of_int_int @ ( abs_abs_int @ X3 ) )
      = ( abs_abs_int @ ( ring_1_of_int_int @ X3 ) ) ) ).

% of_int_abs
thf(fact_6829_of__int__abs,axiom,
    ! [X3: int] :
      ( ( ring_18347121197199848620nteger @ ( abs_abs_int @ X3 ) )
      = ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ X3 ) ) ) ).

% of_int_abs
thf(fact_6830_of__int__abs,axiom,
    ! [X3: int] :
      ( ( ring_1_of_int_real @ ( abs_abs_int @ X3 ) )
      = ( abs_abs_real @ ( ring_1_of_int_real @ X3 ) ) ) ).

% of_int_abs
thf(fact_6831_of__int__abs,axiom,
    ! [X3: int] :
      ( ( ring_1_of_int_rat @ ( abs_abs_int @ X3 ) )
      = ( abs_abs_rat @ ( ring_1_of_int_rat @ X3 ) ) ) ).

% of_int_abs
thf(fact_6832_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_6833_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% abs_bool_eq
thf(fact_6834_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_6835_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_6836_zdvd1__eq,axiom,
    ! [X3: int] :
      ( ( dvd_dvd_int @ X3 @ one_one_int )
      = ( ( abs_abs_int @ X3 )
        = one_one_int ) ) ).

% zdvd1_eq
thf(fact_6837_tanh__0,axiom,
    ( ( tanh_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tanh_0
thf(fact_6838_tanh__0,axiom,
    ( ( tanh_real @ zero_zero_real )
    = zero_zero_real ) ).

% tanh_0
thf(fact_6839_tanh__real__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( tanh_real @ X3 ) @ ( tanh_real @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% tanh_real_less_iff
thf(fact_6840_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [Uu3: nat] : zero_zero_nat
        @ A2 )
      = zero_zero_nat ) ).

% sum.neutral_const
thf(fact_6841_sum_Oneutral__const,axiom,
    ! [A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [Uu3: complex] : zero_zero_complex
        @ A2 )
      = zero_zero_complex ) ).

% sum.neutral_const
thf(fact_6842_sum_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [Uu3: nat] : zero_zero_real
        @ A2 )
      = zero_zero_real ) ).

% sum.neutral_const
thf(fact_6843_of__int__sum,axiom,
    ! [F: complex > int,A2: set_complex] :
      ( ( ring_17405671764205052669omplex @ ( groups5690904116761175830ex_int @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( ring_17405671764205052669omplex @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_6844_of__int__sum,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups3539618377306564664at_int @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( ring_1_of_int_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_sum
thf(fact_6845_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6846_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6847_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6848_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6849_abs__le__self__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% abs_le_self_iff
thf(fact_6850_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_6851_abs__le__self__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ A )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% abs_le_self_iff
thf(fact_6852_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_6853_abs__le__zero__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_le_zero_iff
thf(fact_6854_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_6855_abs__le__zero__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_le_zero_iff
thf(fact_6856_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_6857_zero__less__abs__iff,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) )
      = ( A != zero_z3403309356797280102nteger ) ) ).

% zero_less_abs_iff
thf(fact_6858_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_6859_zero__less__abs__iff,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) )
      = ( A != zero_zero_rat ) ) ).

% zero_less_abs_iff
thf(fact_6860_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_6861_sum_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups5754745047067104278omplex @ G @ bot_bot_set_real )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_6862_sum_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups8097168146408367636l_real @ G @ bot_bot_set_real )
      = zero_zero_real ) ).

% sum.empty
thf(fact_6863_sum_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups1300246762558778688al_rat @ G @ bot_bot_set_real )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6864_sum_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups1935376822645274424al_nat @ G @ bot_bot_set_real )
      = zero_zero_nat ) ).

% sum.empty
thf(fact_6865_sum_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups1932886352136224148al_int @ G @ bot_bot_set_real )
      = zero_zero_int ) ).

% sum.empty
thf(fact_6866_sum_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups2073611262835488442omplex @ G @ bot_bot_set_nat )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_6867_sum_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups2906978787729119204at_rat @ G @ bot_bot_set_nat )
      = zero_zero_rat ) ).

% sum.empty
thf(fact_6868_sum_Oempty,axiom,
    ! [G: nat > int] :
      ( ( groups3539618377306564664at_int @ G @ bot_bot_set_nat )
      = zero_zero_int ) ).

% sum.empty
thf(fact_6869_sum_Oempty,axiom,
    ! [G: int > complex] :
      ( ( groups3049146728041665814omplex @ G @ bot_bot_set_int )
      = zero_zero_complex ) ).

% sum.empty
thf(fact_6870_sum_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups8778361861064173332t_real @ G @ bot_bot_set_int )
      = zero_zero_real ) ).

% sum.empty
thf(fact_6871_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_6872_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.infinite
thf(fact_6873_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_6874_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.infinite
thf(fact_6875_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6876_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6877_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ A2 )
        = zero_zero_rat ) ) ).

% sum.infinite
thf(fact_6878_sum_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_6879_sum_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > nat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.infinite
thf(fact_6880_sum_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups3539618377306564664at_int @ G @ A2 )
        = zero_zero_int ) ) ).

% sum.infinite
thf(fact_6881_sum__eq__0__iff,axiom,
    ! [F3: set_int,F: int > nat] :
      ( ( finite_finite_int @ F3 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X: int] :
              ( ( member_int @ X @ F3 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6882_sum__eq__0__iff,axiom,
    ! [F3: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ F3 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X: complex] :
              ( ( member_complex @ X @ F3 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6883_sum__eq__0__iff,axiom,
    ! [F3: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ F3 )
     => ( ( ( groups977919841031483927at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ F3 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6884_sum__eq__0__iff,axiom,
    ! [F3: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ F3 )
     => ( ( ( groups3542108847815614940at_nat @ F @ F3 )
          = zero_zero_nat )
        = ( ! [X: nat] :
              ( ( member_nat @ X @ F3 )
             => ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% sum_eq_0_iff
thf(fact_6885_artanh__minus__real,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X3 ) )
        = ( uminus_uminus_real @ ( artanh_real @ X3 ) ) ) ) ).

% artanh_minus_real
thf(fact_6886_zabs__less__one__iff,axiom,
    ! [Z2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z2 ) @ one_one_int )
      = ( Z2 = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_6887_tanh__real__pos__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% tanh_real_pos_iff
thf(fact_6888_tanh__real__neg__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( tanh_real @ X3 ) @ zero_zero_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_6889_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_6890_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_6891_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta
thf(fact_6892_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6893_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6894_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta
thf(fact_6895_sum_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6896_sum_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6897_sum_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6898_sum_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta
thf(fact_6899_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups5754745047067104278omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_6900_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2073611262835488442omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_6901_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3049146728041665814omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_complex )
              @ S3 )
            = zero_zero_complex ) ) ) ) ).

% sum.delta'
thf(fact_6902_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6903_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6904_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_real )
              @ S3 )
            = zero_zero_real ) ) ) ) ).

% sum.delta'
thf(fact_6905_sum_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6906_sum_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups2906978787729119204at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6907_sum_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6908_sum_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ zero_zero_rat )
              @ S3 )
            = zero_zero_rat ) ) ) ) ).

% sum.delta'
thf(fact_6909_sum__abs,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( abs_abs_real @ ( F @ I2 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_6910_divide__le__0__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) @ zero_zero_real )
      = ( ( ord_less_eq_real @ A @ zero_zero_real )
        | ( B = zero_zero_real ) ) ) ).

% divide_le_0_abs_iff
thf(fact_6911_divide__le__0__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) @ zero_zero_rat )
      = ( ( ord_less_eq_rat @ A @ zero_zero_rat )
        | ( B = zero_zero_rat ) ) ) ).

% divide_le_0_abs_iff
thf(fact_6912_zero__le__divide__abs__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ A @ ( abs_abs_real @ B ) ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        | ( B = zero_zero_real ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_6913_zero__le__divide__abs__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ A @ ( abs_abs_rat @ B ) ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        | ( B = zero_zero_rat ) ) ) ).

% zero_le_divide_abs_iff
thf(fact_6914_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_6915_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6916_abs__of__nonpos,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6917_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_6918_sum_Oinsert,axiom,
    ! [A2: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X3 @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6919_sum_Oinsert,axiom,
    ! [A2: set_int,X3: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X3 @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6920_sum_Oinsert,axiom,
    ! [A2: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X3 @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6921_sum_Oinsert,axiom,
    ! [A2: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X3 @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6922_sum_Oinsert,axiom,
    ! [A2: set_nat,X3: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X3 @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6923_sum_Oinsert,axiom,
    ! [A2: set_int,X3: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X3 @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6924_sum_Oinsert,axiom,
    ! [A2: set_complex,X3: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X3 @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6925_sum_Oinsert,axiom,
    ! [A2: set_real,X3: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X3 @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6926_sum_Oinsert,axiom,
    ! [A2: set_int,X3: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X3 @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X3 @ A2 ) )
          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6927_sum_Oinsert,axiom,
    ! [A2: set_complex,X3: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X3 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum.insert
thf(fact_6928_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( abs_abs_real @ ( F @ I2 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_6929_zero__less__power__abs__iff,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) )
      = ( ( A != zero_z3403309356797280102nteger )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6930_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_6931_zero__less__power__abs__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) )
      = ( ( A != zero_zero_rat )
        | ( N2 = zero_zero_nat ) ) ) ).

% zero_less_power_abs_iff
thf(fact_6932_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_6933_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_complex @ ( groups2073611262835488442omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6934_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6935_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6936_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6937_sum_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% sum.cl_ivl_Suc
thf(fact_6938_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > complex] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I2: nat] : ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power
thf(fact_6939_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power
thf(fact_6940_sum__zero__power,axiom,
    ! [A2: set_nat,C: nat > real] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
            @ A2 )
          = ( C @ zero_zero_nat ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power
thf(fact_6941_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
            @ ^ [I2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = ( divide1717551699836669952omplex @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2073611262835488442omplex
            @ ^ [I2: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( C @ I2 ) @ ( power_power_complex @ zero_zero_complex @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = zero_zero_complex ) ) ) ).

% sum_zero_power'
thf(fact_6942_sum__zero__power_H,axiom,
    ! [A2: set_nat,C: nat > rat,D: nat > rat] :
      ( ( ( ( finite_finite_nat @ A2 )
          & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = ( divide_divide_rat @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( divide_divide_rat @ ( times_times_rat @ ( C @ I2 ) @ ( power_power_rat @ zero_zero_rat @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = zero_zero_rat ) ) ) ).

% sum_zero_power'
thf(fact_6943_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
            @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = ( divide_divide_real @ ( C @ zero_zero_nat ) @ ( D @ zero_zero_nat ) ) ) )
      & ( ~ ( ( finite_finite_nat @ A2 )
            & ( member_nat @ zero_zero_nat @ A2 ) )
       => ( ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( divide_divide_real @ ( times_times_real @ ( C @ I2 ) @ ( power_power_real @ zero_zero_real @ I2 ) ) @ ( D @ I2 ) )
            @ A2 )
          = zero_zero_real ) ) ) ).

% sum_zero_power'
thf(fact_6944_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A2: set_real] :
      ( ( ( groups5754745047067104278omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6945_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A2: set_nat] :
      ( ( ( groups2073611262835488442omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A4: nat] :
            ( ( member_nat @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6946_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A2: set_int] :
      ( ( ( groups3049146728041665814omplex @ G @ A2 )
       != zero_zero_complex )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_complex ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6947_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_6948_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_6949_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A2: set_real] :
      ( ( ( groups1300246762558778688al_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A4: real] :
            ( ( member_real @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6950_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A2: set_nat] :
      ( ( ( groups2906978787729119204at_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A4: nat] :
            ( ( member_nat @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6951_sum_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > rat,A2: set_int] :
      ( ( ( groups3906332499630173760nt_rat @ G @ A2 )
       != zero_zero_rat )
     => ~ ! [A4: int] :
            ( ( member_int @ A4 @ A2 )
           => ( ( G @ A4 )
              = zero_zero_rat ) ) ) ).

% sum.not_neutral_contains_not_neutral
thf(fact_6952_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_6953_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_6954_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( G @ X4 )
            = zero_zero_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ A2 )
        = zero_zero_nat ) ) ).

% sum.neutral
thf(fact_6955_sum_Oneutral,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ! [X4: complex] :
          ( ( member_complex @ X4 @ A2 )
         => ( ( G @ X4 )
            = zero_zero_complex ) )
     => ( ( groups7754918857620584856omplex @ G @ A2 )
        = zero_zero_complex ) ) ).

% sum.neutral
thf(fact_6956_sum_Oneutral,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ( G @ X4 )
            = zero_zero_real ) )
     => ( ( groups6591440286371151544t_real @ G @ A2 )
        = zero_zero_real ) ) ).

% sum.neutral
thf(fact_6957_abs__le__D1,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ A @ B ) ) ).

% abs_le_D1
thf(fact_6958_abs__le__D1,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ A @ B ) ) ).

% abs_le_D1
thf(fact_6959_abs__le__D1,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ A @ B ) ) ).

% abs_le_D1
thf(fact_6960_abs__le__D1,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ A @ B ) ) ).

% abs_le_D1
thf(fact_6961_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_6962_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_6963_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_6964_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_6965_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_6966_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_6967_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_6968_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_6969_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_6970_abs__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_mult
thf(fact_6971_abs__mult,axiom,
    ! [A: complex,B: complex] :
      ( ( abs_abs_complex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).

% abs_mult
thf(fact_6972_abs__mult,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_mult
thf(fact_6973_abs__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_mult
thf(fact_6974_abs__mult,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_mult
thf(fact_6975_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_6976_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_6977_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_6978_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_6979_abs__minus__commute,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6980_abs__minus__commute,axiom,
    ! [A: real,B: real] :
      ( ( abs_abs_real @ ( minus_minus_real @ A @ B ) )
      = ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6981_abs__minus__commute,axiom,
    ! [A: rat,B: rat] :
      ( ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) )
      = ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6982_abs__minus__commute,axiom,
    ! [A: int,B: int] :
      ( ( abs_abs_int @ ( minus_minus_int @ A @ B ) )
      = ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_minus_commute
thf(fact_6983_abs__eq__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ( abs_abs_int @ X3 )
        = ( abs_abs_int @ Y3 ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus_uminus_int @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_6984_abs__eq__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( abs_abs_real @ X3 )
        = ( abs_abs_real @ Y3 ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus_uminus_real @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_6985_abs__eq__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( abs_abs_rat @ X3 )
        = ( abs_abs_rat @ Y3 ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus_uminus_rat @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_6986_abs__eq__iff,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( ( abs_abs_Code_integer @ X3 )
        = ( abs_abs_Code_integer @ Y3 ) )
      = ( ( X3 = Y3 )
        | ( X3
          = ( uminus1351360451143612070nteger @ Y3 ) ) ) ) ).

% abs_eq_iff
thf(fact_6987_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_6988_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_6989_dvd__if__abs__eq,axiom,
    ! [L: rat,K: rat] :
      ( ( ( abs_abs_rat @ L )
        = ( abs_abs_rat @ K ) )
     => ( dvd_dvd_rat @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6990_dvd__if__abs__eq,axiom,
    ! [L: code_integer,K: code_integer] :
      ( ( ( abs_abs_Code_integer @ L )
        = ( abs_abs_Code_integer @ K ) )
     => ( dvd_dvd_Code_integer @ L @ K ) ) ).

% dvd_if_abs_eq
thf(fact_6991_mem__case__prodE,axiom,
    ! [Z2: real,C: int > int > set_real,P4: product_prod_int_int] :
      ( ( member_real @ Z2 @ ( produc6452406959799940328t_real @ C @ P4 ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( P4
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( member_real @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6992_mem__case__prodE,axiom,
    ! [Z2: nat,C: int > int > set_nat,P4: product_prod_int_int] :
      ( ( member_nat @ Z2 @ ( produc4251311855443802252et_nat @ C @ P4 ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( P4
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6993_mem__case__prodE,axiom,
    ! [Z2: int,C: int > int > set_int,P4: product_prod_int_int] :
      ( ( member_int @ Z2 @ ( produc73460835934605544et_int @ C @ P4 ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( P4
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( member_int @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6994_mem__case__prodE,axiom,
    ! [Z2: set_nat,C: int > int > set_set_nat,P4: product_prod_int_int] :
      ( ( member_set_nat @ Z2 @ ( produc5233655623923918146et_nat @ C @ P4 ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( P4
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( member_set_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6995_mem__case__prodE,axiom,
    ! [Z2: product_prod_nat_nat,C: int > int > set_Pr1261947904930325089at_nat,P4: product_prod_int_int] :
      ( ( member8440522571783428010at_nat @ Z2 @ ( produc1656060378719767003at_nat @ C @ P4 ) )
     => ~ ! [X4: int,Y5: int] :
            ( ( P4
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( member8440522571783428010at_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6996_mem__case__prodE,axiom,
    ! [Z2: real,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_real,P4: produc8763457246119570046nteger] :
      ( ( member_real @ Z2 @ ( produc815715089573277247t_real @ C @ P4 ) )
     => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
           => ~ ( member_real @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6997_mem__case__prodE,axiom,
    ! [Z2: nat,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_nat,P4: produc8763457246119570046nteger] :
      ( ( member_nat @ Z2 @ ( produc3558942015123893603et_nat @ C @ P4 ) )
     => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
           => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6998_mem__case__prodE,axiom,
    ! [Z2: int,C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > set_int,P4: produc8763457246119570046nteger] :
      ( ( member_int @ Z2 @ ( produc8604463032469472703et_int @ C @ P4 ) )
     => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
           => ~ ( member_int @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_6999_mem__case__prodE,axiom,
    ! [Z2: real,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_real,P4: produc7773217078559923341nt_int] :
      ( ( member_real @ Z2 @ ( produc8709739885379107790t_real @ C @ P4 ) )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ~ ( member_real @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7000_mem__case__prodE,axiom,
    ! [Z2: nat,C: ( int > option6357759511663192854e_term ) > product_prod_int_int > set_nat,P4: produc7773217078559923341nt_int] :
      ( ( member_nat @ Z2 @ ( produc8289552606927098482et_nat @ C @ P4 ) )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ~ ( member_nat @ Z2 @ ( C @ X4 @ Y5 ) ) ) ) ).

% mem_case_prodE
thf(fact_7001_zdvd__antisym__abs,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ B )
     => ( ( dvd_dvd_int @ B @ A )
       => ( ( abs_abs_int @ A )
          = ( abs_abs_int @ B ) ) ) ) ).

% zdvd_antisym_abs
thf(fact_7002_case__prodD,axiom,
    ! [F: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: code_integer > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc127349428274296955eger_o @ F @ ( produc6137756002093451184nteger @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_7003_case__prodD,axiom,
    ! [F: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,A: produc6241069584506657477e_term > option6357759511663192854e_term,B: produc8923325533196201883nteger] :
      ( ( produc6253627499356882019eger_o @ F @ ( produc8603105652947943368nteger @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_7004_case__prodD,axiom,
    ! [F: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,A: produc8551481072490612790e_term > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc1573362020775583542_int_o @ F @ ( produc5700946648718959541nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_7005_case__prodD,axiom,
    ! [F: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,A: int > option6357759511663192854e_term,B: product_prod_int_int] :
      ( ( produc2558449545302689196_int_o @ F @ ( produc4305682042979456191nt_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_7006_case__prodD,axiom,
    ! [F: int > int > $o,A: int,B: int] :
      ( ( produc4947309494688390418_int_o @ F @ ( product_Pair_int_int @ A @ B ) )
     => ( F @ A @ B ) ) ).

% case_prodD
thf(fact_7007_case__prodE,axiom,
    ! [C: ( code_integer > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc8763457246119570046nteger] :
      ( ( produc127349428274296955eger_o @ C @ P4 )
     => ~ ! [X4: code_integer > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc6137756002093451184nteger @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7008_case__prodE,axiom,
    ! [C: ( produc6241069584506657477e_term > option6357759511663192854e_term ) > produc8923325533196201883nteger > $o,P4: produc1908205239877642774nteger] :
      ( ( produc6253627499356882019eger_o @ C @ P4 )
     => ~ ! [X4: produc6241069584506657477e_term > option6357759511663192854e_term,Y5: produc8923325533196201883nteger] :
            ( ( P4
              = ( produc8603105652947943368nteger @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7009_case__prodE,axiom,
    ! [C: ( produc8551481072490612790e_term > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc2285326912895808259nt_int] :
      ( ( produc1573362020775583542_int_o @ C @ P4 )
     => ~ ! [X4: produc8551481072490612790e_term > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc5700946648718959541nt_int @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7010_case__prodE,axiom,
    ! [C: ( int > option6357759511663192854e_term ) > product_prod_int_int > $o,P4: produc7773217078559923341nt_int] :
      ( ( produc2558449545302689196_int_o @ C @ P4 )
     => ~ ! [X4: int > option6357759511663192854e_term,Y5: product_prod_int_int] :
            ( ( P4
              = ( produc4305682042979456191nt_int @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7011_case__prodE,axiom,
    ! [C: int > int > $o,P4: product_prod_int_int] :
      ( ( produc4947309494688390418_int_o @ C @ P4 )
     => ~ ! [X4: int,Y5: int] :
            ( ( P4
              = ( product_Pair_int_int @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 ) ) ) ).

% case_prodE
thf(fact_7012_sum__mono,axiom,
    ! [K5: set_real,F: real > rat,G: real > rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7013_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7014_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7015_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7016_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7017_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7018_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7019_sum__mono,axiom,
    ! [K5: set_int,F: int > int,G: int > int] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ K5 )
         => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ K5 ) @ ( groups4538972089207619220nt_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7020_sum__mono,axiom,
    ! [K5: set_nat,F: nat > nat,G: nat > nat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ K5 ) @ ( groups3542108847815614940at_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7021_sum__mono,axiom,
    ! [K5: set_nat,F: nat > real,G: nat > real] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ K5 )
         => ( ord_less_eq_real @ ( F @ I3 ) @ ( G @ I3 ) ) )
     => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ K5 ) @ ( groups6591440286371151544t_real @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_7022_sum_Odistrib,axiom,
    ! [G: nat > nat,H2: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7023_sum_Odistrib,axiom,
    ! [G: complex > complex,H2: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7024_sum_Odistrib,axiom,
    ! [G: nat > real,H2: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H2 @ X ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H2 @ A2 ) ) ) ).

% sum.distrib
thf(fact_7025_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: real > nat > nat,R: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups1935376822645274424al_nat
            @ ^ [X: real] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups1935376822645274424al_nat
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7026_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: int > nat > nat,R: int > nat > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups4541462559716669496nt_nat
            @ ^ [X: int] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups4541462559716669496nt_nat
                @ ^ [X: int] : ( G @ X @ Y )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7027_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: complex > nat > nat,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups5693394587270226106ex_nat
            @ ^ [X: complex] :
                ( groups3542108847815614940at_nat @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups3542108847815614940at_nat
            @ ^ [Y: nat] :
                ( groups5693394587270226106ex_nat
                @ ^ [X: complex] : ( G @ X @ Y )
                @ ( collect_complex
                  @ ^ [X: complex] :
                      ( ( member_complex @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7028_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_complex,G: real > complex > complex,R: real > complex > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups5754745047067104278omplex
            @ ^ [X: real] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups5754745047067104278omplex
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7029_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_complex,G: nat > complex > complex,R: nat > complex > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [X: nat] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups2073611262835488442omplex
                @ ^ [X: nat] : ( G @ X @ Y )
                @ ( collect_nat
                  @ ^ [X: nat] :
                      ( ( member_nat @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7030_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_complex,G: int > complex > complex,R: int > complex > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite3207457112153483333omplex @ B2 )
       => ( ( groups3049146728041665814omplex
            @ ^ [X: int] :
                ( groups7754918857620584856omplex @ ( G @ X )
                @ ( collect_complex
                  @ ^ [Y: complex] :
                      ( ( member_complex @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups7754918857620584856omplex
            @ ^ [Y: complex] :
                ( groups3049146728041665814omplex
                @ ^ [X: int] : ( G @ X @ Y )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7031_sum_Oswap__restrict,axiom,
    ! [A2: set_real,B2: set_nat,G: real > nat > real,R: real > nat > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups8097168146408367636l_real
            @ ^ [X: real] :
                ( groups6591440286371151544t_real @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups6591440286371151544t_real
            @ ^ [Y: nat] :
                ( groups8097168146408367636l_real
                @ ^ [X: real] : ( G @ X @ Y )
                @ ( collect_real
                  @ ^ [X: real] :
                      ( ( member_real @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7032_sum_Oswap__restrict,axiom,
    ! [A2: set_int,B2: set_nat,G: int > nat > real,R: int > nat > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups8778361861064173332t_real
            @ ^ [X: int] :
                ( groups6591440286371151544t_real @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups6591440286371151544t_real
            @ ^ [Y: nat] :
                ( groups8778361861064173332t_real
                @ ^ [X: int] : ( G @ X @ Y )
                @ ( collect_int
                  @ ^ [X: int] :
                      ( ( member_int @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7033_sum_Oswap__restrict,axiom,
    ! [A2: set_complex,B2: set_nat,G: complex > nat > real,R: complex > nat > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( finite_finite_nat @ B2 )
       => ( ( groups5808333547571424918x_real
            @ ^ [X: complex] :
                ( groups6591440286371151544t_real @ ( G @ X )
                @ ( collect_nat
                  @ ^ [Y: nat] :
                      ( ( member_nat @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups6591440286371151544t_real
            @ ^ [Y: nat] :
                ( groups5808333547571424918x_real
                @ ^ [X: complex] : ( G @ X @ Y )
                @ ( collect_complex
                  @ ^ [X: complex] :
                      ( ( member_complex @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7034_sum_Oswap__restrict,axiom,
    ! [A2: set_nat,B2: set_real,G: nat > real > nat,R: nat > real > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( finite_finite_real @ B2 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [X: nat] :
                ( groups1935376822645274424al_nat @ ( G @ X )
                @ ( collect_real
                  @ ^ [Y: real] :
                      ( ( member_real @ Y @ B2 )
                      & ( R @ X @ Y ) ) ) )
            @ A2 )
          = ( groups1935376822645274424al_nat
            @ ^ [Y: real] :
                ( groups3542108847815614940at_nat
                @ ^ [X: nat] : ( G @ X @ Y )
                @ ( collect_nat
                  @ ^ [X: nat] :
                      ( ( member_nat @ X @ A2 )
                      & ( R @ X @ Y ) ) ) )
            @ B2 ) ) ) ) ).

% sum.swap_restrict
thf(fact_7035_case__prodD_H,axiom,
    ! [R: nat > nat > product_prod_nat_nat > $o,A: nat,B: nat,C: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ R @ ( product_Pair_nat_nat @ A @ B ) @ C )
     => ( R @ A @ B @ C ) ) ).

% case_prodD'
thf(fact_7036_case__prodE_H,axiom,
    ! [C: nat > nat > product_prod_nat_nat > $o,P4: product_prod_nat_nat,Z2: product_prod_nat_nat] :
      ( ( produc8739625826339149834_nat_o @ C @ P4 @ Z2 )
     => ~ ! [X4: nat,Y5: nat] :
            ( ( P4
              = ( product_Pair_nat_nat @ X4 @ Y5 ) )
           => ~ ( C @ X4 @ Y5 @ Z2 ) ) ) ).

% case_prodE'
thf(fact_7037_sum__nonneg,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7038_sum__nonneg,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7039_sum__nonneg,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7040_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7041_sum__nonneg,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7042_sum__nonneg,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7043_sum__nonneg,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7044_sum__nonneg,axiom,
    ! [A2: set_real,F: real > int] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1932886352136224148al_int @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7045_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7046_sum__nonneg,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups4538972089207619220nt_int @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_7047_sum__nonpos,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_7048_sum__nonpos,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_real @ ( F @ X4 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_7049_sum__nonpos,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7050_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7051_sum__nonpos,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X4 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_7052_sum__nonpos,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7053_sum__nonpos,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X4 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_7054_sum__nonpos,axiom,
    ! [A2: set_real,F: real > int] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ zero_zero_int ) ) ).

% sum_nonpos
thf(fact_7055_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ zero_zero_int ) ) ).

% sum_nonpos
thf(fact_7056_sum__nonpos,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_int @ ( F @ X4 ) @ zero_zero_int ) )
     => ( ord_less_eq_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ zero_zero_int ) ) ).

% sum_nonpos
thf(fact_7057_sum__mono__inv,axiom,
    ! [F: real > rat,I5: set_real,G: real > rat,I: real] :
      ( ( ( groups1300246762558778688al_rat @ F @ I5 )
        = ( groups1300246762558778688al_rat @ G @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7058_sum__mono__inv,axiom,
    ! [F: nat > rat,I5: set_nat,G: nat > rat,I: nat] :
      ( ( ( groups2906978787729119204at_rat @ F @ I5 )
        = ( groups2906978787729119204at_rat @ G @ I5 ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7059_sum__mono__inv,axiom,
    ! [F: int > rat,I5: set_int,G: int > rat,I: int] :
      ( ( ( groups3906332499630173760nt_rat @ F @ I5 )
        = ( groups3906332499630173760nt_rat @ G @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7060_sum__mono__inv,axiom,
    ! [F: complex > rat,I5: set_complex,G: complex > rat,I: complex] :
      ( ( ( groups5058264527183730370ex_rat @ F @ I5 )
        = ( groups5058264527183730370ex_rat @ G @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_rat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7061_sum__mono__inv,axiom,
    ! [F: real > nat,I5: set_real,G: real > nat,I: real] :
      ( ( ( groups1935376822645274424al_nat @ F @ I5 )
        = ( groups1935376822645274424al_nat @ G @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7062_sum__mono__inv,axiom,
    ! [F: int > nat,I5: set_int,G: int > nat,I: int] :
      ( ( ( groups4541462559716669496nt_nat @ F @ I5 )
        = ( groups4541462559716669496nt_nat @ G @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7063_sum__mono__inv,axiom,
    ! [F: complex > nat,I5: set_complex,G: complex > nat,I: complex] :
      ( ( ( groups5693394587270226106ex_nat @ F @ I5 )
        = ( groups5693394587270226106ex_nat @ G @ I5 ) )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ I5 )
           => ( ord_less_eq_nat @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_complex @ I @ I5 )
         => ( ( finite3207457112153483333omplex @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7064_sum__mono__inv,axiom,
    ! [F: real > int,I5: set_real,G: real > int,I: real] :
      ( ( ( groups1932886352136224148al_int @ F @ I5 )
        = ( groups1932886352136224148al_int @ G @ I5 ) )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_real @ I @ I5 )
         => ( ( finite_finite_real @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7065_sum__mono__inv,axiom,
    ! [F: nat > int,I5: set_nat,G: nat > int,I: nat] :
      ( ( ( groups3539618377306564664at_int @ F @ I5 )
        = ( groups3539618377306564664at_int @ G @ I5 ) )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_nat @ I @ I5 )
         => ( ( finite_finite_nat @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7066_sum__mono__inv,axiom,
    ! [F: int > int,I5: set_int,G: int > int,I: int] :
      ( ( ( groups4538972089207619220nt_int @ F @ I5 )
        = ( groups4538972089207619220nt_int @ G @ I5 ) )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ I5 )
           => ( ord_less_eq_int @ ( F @ I3 ) @ ( G @ I3 ) ) )
       => ( ( member_int @ I @ I5 )
         => ( ( finite_finite_int @ I5 )
           => ( ( F @ I )
              = ( G @ I ) ) ) ) ) ) ).

% sum_mono_inv
thf(fact_7067_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_7068_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_7069_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_7070_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_7071_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_7072_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_7073_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_7074_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_7075_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7076_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7077_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7078_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_7079_abs__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_7080_abs__triangle__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_7081_abs__triangle__ineq,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_7082_abs__triangle__ineq,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( plus_plus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq
thf(fact_7083_abs__mult__less,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer,D: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ C )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ B ) @ D )
       => ( ord_le6747313008572928689nteger @ ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( times_3573771949741848930nteger @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7084_abs__mult__less,axiom,
    ! [A: real,C: real,B: real,D: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ C )
     => ( ( ord_less_real @ ( abs_abs_real @ B ) @ D )
       => ( ord_less_real @ ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( times_times_real @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7085_abs__mult__less,axiom,
    ! [A: rat,C: rat,B: rat,D: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ C )
     => ( ( ord_less_rat @ ( abs_abs_rat @ B ) @ D )
       => ( ord_less_rat @ ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( times_times_rat @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7086_abs__mult__less,axiom,
    ! [A: int,C: int,B: int,D: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ C )
     => ( ( ord_less_int @ ( abs_abs_int @ B ) @ D )
       => ( ord_less_int @ ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( times_times_int @ C @ D ) ) ) ) ).

% abs_mult_less
thf(fact_7087_abs__triangle__ineq2,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_7088_abs__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_7089_abs__triangle__ineq2,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_7090_abs__triangle__ineq2,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq2
thf(fact_7091_abs__triangle__ineq3,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_7092_abs__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_7093_abs__triangle__ineq3,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_7094_abs__triangle__ineq3,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) ) ).

% abs_triangle_ineq3
thf(fact_7095_abs__triangle__ineq2__sym,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7096_abs__triangle__ineq2__sym,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7097_abs__triangle__ineq2__sym,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7098_abs__triangle__ineq2__sym,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( minus_minus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ A ) ) ) ).

% abs_triangle_ineq2_sym
thf(fact_7099_nonzero__abs__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( abs_abs_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_7100_nonzero__abs__divide,axiom,
    ! [B: rat,A: rat] :
      ( ( B != zero_zero_rat )
     => ( ( abs_abs_rat @ ( divide_divide_rat @ A @ B ) )
        = ( divide_divide_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% nonzero_abs_divide
thf(fact_7101_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_7102_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_7103_abs__ge__minus__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ ( abs_abs_rat @ A ) ) ).

% abs_ge_minus_self
thf(fact_7104_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_7105_abs__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_eq_real @ A @ B )
        & ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_7106_abs__le__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le3102999989581377725nteger @ A @ B )
        & ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_7107_abs__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_eq_rat @ A @ B )
        & ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_7108_abs__le__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_eq_int @ A @ B )
        & ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_le_iff
thf(fact_7109_abs__le__D2,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_7110_abs__le__D2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B )
     => ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_7111_abs__le__D2,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B )
     => ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_7112_abs__le__D2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B )
     => ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% abs_le_D2
thf(fact_7113_abs__leI,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ A ) @ B )
       => ( ord_less_eq_real @ ( abs_abs_real @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_7114_abs__leI,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ B )
     => ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
       => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_7115_abs__leI,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ A ) @ B )
       => ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_7116_abs__leI,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ ( uminus_uminus_int @ A ) @ B )
       => ( ord_less_eq_int @ ( abs_abs_int @ A ) @ B ) ) ) ).

% abs_leI
thf(fact_7117_abs__less__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ ( abs_abs_int @ A ) @ B )
      = ( ( ord_less_int @ A @ B )
        & ( ord_less_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_7118_abs__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( abs_abs_real @ A ) @ B )
      = ( ( ord_less_real @ A @ B )
        & ( ord_less_real @ ( uminus_uminus_real @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_7119_abs__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ A ) @ B )
      = ( ( ord_less_rat @ A @ B )
        & ( ord_less_rat @ ( uminus_uminus_rat @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_7120_abs__less__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ B )
      = ( ( ord_le6747313008572928689nteger @ A @ B )
        & ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% abs_less_iff
thf(fact_7121_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ ( suc @ X4 ) @ A2 )
           => ( ( F @ ( suc @ X4 ) )
              = ( G @ ( suc @ X4 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_7122_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ ( suc @ X4 ) @ A2 )
           => ( ( F @ ( suc @ X4 ) )
              = ( G @ ( suc @ X4 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_7123_Collect__case__prod__mono,axiom,
    ! [A2: int > int > $o,B2: int > int > $o] :
      ( ( ord_le6741204236512500942_int_o @ A2 @ B2 )
     => ( ord_le2843351958646193337nt_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ A2 ) ) @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ B2 ) ) ) ) ).

% Collect_case_prod_mono
thf(fact_7124_abs__zmult__eq__1,axiom,
    ! [M: int,N2: int] :
      ( ( ( abs_abs_int @ ( times_times_int @ M @ N2 ) )
        = one_one_int )
     => ( ( abs_abs_int @ M )
        = one_one_int ) ) ).

% abs_zmult_eq_1
thf(fact_7125_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > complex,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups5754745047067104278omplex @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups5754745047067104278omplex
          @ ^ [X: real] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7126_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > complex,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2073611262835488442omplex @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups2073611262835488442omplex
          @ ^ [X: nat] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7127_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > complex,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups3049146728041665814omplex
          @ ^ [X: int] : ( if_complex @ ( P @ X ) @ ( G @ X ) @ zero_zero_complex )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7128_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > real,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups8097168146408367636l_real
          @ ^ [X: real] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7129_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > real,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups8778361861064173332t_real
          @ ^ [X: int] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7130_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > real,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups5808333547571424918x_real
          @ ^ [X: complex] : ( if_real @ ( P @ X ) @ ( G @ X ) @ zero_zero_real )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7131_sum_Ointer__filter,axiom,
    ! [A2: set_real,G: real > rat,P: real > $o] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( collect_real
            @ ^ [X: real] :
                ( ( member_real @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups1300246762558778688al_rat
          @ ^ [X: real] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7132_sum_Ointer__filter,axiom,
    ! [A2: set_nat,G: nat > rat,P: nat > $o] :
      ( ( finite_finite_nat @ A2 )
     => ( ( groups2906978787729119204at_rat @ G
          @ ( collect_nat
            @ ^ [X: nat] :
                ( ( member_nat @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups2906978787729119204at_rat
          @ ^ [X: nat] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7133_sum_Ointer__filter,axiom,
    ! [A2: set_int,G: int > rat,P: int > $o] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( collect_int
            @ ^ [X: int] :
                ( ( member_int @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups3906332499630173760nt_rat
          @ ^ [X: int] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7134_sum_Ointer__filter,axiom,
    ! [A2: set_complex,G: complex > rat,P: complex > $o] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( collect_complex
            @ ^ [X: complex] :
                ( ( member_complex @ X @ A2 )
                & ( P @ X ) ) ) )
        = ( groups5058264527183730370ex_rat
          @ ^ [X: complex] : ( if_rat @ ( P @ X ) @ ( G @ X ) @ zero_zero_rat )
          @ A2 ) ) ) ).

% sum.inter_filter
thf(fact_7135_sum__subtractf__nat,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
      ( ! [X4: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups977919841031483927at_nat
          @ ^ [X: product_prod_nat_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7136_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: real > nat,F: real > nat] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [X: real] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7137_sum__subtractf__nat,axiom,
    ! [A2: set_set_nat,G: set_nat > nat,F: set_nat > nat] :
      ( ! [X4: set_nat] :
          ( ( member_set_nat @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups8294997508430121362at_nat
          @ ^ [X: set_nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups8294997508430121362at_nat @ F @ A2 ) @ ( groups8294997508430121362at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7138_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: int > nat,F: int > nat] :
      ( ! [X4: int] :
          ( ( member_int @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7139_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X4 ) @ ( F @ X4 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_7140_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X3 @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I2: real] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7141_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X3: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X3 @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I2: nat] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7142_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X3 @ I3 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X3 @ I5 )
          = one_one_Code_integer )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I2: int] : ( times_3573771949741848930nteger @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7143_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X3 @ I5 )
          = one_one_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I2: real] : ( times_times_real @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7144_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X3 @ I3 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X3 @ I5 )
          = one_one_real )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I2: int] : ( times_times_real @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7145_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X3 @ I5 )
          = one_one_rat )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I2: real] : ( times_times_rat @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7146_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X3: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X3 @ I5 )
          = one_one_rat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I2: nat] : ( times_times_rat @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7147_convex__sum__bound__le,axiom,
    ! [I5: set_int,X3: int > rat,A: int > rat,B: rat,Delta: rat] :
      ( ! [I3: int] :
          ( ( member_int @ I3 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X3 @ I3 ) ) )
     => ( ( ( groups3906332499630173760nt_rat @ X3 @ I5 )
          = one_one_rat )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups3906332499630173760nt_rat
                  @ ^ [I2: int] : ( times_times_rat @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7148_convex__sum__bound__le,axiom,
    ! [I5: set_real,X3: real > int,A: real > int,B: int,Delta: int] :
      ( ! [I3: real] :
          ( ( member_real @ I3 @ I5 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X3 @ I3 ) ) )
     => ( ( ( groups1932886352136224148al_int @ X3 @ I5 )
          = one_one_int )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups1932886352136224148al_int
                  @ ^ [I2: real] : ( times_times_int @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7149_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X3: nat > int,A: nat > int,B: int,Delta: int] :
      ( ! [I3: nat] :
          ( ( member_nat @ I3 @ I5 )
         => ( ord_less_eq_int @ zero_zero_int @ ( X3 @ I3 ) ) )
     => ( ( ( groups3539618377306564664at_int @ X3 @ I5 )
          = one_one_int )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( A @ I3 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_int
            @ ( abs_abs_int
              @ ( minus_minus_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I2: nat] : ( times_times_int @ ( A @ I2 ) @ ( X3 @ I2 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_7150_infinite__int__iff__unbounded__le,axiom,
    ! [S3: set_int] :
      ( ( ~ ( finite_finite_int @ S3 ) )
      = ( ! [M2: int] :
          ? [N: int] :
            ( ( ord_less_eq_int @ M2 @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S3 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_7151_infinite__int__iff__unbounded,axiom,
    ! [S3: set_int] :
      ( ( ~ ( finite_finite_int @ S3 ) )
      = ( ! [M2: int] :
          ? [N: int] :
            ( ( ord_less_int @ M2 @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S3 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_7152_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_7153_sum_Oshift__bounds__cl__Suc__ivl,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_7154_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > nat,M: nat,K: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_7155_sum_Oshift__bounds__cl__nat__ivl,axiom,
    ! [G: nat > real,M: nat,K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ M @ K ) @ ( plus_plus_nat @ N2 @ K ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ I2 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_7156_tanh__real__lt__1,axiom,
    ! [X3: real] : ( ord_less_real @ ( tanh_real @ X3 ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_7157_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > real,I: int > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7158_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > real,I: complex > int,F: int > real] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7159_sum__le__included,axiom,
    ! [S: set_complex,T: set_int,G: int > real,I: int > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups8778361861064173332t_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7160_sum__le__included,axiom,
    ! [S: set_complex,T: set_complex,G: complex > real,I: complex > complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ T )
             => ( ord_less_eq_real @ zero_zero_real @ ( G @ X4 ) ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ S ) @ ( groups5808333547571424918x_real @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7161_sum__le__included,axiom,
    ! [S: set_nat,T: set_nat,G: nat > rat,I: nat > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7162_sum__le__included,axiom,
    ! [S: set_nat,T: set_int,G: int > rat,I: int > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7163_sum__le__included,axiom,
    ! [S: set_nat,T: set_complex,G: complex > rat,I: complex > nat,F: nat > rat] :
      ( ( finite_finite_nat @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7164_sum__le__included,axiom,
    ! [S: set_int,T: set_nat,G: nat > rat,I: nat > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_nat @ T )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S )
               => ? [Xa: nat] :
                    ( ( member_nat @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups2906978787729119204at_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7165_sum__le__included,axiom,
    ! [S: set_int,T: set_int,G: int > rat,I: int > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite_finite_int @ T )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S )
               => ? [Xa: int] :
                    ( ( member_int @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups3906332499630173760nt_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7166_sum__le__included,axiom,
    ! [S: set_int,T: set_complex,G: complex > rat,I: complex > int,F: int > rat] :
      ( ( finite_finite_int @ S )
     => ( ( finite3207457112153483333omplex @ T )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ T )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( G @ X4 ) ) )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S )
               => ? [Xa: complex] :
                    ( ( member_complex @ Xa @ T )
                    & ( ( I @ Xa )
                      = X4 )
                    & ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ Xa ) ) ) )
           => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ S ) @ ( groups5058264527183730370ex_rat @ G @ T ) ) ) ) ) ) ).

% sum_le_included
thf(fact_7167_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X: real] :
                ( ( member_real @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7168_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X: int] :
                ( ( member_int @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7169_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ A2 )
            = zero_zero_real )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_real ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7170_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X: real] :
                ( ( member_real @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7171_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X: nat] :
                ( ( member_nat @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7172_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X: int] :
                ( ( member_int @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7173_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ A2 )
            = zero_zero_rat )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_rat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7174_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X: real] :
                ( ( member_real @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7175_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X: int] :
                ( ( member_int @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7176_sum__nonneg__eq__0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
            = zero_zero_nat )
          = ( ! [X: complex] :
                ( ( member_complex @ X @ A2 )
               => ( ( F @ X )
                  = zero_zero_nat ) ) ) ) ) ) ).

% sum_nonneg_eq_0_iff
thf(fact_7177_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [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_7178_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [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_7179_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: nat] :
              ( ( member_nat @ X2 @ A2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7180_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: int] :
              ( ( member_int @ X2 @ A2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7181_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [X2: complex] :
              ( ( member_complex @ X2 @ A2 )
              & ( ord_less_rat @ ( F @ X2 ) @ ( G @ X2 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono_ex1
thf(fact_7182_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [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_7183_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [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_7184_sum__strict__mono__ex1,axiom,
    ! [A2: set_nat,F: nat > int,G: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ A2 )
           => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [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_7185_sum__strict__mono__ex1,axiom,
    ! [A2: set_int,F: int > int,G: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ A2 )
           => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [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_7186_sum__strict__mono__ex1,axiom,
    ! [A2: set_complex,F: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ A2 )
           => ( ord_less_eq_int @ ( F @ X4 ) @ ( G @ X4 ) ) )
       => ( ? [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_7187_sum_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( R @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups2073611262835488442omplex @ H2 @ S3 ) @ ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7188_sum_Orelated,axiom,
    ! [R: complex > complex > $o,S3: set_int,H2: int > complex,G: int > complex] :
      ( ( R @ zero_zero_complex @ zero_zero_complex )
     => ( ! [X16: complex,Y15: complex,X23: complex,Y23: complex] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_complex @ X16 @ Y15 ) @ ( plus_plus_complex @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups3049146728041665814omplex @ H2 @ S3 ) @ ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7189_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_int,H2: int > real,G: int > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups8778361861064173332t_real @ H2 @ S3 ) @ ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7190_sum_Orelated,axiom,
    ! [R: real > real > $o,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( R @ zero_zero_real @ zero_zero_real )
     => ( ! [X16: real,Y15: real,X23: real,Y23: real] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_real @ X16 @ Y15 ) @ ( plus_plus_real @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups5808333547571424918x_real @ H2 @ S3 ) @ ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7191_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_nat,H2: nat > rat,G: nat > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups2906978787729119204at_rat @ H2 @ S3 ) @ ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7192_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_int,H2: int > rat,G: int > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups3906332499630173760nt_rat @ H2 @ S3 ) @ ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7193_sum_Orelated,axiom,
    ! [R: rat > rat > $o,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( R @ zero_zero_rat @ zero_zero_rat )
     => ( ! [X16: rat,Y15: rat,X23: rat,Y23: rat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_rat @ X16 @ Y15 ) @ ( plus_plus_rat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups5058264527183730370ex_rat @ H2 @ S3 ) @ ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7194_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_int,H2: int > nat,G: int > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite_finite_int @ S3 )
         => ( ! [X4: int] :
                ( ( member_int @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups4541462559716669496nt_nat @ H2 @ S3 ) @ ( groups4541462559716669496nt_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7195_sum_Orelated,axiom,
    ! [R: nat > nat > $o,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( R @ zero_zero_nat @ zero_zero_nat )
     => ( ! [X16: nat,Y15: nat,X23: nat,Y23: nat] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_nat @ X16 @ Y15 ) @ ( plus_plus_nat @ X23 @ Y23 ) ) )
       => ( ( finite3207457112153483333omplex @ S3 )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups5693394587270226106ex_nat @ H2 @ S3 ) @ ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7196_sum_Orelated,axiom,
    ! [R: int > int > $o,S3: set_nat,H2: nat > int,G: nat > int] :
      ( ( R @ zero_zero_int @ zero_zero_int )
     => ( ! [X16: int,Y15: int,X23: int,Y23: int] :
            ( ( ( R @ X16 @ X23 )
              & ( R @ Y15 @ Y23 ) )
           => ( R @ ( plus_plus_int @ X16 @ Y15 ) @ ( plus_plus_int @ X23 @ Y23 ) ) )
       => ( ( finite_finite_nat @ S3 )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S3 )
               => ( R @ ( H2 @ X4 ) @ ( G @ X4 ) ) )
           => ( R @ ( groups3539618377306564664at_int @ H2 @ S3 ) @ ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ) ).

% sum.related
thf(fact_7197_dense__eq0__I,axiom,
    ! [X3: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ E ) )
     => ( X3 = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_7198_dense__eq0__I,axiom,
    ! [X3: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ E ) )
     => ( X3 = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_7199_abs__mult__pos,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X3 )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y3 ) @ X3 )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y3 @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_7200_abs__mult__pos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( times_times_real @ ( abs_abs_real @ Y3 ) @ X3 )
        = ( abs_abs_real @ ( times_times_real @ Y3 @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_7201_abs__mult__pos,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y3 ) @ X3 )
        = ( abs_abs_rat @ ( times_times_rat @ Y3 @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_7202_abs__mult__pos,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( times_times_int @ ( abs_abs_int @ Y3 ) @ X3 )
        = ( abs_abs_int @ ( times_times_int @ Y3 @ X3 ) ) ) ) ).

% abs_mult_pos
thf(fact_7203_abs__eq__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
          | ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) )
        & ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
          | ( ord_le3102999989581377725nteger @ B @ zero_z3403309356797280102nteger ) ) )
     => ( ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
        = ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_7204_abs__eq__mult,axiom,
    ! [A: real,B: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          | ( ord_less_eq_real @ A @ zero_zero_real ) )
        & ( ( ord_less_eq_real @ zero_zero_real @ B )
          | ( ord_less_eq_real @ B @ zero_zero_real ) ) )
     => ( ( abs_abs_real @ ( times_times_real @ A @ B ) )
        = ( times_times_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_7205_abs__eq__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( ( ( ord_less_eq_rat @ zero_zero_rat @ A )
          | ( ord_less_eq_rat @ A @ zero_zero_rat ) )
        & ( ( ord_less_eq_rat @ zero_zero_rat @ B )
          | ( ord_less_eq_rat @ B @ zero_zero_rat ) ) )
     => ( ( abs_abs_rat @ ( times_times_rat @ A @ B ) )
        = ( times_times_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_7206_abs__eq__mult,axiom,
    ! [A: int,B: int] :
      ( ( ( ( ord_less_eq_int @ zero_zero_int @ A )
          | ( ord_less_eq_int @ A @ zero_zero_int ) )
        & ( ( ord_less_eq_int @ zero_zero_int @ B )
          | ( ord_less_eq_int @ B @ zero_zero_int ) ) )
     => ( ( abs_abs_int @ ( times_times_int @ A @ B ) )
        = ( times_times_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% abs_eq_mult
thf(fact_7207_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7208_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A2 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7209_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > real,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ( ord_less_real @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7210_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7211_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > rat,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A2 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7212_sum__strict__mono,axiom,
    ! [A2: set_nat,F: nat > rat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( A2 != bot_bot_set_nat )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ A2 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7213_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > rat,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ( ord_less_rat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7214_sum__strict__mono,axiom,
    ! [A2: set_complex,F: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( A2 != bot_bot_set_complex )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ A2 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7215_sum__strict__mono,axiom,
    ! [A2: set_real,F: real > nat,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( A2 != bot_bot_set_real )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ A2 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7216_sum__strict__mono,axiom,
    ! [A2: set_int,F: int > nat,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( A2 != bot_bot_set_int )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ A2 )
             => ( ord_less_nat @ ( F @ X4 ) @ ( G @ X4 ) ) )
         => ( ord_less_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ).

% sum_strict_mono
thf(fact_7217_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_7218_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_7219_abs__minus__le__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( abs_abs_rat @ A ) ) @ zero_zero_rat ) ).

% abs_minus_le_zero
thf(fact_7220_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_7221_abs__eq__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( ( abs_abs_real @ A )
        = B )
      = ( ( ord_less_eq_real @ zero_zero_real @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_real @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7222_abs__eq__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = B )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ B )
        & ( ( A = B )
          | ( A
            = ( uminus1351360451143612070nteger @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7223_abs__eq__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ( abs_abs_rat @ A )
        = B )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_rat @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7224_abs__eq__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( ( abs_abs_int @ A )
        = B )
      = ( ( ord_less_eq_int @ zero_zero_int @ B )
        & ( ( A = B )
          | ( A
            = ( uminus_uminus_int @ B ) ) ) ) ) ).

% abs_eq_iff'
thf(fact_7225_eq__abs__iff_H,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( abs_abs_real @ B ) )
      = ( ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_real @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7226_eq__abs__iff_H,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( abs_abs_Code_integer @ B ) )
      = ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
        & ( ( B = A )
          | ( B
            = ( uminus1351360451143612070nteger @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7227_eq__abs__iff_H,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( abs_abs_rat @ B ) )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_rat @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7228_eq__abs__iff_H,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( abs_abs_int @ B ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ A )
        & ( ( B = A )
          | ( B
            = ( uminus_uminus_int @ A ) ) ) ) ) ).

% eq_abs_iff'
thf(fact_7229_zero__le__power__abs,axiom,
    ! [A: code_integer,N2: nat] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7230_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_7231_zero__le__power__abs,axiom,
    ! [A: rat,N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).

% zero_le_power_abs
thf(fact_7232_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_7233_abs__div__pos,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ( ( divide_divide_real @ ( abs_abs_real @ X3 ) @ Y3 )
        = ( abs_abs_real @ ( divide_divide_real @ X3 @ Y3 ) ) ) ) ).

% abs_div_pos
thf(fact_7234_abs__div__pos,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y3 )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X3 ) @ Y3 )
        = ( abs_abs_rat @ ( divide_divide_rat @ X3 @ Y3 ) ) ) ) ).

% abs_div_pos
thf(fact_7235_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_7236_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_7237_abs__if__raw,axiom,
    ( abs_abs_rat
    = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_7238_abs__if__raw,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).

% abs_if_raw
thf(fact_7239_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_7240_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_7241_abs__of__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( abs_abs_rat @ A )
        = ( uminus_uminus_rat @ A ) ) ) ).

% abs_of_neg
thf(fact_7242_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_7243_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_7244_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_7245_abs__if,axiom,
    ( abs_abs_rat
    = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_7246_abs__if,axiom,
    ( abs_abs_Code_integer
    = ( ^ [A3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ A3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ A3 ) @ A3 ) ) ) ).

% abs_if
thf(fact_7247_abs__diff__triangle__ineq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer,D: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ ( plus_p5714425477246183910nteger @ C @ D ) ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ C ) ) @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7248_abs__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ ( minus_minus_real @ A @ C ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7249_abs__diff__triangle__ineq,axiom,
    ! [A: rat,B: rat,C: rat,D: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ ( plus_plus_rat @ C @ D ) ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ C ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7250_abs__diff__triangle__ineq,axiom,
    ! [A: int,B: int,C: int,D: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ ( plus_plus_int @ C @ D ) ) ) @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ A @ C ) ) @ ( abs_abs_int @ ( minus_minus_int @ B @ D ) ) ) ) ).

% abs_diff_triangle_ineq
thf(fact_7251_abs__triangle__ineq4,axiom,
    ! [A: code_integer,B: code_integer] : ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ A @ B ) ) @ ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_7252_abs__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_7253_abs__triangle__ineq4,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ A @ B ) ) @ ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_7254_abs__triangle__ineq4,axiom,
    ! [A: int,B: int] : ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ A @ B ) ) @ ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ).

% abs_triangle_ineq4
thf(fact_7255_abs__diff__le__iff,axiom,
    ! [X3: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
        & ( ord_le3102999989581377725nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7256_abs__diff__le__iff,axiom,
    ! [X3: real,A: real,R2: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
        & ( ord_less_eq_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7257_abs__diff__le__iff,axiom,
    ! [X3: rat,A: rat,R2: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
        & ( ord_less_eq_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7258_abs__diff__le__iff,axiom,
    ! [X3: int,A: int,R2: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
        & ( ord_less_eq_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_7259_sum_Oinsert__if,axiom,
    ! [A2: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X3 @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
            = ( groups8097168146408367636l_real @ G @ A2 ) ) )
        & ( ~ ( member_real @ X3 @ A2 )
         => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7260_sum_Oinsert__if,axiom,
    ! [A2: set_int,X3: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X3 @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
            = ( groups8778361861064173332t_real @ G @ A2 ) ) )
        & ( ~ ( member_int @ X3 @ A2 )
         => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
            = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7261_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X3 @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
            = ( groups5808333547571424918x_real @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X3 @ A2 )
         => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
            = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7262_sum_Oinsert__if,axiom,
    ! [A2: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X3 @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
            = ( groups1300246762558778688al_rat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X3 @ A2 )
         => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7263_sum_Oinsert__if,axiom,
    ! [A2: set_nat,X3: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ X3 @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
            = ( groups2906978787729119204at_rat @ G @ A2 ) ) )
        & ( ~ ( member_nat @ X3 @ A2 )
         => ( ( groups2906978787729119204at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups2906978787729119204at_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7264_sum_Oinsert__if,axiom,
    ! [A2: set_int,X3: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X3 @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
            = ( groups3906332499630173760nt_rat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X3 @ A2 )
         => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7265_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X3: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X3 @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
            = ( groups5058264527183730370ex_rat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X3 @ A2 )
         => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
            = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7266_sum_Oinsert__if,axiom,
    ! [A2: set_real,X3: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ X3 @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
            = ( groups1935376822645274424al_nat @ G @ A2 ) ) )
        & ( ~ ( member_real @ X3 @ A2 )
         => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7267_sum_Oinsert__if,axiom,
    ! [A2: set_int,X3: int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ X3 @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X3 @ A2 ) )
            = ( groups4541462559716669496nt_nat @ G @ A2 ) ) )
        & ( ~ ( member_int @ X3 @ A2 )
         => ( ( groups4541462559716669496nt_nat @ G @ ( insert_int @ X3 @ A2 ) )
            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7268_sum_Oinsert__if,axiom,
    ! [A2: set_complex,X3: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ X3 @ A2 )
         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
            = ( groups5693394587270226106ex_nat @ G @ A2 ) ) )
        & ( ~ ( member_complex @ X3 @ A2 )
         => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
            = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ) ) ).

% sum.insert_if
thf(fact_7269_abs__diff__less__iff,axiom,
    ! [X3: code_integer,A: code_integer,R2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X3 @ A ) ) @ R2 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R2 ) @ X3 )
        & ( ord_le6747313008572928689nteger @ X3 @ ( plus_p5714425477246183910nteger @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7270_abs__diff__less__iff,axiom,
    ! [X3: real,A: real,R2: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R2 ) @ X3 )
        & ( ord_less_real @ X3 @ ( plus_plus_real @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7271_abs__diff__less__iff,axiom,
    ! [X3: rat,A: rat,R2: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R2 ) @ X3 )
        & ( ord_less_rat @ X3 @ ( plus_plus_rat @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7272_abs__diff__less__iff,axiom,
    ! [X3: int,A: int,R2: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ A ) ) @ R2 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R2 ) @ X3 )
        & ( ord_less_int @ X3 @ ( plus_plus_int @ A @ R2 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_7273_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ S6 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
               => ( 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 @ S3 @ S6 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S3 )
                        = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7274_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > complex,H2: int > complex] :
      ( ( finite_finite_real @ S6 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
               => ( 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 @ S3 @ S6 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_complex ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups5754745047067104278omplex @ G @ S3 )
                        = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7275_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_real,S3: set_int,I: real > int,J: int > real,T3: set_real,G: int > complex,H2: real > complex] :
      ( ( finite_finite_int @ S6 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
               => ( 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 @ S3 @ S6 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_complex ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S3 )
                        = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7276_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > complex,H2: int > complex] :
      ( ( finite_finite_int @ S6 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
               => ( 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_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_complex ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_complex ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups3049146728041665814omplex @ G @ S3 )
                        = ( groups3049146728041665814omplex @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7277_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_real,S3: set_real,I: real > real,J: real > real,T3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ S6 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
               => ( 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 @ S3 @ S6 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7278_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_int,S3: set_real,I: int > real,J: real > int,T3: set_int,G: real > real,H2: int > real] :
      ( ( finite_finite_real @ S6 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
               => ( 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 @ S3 @ S6 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7279_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_real,T5: set_complex,S3: set_real,I: complex > real,J: real > complex,T3: set_complex,G: real > real,H2: complex > real] :
      ( ( finite_finite_real @ S6 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A4: real] :
              ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ S3 @ S6 ) )
               => ( 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 @ S3 @ S6 ) ) )
               => ( ! [A4: real] :
                      ( ( member_real @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: real] :
                          ( ( member_real @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8097168146408367636l_real @ G @ S3 )
                        = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7280_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_real,S3: set_int,I: real > int,J: int > real,T3: set_real,G: int > real,H2: real > real] :
      ( ( finite_finite_int @ S6 )
     => ( ( finite_finite_real @ T5 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
               => ( 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 @ S3 @ S6 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: real] :
                        ( ( member_real @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7281_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_int,S3: set_int,I: int > int,J: int > int,T3: set_int,G: int > real,H2: int > real] :
      ( ( finite_finite_int @ S6 )
     => ( ( finite_finite_int @ T5 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
               => ( 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_int @ ( I @ B4 ) @ ( minus_minus_set_int @ S3 @ S6 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: int] :
                        ( ( member_int @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups8778361861064173332t_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7282_sum_Oreindex__bij__witness__not__neutral,axiom,
    ! [S6: set_int,T5: set_complex,S3: set_int,I: complex > int,J: int > complex,T3: set_complex,G: int > real,H2: complex > real] :
      ( ( finite_finite_int @ S6 )
     => ( ( finite3207457112153483333omplex @ T5 )
       => ( ! [A4: int] :
              ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
             => ( ( I @ ( J @ A4 ) )
                = A4 ) )
         => ( ! [A4: int] :
                ( ( member_int @ A4 @ ( minus_minus_set_int @ S3 @ S6 ) )
               => ( 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 @ S3 @ S6 ) ) )
               => ( ! [A4: int] :
                      ( ( member_int @ A4 @ S6 )
                     => ( ( G @ A4 )
                        = zero_zero_real ) )
                 => ( ! [B4: complex] :
                        ( ( member_complex @ B4 @ T5 )
                       => ( ( H2 @ B4 )
                          = zero_zero_real ) )
                   => ( ! [A4: int] :
                          ( ( member_int @ A4 @ S3 )
                         => ( ( H2 @ ( J @ A4 ) )
                            = ( G @ A4 ) ) )
                     => ( ( groups8778361861064173332t_real @ G @ S3 )
                        = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ) ) ) ) ) ).

% sum.reindex_bij_witness_not_neutral
thf(fact_7283_sum__eq__Suc0__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: int] :
              ( ( member_int @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: int] :
                  ( ( member_int @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7284_sum__eq__Suc0__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: complex] :
                  ( ( member_complex @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7285_sum__eq__Suc0__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ( groups977919841031483927at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7286_sum__eq__Suc0__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( suc @ zero_zero_nat ) )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( ( F @ X )
                = ( suc @ zero_zero_nat ) )
              & ! [Y: nat] :
                  ( ( member_nat @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_Suc0_iff
thf(fact_7287_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N2 ) )
     => ? [X4: nat] :
          ( ( member_nat @ X4 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X4 ) ) ) ) ).

% sum_SucD
thf(fact_7288_sum__eq__1__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups4541462559716669496nt_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: int] :
              ( ( member_int @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: int] :
                  ( ( member_int @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7289_sum__eq__1__iff,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups5693394587270226106ex_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: complex] :
                  ( ( member_complex @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7290_sum__eq__1__iff,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ A2 )
     => ( ( ( groups977919841031483927at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: product_prod_nat_nat] :
              ( ( member8440522571783428010at_nat @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: product_prod_nat_nat] :
                  ( ( member8440522571783428010at_nat @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7291_sum__eq__1__iff,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups3542108847815614940at_nat @ F @ A2 )
          = one_one_nat )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( ( F @ X )
                = one_one_nat )
              & ! [Y: nat] :
                  ( ( member_nat @ Y @ A2 )
                 => ( ( X != Y )
                   => ( ( F @ Y )
                      = zero_zero_nat ) ) ) ) ) ) ) ).

% sum_eq_1_iff
thf(fact_7292_abs__real__def,axiom,
    ( abs_abs_real
    = ( ^ [A3: real] : ( if_real @ ( ord_less_real @ A3 @ zero_zero_real ) @ ( uminus_uminus_real @ A3 ) @ A3 ) ) ) ).

% abs_real_def
thf(fact_7293_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I2: int] : ( if_int @ ( ord_less_int @ I2 @ zero_zero_int ) @ ( uminus_uminus_int @ I2 ) @ I2 ) ) ) ).

% zabs_def
thf(fact_7294_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > real,B2: real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = B2 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7295_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > real,B2: real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = B2 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7296_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > real,B2: real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = B2 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_real @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7297_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > rat,B2: rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = B2 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7298_sum__nonneg__leq__bound,axiom,
    ! [S: set_nat,F: nat > rat,B2: rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = B2 )
         => ( ( member_nat @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7299_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > rat,B2: rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = B2 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7300_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > rat,B2: rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = B2 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_rat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7301_sum__nonneg__leq__bound,axiom,
    ! [S: set_real,F: real > nat,B2: nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = B2 )
         => ( ( member_real @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7302_sum__nonneg__leq__bound,axiom,
    ! [S: set_int,F: int > nat,B2: nat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = B2 )
         => ( ( member_int @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7303_sum__nonneg__leq__bound,axiom,
    ! [S: set_complex,F: complex > nat,B2: nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = B2 )
         => ( ( member_complex @ I @ S )
           => ( ord_less_eq_nat @ ( F @ I ) @ B2 ) ) ) ) ) ).

% sum_nonneg_leq_bound
thf(fact_7304_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > real,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8097168146408367636l_real @ F @ S )
            = zero_zero_real )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7305_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > real,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups8778361861064173332t_real @ F @ S )
            = zero_zero_real )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7306_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > real,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
       => ( ( ( groups5808333547571424918x_real @ F @ S )
            = zero_zero_real )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_real ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7307_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > rat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups1300246762558778688al_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7308_sum__nonneg__0,axiom,
    ! [S: set_nat,F: nat > rat,I: nat] :
      ( ( finite_finite_nat @ S )
     => ( ! [I3: nat] :
            ( ( member_nat @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups2906978787729119204at_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_nat @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7309_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > rat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups3906332499630173760nt_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7310_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > rat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
       => ( ( ( groups5058264527183730370ex_rat @ F @ S )
            = zero_zero_rat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_rat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7311_sum__nonneg__0,axiom,
    ! [S: set_real,F: real > nat,I: real] :
      ( ( finite_finite_real @ S )
     => ( ! [I3: real] :
            ( ( member_real @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups1935376822645274424al_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_real @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7312_sum__nonneg__0,axiom,
    ! [S: set_int,F: int > nat,I: int] :
      ( ( finite_finite_int @ S )
     => ( ! [I3: int] :
            ( ( member_int @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups4541462559716669496nt_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_int @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7313_sum__nonneg__0,axiom,
    ! [S: set_complex,F: complex > nat,I: complex] :
      ( ( finite3207457112153483333omplex @ S )
     => ( ! [I3: complex] :
            ( ( member_complex @ I3 @ S )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
       => ( ( ( groups5693394587270226106ex_nat @ F @ S )
            = zero_zero_nat )
         => ( ( member_complex @ I @ S )
           => ( ( F @ I )
              = zero_zero_nat ) ) ) ) ) ).

% sum_nonneg_0
thf(fact_7314_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_7315_abs__mod__less,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ord_less_int @ ( abs_abs_int @ ( modulo_modulo_int @ K @ L ) ) @ ( abs_abs_int @ L ) ) ) ).

% abs_mod_less
thf(fact_7316_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > complex] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups5754745047067104278omplex @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G @ X )
                  = zero_zero_complex ) ) ) )
        = ( groups5754745047067104278omplex @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7317_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3049146728041665814omplex @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_complex ) ) ) )
        = ( groups3049146728041665814omplex @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7318_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G @ X )
                  = zero_zero_real ) ) ) )
        = ( groups8097168146408367636l_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7319_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_real ) ) ) )
        = ( groups8778361861064173332t_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7320_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = zero_zero_real ) ) ) )
        = ( groups5808333547571424918x_real @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7321_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups1300246762558778688al_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7322_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups3906332499630173760nt_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7323_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G
          @ ( minus_811609699411566653omplex @ A2
            @ ( collect_complex
              @ ^ [X: complex] :
                  ( ( G @ X )
                  = zero_zero_rat ) ) ) )
        = ( groups5058264527183730370ex_rat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7324_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G
          @ ( minus_minus_set_real @ A2
            @ ( collect_real
              @ ^ [X: real] :
                  ( ( G @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups1935376822645274424al_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7325_sum_Osetdiff__irrelevant,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups4541462559716669496nt_nat @ G
          @ ( minus_minus_set_int @ A2
            @ ( collect_int
              @ ^ [X: int] :
                  ( ( G @ X )
                  = zero_zero_nat ) ) ) )
        = ( groups4541462559716669496nt_nat @ G @ A2 ) ) ) ).

% sum.setdiff_irrelevant
thf(fact_7326_sum__power__add,axiom,
    ! [X3: complex,M: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I2: nat] : ( power_power_complex @ X3 @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7327_sum__power__add,axiom,
    ! [X3: rat,M: nat,I5: set_nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I2: nat] : ( power_power_rat @ X3 @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7328_sum__power__add,axiom,
    ! [X3: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( power_power_int @ X3 @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X3 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7329_sum__power__add,axiom,
    ! [X3: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( power_power_real @ X3 @ ( plus_plus_nat @ M @ I2 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_7330_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_7331_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N2: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I2 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_7332_tanh__real__gt__neg1,axiom,
    ! [X3: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X3 ) ) ).

% tanh_real_gt_neg1
thf(fact_7333_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7334_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7335_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) ) )
           => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7336_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7337_sum__pos2,axiom,
    ! [I5: set_nat,I: nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( member_nat @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: nat] :
                ( ( member_nat @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7338_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7339_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_rat @ zero_zero_rat @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ I3 ) ) )
           => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7340_sum__pos2,axiom,
    ! [I5: set_real,I: real,F: real > nat] :
      ( ( finite_finite_real @ I5 )
     => ( ( member_real @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I3: real] :
                ( ( member_real @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7341_sum__pos2,axiom,
    ! [I5: set_int,I: int,F: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( member_int @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I3: int] :
                ( ( member_int @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7342_sum__pos2,axiom,
    ! [I5: set_complex,I: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( member_complex @ I @ I5 )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I ) )
         => ( ! [I3: complex] :
                ( ( member_complex @ I3 @ I5 )
               => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ I3 ) ) )
           => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ) ).

% sum_pos2
thf(fact_7343_sum__pos,axiom,
    ! [I5: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7344_sum__pos,axiom,
    ! [I5: set_real,F: real > real] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7345_sum__pos,axiom,
    ! [I5: set_int,F: int > real] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) )
         => ( ord_less_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7346_sum__pos,axiom,
    ! [I5: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7347_sum__pos,axiom,
    ! [I5: set_real,F: real > rat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7348_sum__pos,axiom,
    ! [I5: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ I5 )
     => ( ( I5 != bot_bot_set_nat )
       => ( ! [I3: nat] :
              ( ( member_nat @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7349_sum__pos,axiom,
    ! [I5: set_int,F: int > rat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_rat @ zero_zero_rat @ ( F @ I3 ) ) )
         => ( ord_less_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7350_sum__pos,axiom,
    ! [I5: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ I5 )
     => ( ( I5 != bot_bot_set_complex )
       => ( ! [I3: complex] :
              ( ( member_complex @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7351_sum__pos,axiom,
    ! [I5: set_real,F: real > nat] :
      ( ( finite_finite_real @ I5 )
     => ( ( I5 != bot_bot_set_real )
       => ( ! [I3: real] :
              ( ( member_real @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7352_sum__pos,axiom,
    ! [I5: set_int,F: int > nat] :
      ( ( finite_finite_int @ I5 )
     => ( ( I5 != bot_bot_set_int )
       => ( ! [I3: int] :
              ( ( member_int @ I3 @ I5 )
             => ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) )
         => ( ord_less_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ I5 ) ) ) ) ) ).

% sum_pos
thf(fact_7353_abs__add__one__gt__zero,axiom,
    ! [X3: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X3 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7354_abs__add__one__gt__zero,axiom,
    ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X3 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7355_abs__add__one__gt__zero,axiom,
    ! [X3: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X3 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7356_abs__add__one__gt__zero,axiom,
    ! [X3: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X3 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_7357_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ T3 )
              = ( groups5754745047067104278omplex @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7358_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ T3 )
              = ( groups8097168146408367636l_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7359_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ T3 )
              = ( groups5808333547571424918x_real @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7360_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ T3 )
              = ( groups1300246762558778688al_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7361_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ T3 )
              = ( groups5058264527183730370ex_rat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7362_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ T3 )
              = ( groups1935376822645274424al_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7363_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ T3 )
              = ( groups5693394587270226106ex_nat @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7364_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_real,S3: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_int ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1932886352136224148al_int @ G @ T3 )
              = ( groups1932886352136224148al_int @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7365_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_int ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5690904116761175830ex_int @ G @ T3 )
              = ( groups5690904116761175830ex_int @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7366_sum_Omono__neutral__cong__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_complex ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups2073611262835488442omplex @ G @ T3 )
              = ( groups2073611262835488442omplex @ H2 @ S3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_right
thf(fact_7367_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > complex,G: real > complex] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_complex ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5754745047067104278omplex @ G @ S3 )
              = ( groups5754745047067104278omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7368_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > real,G: real > real] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups8097168146408367636l_real @ G @ S3 )
              = ( groups8097168146408367636l_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7369_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > real,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_real ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5808333547571424918x_real @ G @ S3 )
              = ( groups5808333547571424918x_real @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7370_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > rat,G: real > rat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1300246762558778688al_rat @ G @ S3 )
              = ( groups1300246762558778688al_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7371_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > rat,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_rat ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5058264527183730370ex_rat @ G @ S3 )
              = ( groups5058264527183730370ex_rat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7372_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > nat,G: real > nat] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_nat ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1935376822645274424al_nat @ G @ S3 )
              = ( groups1935376822645274424al_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7373_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > nat,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_nat ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5693394587270226106ex_nat @ G @ S3 )
              = ( groups5693394587270226106ex_nat @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7374_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_real,S3: set_real,H2: real > int,G: real > int] :
      ( ( finite_finite_real @ T3 )
     => ( ( ord_less_eq_set_real @ S3 @ T3 )
       => ( ! [X4: real] :
              ( ( member_real @ X4 @ ( minus_minus_set_real @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_int ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups1932886352136224148al_int @ G @ S3 )
              = ( groups1932886352136224148al_int @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7375_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_complex,S3: set_complex,H2: complex > int,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_int ) )
         => ( ! [X4: complex] :
                ( ( member_complex @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups5690904116761175830ex_int @ G @ S3 )
              = ( groups5690904116761175830ex_int @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7376_sum_Omono__neutral__cong__left,axiom,
    ! [T3: set_nat,S3: set_nat,H2: nat > complex,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( H2 @ X4 )
                = zero_zero_complex ) )
         => ( ! [X4: nat] :
                ( ( member_nat @ X4 @ S3 )
               => ( ( G @ X4 )
                  = ( H2 @ X4 ) ) )
           => ( ( groups2073611262835488442omplex @ G @ S3 )
              = ( groups2073611262835488442omplex @ H2 @ T3 ) ) ) ) ) ) ).

% sum.mono_neutral_cong_left
thf(fact_7377_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ T3 )
            = ( groups5808333547571424918x_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7378_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ T3 )
            = ( groups5058264527183730370ex_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7379_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ T3 )
            = ( groups5693394587270226106ex_nat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7380_sum_Omono__neutral__right,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ T3 )
            = ( groups5690904116761175830ex_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7381_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ T3 )
            = ( groups2073611262835488442omplex @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7382_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ T3 )
            = ( groups2906978787729119204at_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7383_sum_Omono__neutral__right,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ T3 )
            = ( groups3539618377306564664at_int @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7384_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ T3 )
            = ( groups3049146728041665814omplex @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7385_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ T3 )
            = ( groups8778361861064173332t_real @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7386_sum_Omono__neutral__right,axiom,
    ! [T3: set_int,S3: set_int,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ T3 )
            = ( groups3906332499630173760nt_rat @ G @ S3 ) ) ) ) ) ).

% sum.mono_neutral_right
thf(fact_7387_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups5808333547571424918x_real @ G @ S3 )
            = ( groups5808333547571424918x_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7388_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups5058264527183730370ex_rat @ G @ S3 )
            = ( groups5058264527183730370ex_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7389_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_nat ) )
         => ( ( groups5693394587270226106ex_nat @ G @ S3 )
            = ( groups5693394587270226106ex_nat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7390_sum_Omono__neutral__left,axiom,
    ! [T3: set_complex,S3: set_complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ T3 )
     => ( ( ord_le211207098394363844omplex @ S3 @ T3 )
       => ( ! [X4: complex] :
              ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_int ) )
         => ( ( groups5690904116761175830ex_int @ G @ S3 )
            = ( groups5690904116761175830ex_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7391_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > complex] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_complex ) )
         => ( ( groups2073611262835488442omplex @ G @ S3 )
            = ( groups2073611262835488442omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7392_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > rat] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups2906978787729119204at_rat @ G @ S3 )
            = ( groups2906978787729119204at_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7393_sum_Omono__neutral__left,axiom,
    ! [T3: set_nat,S3: set_nat,G: nat > int] :
      ( ( finite_finite_nat @ T3 )
     => ( ( ord_less_eq_set_nat @ S3 @ T3 )
       => ( ! [X4: nat] :
              ( ( member_nat @ X4 @ ( minus_minus_set_nat @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_int ) )
         => ( ( groups3539618377306564664at_int @ G @ S3 )
            = ( groups3539618377306564664at_int @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7394_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > complex] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_complex ) )
         => ( ( groups3049146728041665814omplex @ G @ S3 )
            = ( groups3049146728041665814omplex @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7395_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > real] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_real ) )
         => ( ( groups8778361861064173332t_real @ G @ S3 )
            = ( groups8778361861064173332t_real @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7396_sum_Omono__neutral__left,axiom,
    ! [T3: set_int,S3: set_int,G: int > rat] :
      ( ( finite_finite_int @ T3 )
     => ( ( ord_less_eq_set_int @ S3 @ T3 )
       => ( ! [X4: int] :
              ( ( member_int @ X4 @ ( minus_minus_set_int @ T3 @ S3 ) )
             => ( ( G @ X4 )
                = zero_zero_rat ) )
         => ( ( groups3906332499630173760nt_rat @ G @ S3 )
            = ( groups3906332499630173760nt_rat @ G @ T3 ) ) ) ) ) ).

% sum.mono_neutral_left
thf(fact_7397_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) )
               => ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7398_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) )
               => ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7399_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) )
               => ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7400_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) )
               => ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7401_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) )
               => ( ( groups5058264527183730370ex_rat @ G @ A2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7402_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) )
               => ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7403_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) )
               => ( ( groups5693394587270226106ex_nat @ G @ A2 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7404_sum_Osame__carrierI,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups1932886352136224148al_int @ G @ C4 )
                  = ( groups1932886352136224148al_int @ H2 @ C4 ) )
               => ( ( groups1932886352136224148al_int @ G @ A2 )
                  = ( groups1932886352136224148al_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7405_sum_Osame__carrierI,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups5690904116761175830ex_int @ G @ C4 )
                  = ( groups5690904116761175830ex_int @ H2 @ C4 ) )
               => ( ( groups5690904116761175830ex_int @ G @ A2 )
                  = ( groups5690904116761175830ex_int @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7406_sum_Osame__carrierI,axiom,
    ! [C4: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A2 @ C4 )
       => ( ( ord_less_eq_set_nat @ B2 @ C4 )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_complex ) )
           => ( ! [B4: nat] :
                  ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_complex ) )
             => ( ( ( groups2073611262835488442omplex @ G @ C4 )
                  = ( groups2073611262835488442omplex @ H2 @ C4 ) )
               => ( ( groups2073611262835488442omplex @ G @ A2 )
                  = ( groups2073611262835488442omplex @ H2 @ B2 ) ) ) ) ) ) ) ) ).

% sum.same_carrierI
thf(fact_7407_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > complex,H2: real > complex] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_complex ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_complex ) )
             => ( ( ( groups5754745047067104278omplex @ G @ A2 )
                  = ( groups5754745047067104278omplex @ H2 @ B2 ) )
                = ( ( groups5754745047067104278omplex @ G @ C4 )
                  = ( groups5754745047067104278omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7408_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > real,H2: real > real] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups8097168146408367636l_real @ G @ A2 )
                  = ( groups8097168146408367636l_real @ H2 @ B2 ) )
                = ( ( groups8097168146408367636l_real @ G @ C4 )
                  = ( groups8097168146408367636l_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7409_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > real,H2: complex > real] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_real ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_real ) )
             => ( ( ( groups5808333547571424918x_real @ G @ A2 )
                  = ( groups5808333547571424918x_real @ H2 @ B2 ) )
                = ( ( groups5808333547571424918x_real @ G @ C4 )
                  = ( groups5808333547571424918x_real @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7410_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > rat,H2: real > rat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups1300246762558778688al_rat @ G @ A2 )
                  = ( groups1300246762558778688al_rat @ H2 @ B2 ) )
                = ( ( groups1300246762558778688al_rat @ G @ C4 )
                  = ( groups1300246762558778688al_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7411_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > rat,H2: complex > rat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_rat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_rat ) )
             => ( ( ( groups5058264527183730370ex_rat @ G @ A2 )
                  = ( groups5058264527183730370ex_rat @ H2 @ B2 ) )
                = ( ( groups5058264527183730370ex_rat @ G @ C4 )
                  = ( groups5058264527183730370ex_rat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7412_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > nat,H2: real > nat] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups1935376822645274424al_nat @ G @ A2 )
                  = ( groups1935376822645274424al_nat @ H2 @ B2 ) )
                = ( ( groups1935376822645274424al_nat @ G @ C4 )
                  = ( groups1935376822645274424al_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7413_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > nat,H2: complex > nat] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_nat ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_nat ) )
             => ( ( ( groups5693394587270226106ex_nat @ G @ A2 )
                  = ( groups5693394587270226106ex_nat @ H2 @ B2 ) )
                = ( ( groups5693394587270226106ex_nat @ G @ C4 )
                  = ( groups5693394587270226106ex_nat @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7414_sum_Osame__carrier,axiom,
    ! [C4: set_real,A2: set_real,B2: set_real,G: real > int,H2: real > int] :
      ( ( finite_finite_real @ C4 )
     => ( ( ord_less_eq_set_real @ A2 @ C4 )
       => ( ( ord_less_eq_set_real @ B2 @ C4 )
         => ( ! [A4: real] :
                ( ( member_real @ A4 @ ( minus_minus_set_real @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: real] :
                  ( ( member_real @ B4 @ ( minus_minus_set_real @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups1932886352136224148al_int @ G @ A2 )
                  = ( groups1932886352136224148al_int @ H2 @ B2 ) )
                = ( ( groups1932886352136224148al_int @ G @ C4 )
                  = ( groups1932886352136224148al_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7415_sum_Osame__carrier,axiom,
    ! [C4: set_complex,A2: set_complex,B2: set_complex,G: complex > int,H2: complex > int] :
      ( ( finite3207457112153483333omplex @ C4 )
     => ( ( ord_le211207098394363844omplex @ A2 @ C4 )
       => ( ( ord_le211207098394363844omplex @ B2 @ C4 )
         => ( ! [A4: complex] :
                ( ( member_complex @ A4 @ ( minus_811609699411566653omplex @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_int ) )
           => ( ! [B4: complex] :
                  ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_int ) )
             => ( ( ( groups5690904116761175830ex_int @ G @ A2 )
                  = ( groups5690904116761175830ex_int @ H2 @ B2 ) )
                = ( ( groups5690904116761175830ex_int @ G @ C4 )
                  = ( groups5690904116761175830ex_int @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7416_sum_Osame__carrier,axiom,
    ! [C4: set_nat,A2: set_nat,B2: set_nat,G: nat > complex,H2: nat > complex] :
      ( ( finite_finite_nat @ C4 )
     => ( ( ord_less_eq_set_nat @ A2 @ C4 )
       => ( ( ord_less_eq_set_nat @ B2 @ C4 )
         => ( ! [A4: nat] :
                ( ( member_nat @ A4 @ ( minus_minus_set_nat @ C4 @ A2 ) )
               => ( ( G @ A4 )
                  = zero_zero_complex ) )
           => ( ! [B4: nat] :
                  ( ( member_nat @ B4 @ ( minus_minus_set_nat @ C4 @ B2 ) )
                 => ( ( H2 @ B4 )
                    = zero_zero_complex ) )
             => ( ( ( groups2073611262835488442omplex @ G @ A2 )
                  = ( groups2073611262835488442omplex @ H2 @ B2 ) )
                = ( ( groups2073611262835488442omplex @ G @ C4 )
                  = ( groups2073611262835488442omplex @ H2 @ C4 ) ) ) ) ) ) ) ) ).

% sum.same_carrier
thf(fact_7417_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > real] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5808333547571424918x_real @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7418_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > rat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5058264527183730370ex_rat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7419_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > nat] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5693394587270226106ex_nat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7420_sum_Osubset__diff,axiom,
    ! [B2: set_complex,A2: set_complex,G: complex > int] :
      ( ( ord_le211207098394363844omplex @ B2 @ A2 )
     => ( ( finite3207457112153483333omplex @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ B2 ) ) @ ( groups5690904116761175830ex_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7421_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > rat] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups2906978787729119204at_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups2906978787729119204at_rat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7422_sum_Osubset__diff,axiom,
    ! [B2: set_nat,A2: set_nat,G: nat > int] :
      ( ( ord_less_eq_set_nat @ B2 @ A2 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( groups3539618377306564664at_int @ G @ A2 )
          = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( minus_minus_set_nat @ A2 @ B2 ) ) @ ( groups3539618377306564664at_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7423_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > real] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups8778361861064173332t_real @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7424_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > rat] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups3906332499630173760nt_rat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7425_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > nat] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4541462559716669496nt_nat @ G @ A2 )
          = ( plus_plus_nat @ ( groups4541462559716669496nt_nat @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4541462559716669496nt_nat @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7426_sum_Osubset__diff,axiom,
    ! [B2: set_int,A2: set_int,G: int > int] :
      ( ( ord_less_eq_set_int @ B2 @ A2 )
     => ( ( finite_finite_int @ A2 )
       => ( ( groups4538972089207619220nt_int @ G @ A2 )
          = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ ( minus_minus_set_int @ A2 @ B2 ) ) @ ( groups4538972089207619220nt_int @ G @ B2 ) ) ) ) ) ).

% sum.subset_diff
thf(fact_7427_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5808333547571424918x_real @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7428_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7429_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5690904116761175830ex_int @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7430_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7431_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups3539618377306564664at_int @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7432_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( ( groups8778361861064173332t_real @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
          = ( minus_minus_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ ( groups8778361861064173332t_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7433_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
          = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( groups3906332499630173760nt_rat @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7434_sum__diff,axiom,
    ! [A2: set_int,B2: set_int,F: int > int] :
      ( ( finite_finite_int @ A2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( ( groups4538972089207619220nt_int @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
          = ( minus_minus_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ ( groups4538972089207619220nt_int @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7435_sum__diff,axiom,
    ! [A2: set_complex,B2: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups7754918857620584856omplex @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_complex @ ( groups7754918857620584856omplex @ F @ A2 ) @ ( groups7754918857620584856omplex @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7436_sum__diff,axiom,
    ! [A2: set_nat,B2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups6591440286371151544t_real @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ ( groups6591440286371151544t_real @ F @ B2 ) ) ) ) ) ).

% sum_diff
thf(fact_7437_of__int__leD,axiom,
    ! [N2: int,X3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X3 )
     => ( ( N2 = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X3 ) ) ) ).

% of_int_leD
thf(fact_7438_of__int__leD,axiom,
    ! [N2: int,X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X3 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ).

% of_int_leD
thf(fact_7439_of__int__leD,axiom,
    ! [N2: int,X3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X3 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ) ).

% of_int_leD
thf(fact_7440_of__int__leD,axiom,
    ! [N2: int,X3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X3 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X3 ) ) ) ).

% of_int_leD
thf(fact_7441_sum__diff__nat,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ B2 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ F @ ( minus_811609699411566653omplex @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7442_sum__diff__nat,axiom,
    ! [B2: set_Pr1261947904930325089at_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( finite6177210948735845034at_nat @ B2 )
     => ( ( ord_le3146513528884898305at_nat @ B2 @ A2 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( groups977919841031483927at_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7443_sum__diff__nat,axiom,
    ! [B2: set_int,A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ B2 )
     => ( ( ord_less_eq_set_int @ B2 @ A2 )
       => ( ( groups4541462559716669496nt_nat @ F @ ( minus_minus_set_int @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ ( groups4541462559716669496nt_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7444_sum__diff__nat,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ B2 @ A2 )
       => ( ( groups3542108847815614940at_nat @ F @ ( minus_minus_set_nat @ A2 @ B2 ) )
          = ( minus_minus_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ ( groups3542108847815614940at_nat @ F @ B2 ) ) ) ) ) ).

% sum_diff_nat
thf(fact_7445_of__int__lessD,axiom,
    ! [N2: int,X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X3 )
     => ( ( N2 = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X3 ) ) ) ).

% of_int_lessD
thf(fact_7446_of__int__lessD,axiom,
    ! [N2: int,X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X3 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X3 ) ) ) ).

% of_int_lessD
thf(fact_7447_of__int__lessD,axiom,
    ! [N2: int,X3: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X3 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X3 ) ) ) ).

% of_int_lessD
thf(fact_7448_of__int__lessD,axiom,
    ! [N2: int,X3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X3 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X3 ) ) ) ).

% of_int_lessD
thf(fact_7449_sum__diff1__nat,axiom,
    ! [A: produc3843707927480180839at_nat,A2: set_Pr4329608150637261639at_nat,F: produc3843707927480180839at_nat > nat] :
      ( ( ( member8757157785044589968at_nat @ A @ A2 )
       => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
          = ( minus_minus_nat @ ( groups3860910324918113789at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member8757157785044589968at_nat @ A @ A2 )
       => ( ( groups3860910324918113789at_nat @ F @ ( minus_3314409938677909166at_nat @ A2 @ ( insert9069300056098147895at_nat @ A @ bot_bo228742789529271731at_nat ) ) )
          = ( groups3860910324918113789at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_7450_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_7451_sum__diff1__nat,axiom,
    ! [A: product_prod_nat_nat,A2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > nat] :
      ( ( ( member8440522571783428010at_nat @ A @ A2 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
          = ( minus_minus_nat @ ( groups977919841031483927at_nat @ F @ A2 ) @ ( F @ A ) ) ) )
      & ( ~ ( member8440522571783428010at_nat @ A @ A2 )
       => ( ( groups977919841031483927at_nat @ F @ ( minus_1356011639430497352at_nat @ A2 @ ( insert8211810215607154385at_nat @ A @ bot_bo2099793752762293965at_nat ) ) )
          = ( groups977919841031483927at_nat @ F @ A2 ) ) ) ) ).

% sum_diff1_nat
thf(fact_7452_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_7453_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_7454_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_7455_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_7456_sum__shift__lb__Suc0__0,axiom,
    ! [F: nat > rat,K: nat] :
      ( ( ( F @ zero_zero_nat )
        = zero_zero_rat )
     => ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ K ) )
        = ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ K ) ) ) ) ).

% sum_shift_lb_Suc0_0
thf(fact_7457_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_7458_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_7459_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_7460_sum_OatLeast0__atMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atLeast0_atMost_Suc
thf(fact_7461_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_7462_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_7463_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_7464_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_rat @ ( G @ M ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_7465_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_int @ ( G @ M ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_7466_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_nat @ ( G @ M ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_7467_sum_OatLeast__Suc__atMost,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( plus_plus_real @ ( G @ M ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) ) ) ) ) ).

% sum.atLeast_Suc_atMost
thf(fact_7468_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ ( suc @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_7469_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ ( suc @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_7470_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ ( suc @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_7471_sum_Onat__ivl__Suc_H,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ ( suc @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.nat_ivl_Suc'
thf(fact_7472_round__diff__minimal,axiom,
    ! [Z2: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z2 ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ ( ring_1_of_int_real @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_7473_round__diff__minimal,axiom,
    ! [Z2: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z2 ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z2 @ ( ring_1_of_int_rat @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_7474_zdvd__mult__cancel1,axiom,
    ! [M: int,N2: int] :
      ( ( M != zero_zero_int )
     => ( ( dvd_dvd_int @ ( times_times_int @ M @ N2 ) @ M )
        = ( ( abs_abs_int @ N2 )
          = one_one_int ) ) ) ).

% zdvd_mult_cancel1
thf(fact_7475_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_rat @ ( G @ M )
          @ ( groups2906978787729119204at_rat
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7476_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_int @ ( G @ M )
          @ ( groups3539618377306564664at_int
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7477_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_nat @ ( G @ M )
          @ ( groups3542108847815614940at_nat
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7478_sum_OSuc__reindex__ivl,axiom,
    ! [M: nat,N2: nat,G: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) )
        = ( plus_plus_real @ ( G @ M )
          @ ( groups6591440286371151544t_real
            @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_7479_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( minus_minus_rat @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7480_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( minus_minus_int @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7481_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( minus_minus_real @ ( F @ ( suc @ I2 ) ) @ ( F @ I2 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_7482_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7483_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ B4 ) ) )
         => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7484_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > rat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7485_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7486_sum__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [B4: nat] :
              ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ B4 ) ) )
         => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7487_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
         => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7488_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ B4 ) ) )
         => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7489_sum__mono2,axiom,
    ! [B2: set_real,A2: set_real,F: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ! [B4: real] :
              ( ( member_real @ B4 @ ( minus_minus_set_real @ B2 @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
         => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7490_sum__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ! [B4: complex] :
              ( ( member_complex @ B4 @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
         => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7491_sum__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ! [B4: nat] :
              ( ( member_nat @ B4 @ ( minus_minus_set_nat @ B2 @ A2 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ B4 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ).

% sum_mono2
thf(fact_7492_abs__le__square__iff,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ ( abs_abs_Code_integer @ Y3 ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7493_abs__le__square__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ ( abs_abs_real @ Y3 ) )
      = ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7494_abs__le__square__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ ( abs_abs_rat @ Y3 ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7495_abs__le__square__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ ( abs_abs_int @ Y3 ) )
      = ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_7496_sum_Oremove,axiom,
    ! [A2: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X3 @ A2 )
       => ( ( groups5808333547571424918x_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7497_sum_Oremove,axiom,
    ! [A2: set_complex,X3: complex,G: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X3 @ A2 )
       => ( ( groups5058264527183730370ex_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7498_sum_Oremove,axiom,
    ! [A2: set_complex,X3: complex,G: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X3 @ A2 )
       => ( ( groups5693394587270226106ex_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7499_sum_Oremove,axiom,
    ! [A2: set_complex,X3: complex,G: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ X3 @ A2 )
       => ( ( groups5690904116761175830ex_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X3 ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7500_sum_Oremove,axiom,
    ! [A2: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X3 @ A2 )
       => ( ( groups8097168146408367636l_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7501_sum_Oremove,axiom,
    ! [A2: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X3 @ A2 )
       => ( ( groups1300246762558778688al_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7502_sum_Oremove,axiom,
    ! [A2: set_real,X3: real,G: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X3 @ A2 )
       => ( ( groups1935376822645274424al_nat @ G @ A2 )
          = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7503_sum_Oremove,axiom,
    ! [A2: set_real,X3: real,G: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ X3 @ A2 )
       => ( ( groups1932886352136224148al_int @ G @ A2 )
          = ( plus_plus_int @ ( G @ X3 ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7504_sum_Oremove,axiom,
    ! [A2: set_int,X3: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X3 @ A2 )
       => ( ( groups8778361861064173332t_real @ G @ A2 )
          = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7505_sum_Oremove,axiom,
    ! [A2: set_int,X3: int,G: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ X3 @ A2 )
       => ( ( groups3906332499630173760nt_rat @ G @ A2 )
          = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.remove
thf(fact_7506_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > real,X3: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5808333547571424918x_real @ G @ ( insert_complex @ X3 @ A2 ) )
        = ( plus_plus_real @ ( G @ X3 ) @ ( groups5808333547571424918x_real @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7507_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > rat,X3: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5058264527183730370ex_rat @ G @ ( insert_complex @ X3 @ A2 ) )
        = ( plus_plus_rat @ ( G @ X3 ) @ ( groups5058264527183730370ex_rat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7508_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > nat,X3: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5693394587270226106ex_nat @ G @ ( insert_complex @ X3 @ A2 ) )
        = ( plus_plus_nat @ ( G @ X3 ) @ ( groups5693394587270226106ex_nat @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7509_sum_Oinsert__remove,axiom,
    ! [A2: set_complex,G: complex > int,X3: complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( groups5690904116761175830ex_int @ G @ ( insert_complex @ X3 @ A2 ) )
        = ( plus_plus_int @ ( G @ X3 ) @ ( groups5690904116761175830ex_int @ G @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ X3 @ bot_bot_set_complex ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7510_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > real,X3: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups8097168146408367636l_real @ G @ ( insert_real @ X3 @ A2 ) )
        = ( plus_plus_real @ ( G @ X3 ) @ ( groups8097168146408367636l_real @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7511_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > rat,X3: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1300246762558778688al_rat @ G @ ( insert_real @ X3 @ A2 ) )
        = ( plus_plus_rat @ ( G @ X3 ) @ ( groups1300246762558778688al_rat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7512_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > nat,X3: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1935376822645274424al_nat @ G @ ( insert_real @ X3 @ A2 ) )
        = ( plus_plus_nat @ ( G @ X3 ) @ ( groups1935376822645274424al_nat @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7513_sum_Oinsert__remove,axiom,
    ! [A2: set_real,G: real > int,X3: real] :
      ( ( finite_finite_real @ A2 )
     => ( ( groups1932886352136224148al_int @ G @ ( insert_real @ X3 @ A2 ) )
        = ( plus_plus_int @ ( G @ X3 ) @ ( groups1932886352136224148al_int @ G @ ( minus_minus_set_real @ A2 @ ( insert_real @ X3 @ bot_bot_set_real ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7514_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > real,X3: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups8778361861064173332t_real @ G @ ( insert_int @ X3 @ A2 ) )
        = ( plus_plus_real @ ( G @ X3 ) @ ( groups8778361861064173332t_real @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7515_sum_Oinsert__remove,axiom,
    ! [A2: set_int,G: int > rat,X3: int] :
      ( ( finite_finite_int @ A2 )
     => ( ( groups3906332499630173760nt_rat @ G @ ( insert_int @ X3 @ A2 ) )
        = ( plus_plus_rat @ ( G @ X3 ) @ ( groups3906332499630173760nt_rat @ G @ ( minus_minus_set_int @ A2 @ ( insert_int @ X3 @ bot_bot_set_int ) ) ) ) ) ) ).

% sum.insert_remove
thf(fact_7516_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_7517_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_7518_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_7519_sum__diff1,axiom,
    ! [A2: set_complex,A: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( minus_minus_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_complex @ A @ A2 )
         => ( ( groups5058264527183730370ex_rat @ F @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ A @ bot_bot_set_complex ) ) )
            = ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7520_sum__diff1,axiom,
    ! [A2: set_real,A: real,F: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ( ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( minus_minus_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_real @ A @ A2 )
         => ( ( groups1300246762558778688al_rat @ F @ ( minus_minus_set_real @ A2 @ ( insert_real @ A @ bot_bot_set_real ) ) )
            = ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7521_sum__diff1,axiom,
    ! [A2: set_int,A: int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( minus_minus_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_int @ A @ A2 )
         => ( ( groups3906332499630173760nt_rat @ F @ ( minus_minus_set_int @ A2 @ ( insert_int @ A @ bot_bot_set_int ) ) )
            = ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7522_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_7523_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_7524_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_7525_sum__diff1,axiom,
    ! [A2: set_nat,A: nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( member_nat @ A @ A2 )
         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
            = ( minus_minus_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( F @ A ) ) ) )
        & ( ~ ( member_nat @ A @ A2 )
         => ( ( groups2906978787729119204at_rat @ F @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ A @ bot_bot_set_nat ) ) )
            = ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).

% sum_diff1
thf(fact_7526_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_7527_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_int @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_7528_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_7529_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P4: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P4 ) ) )
        = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ ( plus_plus_nat @ N2 @ P4 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_7530_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > real,C: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5808333547571424918x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5808333547571424918x_real @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7531_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > rat,C: complex > rat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5058264527183730370ex_rat
              @ ^ [K3: complex] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5058264527183730370ex_rat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7532_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > nat,C: complex > nat] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5693394587270226106ex_nat
              @ ^ [K3: complex] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5693394587270226106ex_nat @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7533_sum_Odelta__remove,axiom,
    ! [S3: set_complex,A: complex,B: complex > int,C: complex > int] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups5690904116761175830ex_int
              @ ^ [K3: complex] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups5690904116761175830ex_int @ C @ ( minus_811609699411566653omplex @ S3 @ ( insert_complex @ A @ bot_bot_set_complex ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7534_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > real,C: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups8097168146408367636l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups8097168146408367636l_real @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7535_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > rat,C: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1300246762558778688al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1300246762558778688al_rat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7536_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > nat,C: real > nat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_nat @ ( B @ A ) @ ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1935376822645274424al_nat
              @ ^ [K3: real] : ( if_nat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1935376822645274424al_nat @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7537_sum_Odelta__remove,axiom,
    ! [S3: set_real,A: real,B: real > int,C: real > int] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_int @ ( B @ A ) @ ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1932886352136224148al_int
              @ ^ [K3: real] : ( if_int @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups1932886352136224148al_int @ C @ ( minus_minus_set_real @ S3 @ ( insert_real @ A @ bot_bot_set_real ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7538_sum_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > real,C: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_real @ ( B @ A ) @ ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups8778361861064173332t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups8778361861064173332t_real @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7539_sum_Odelta__remove,axiom,
    ! [S3: set_int,A: int,B: int > rat,C: int > rat] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( plus_plus_rat @ ( B @ A ) @ ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups3906332499630173760nt_rat
              @ ^ [K3: int] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ ( C @ K3 ) )
              @ S3 )
            = ( groups3906332499630173760nt_rat @ C @ ( minus_minus_set_int @ S3 @ ( insert_int @ A @ bot_bot_set_int ) ) ) ) ) ) ) ).

% sum.delta_remove
thf(fact_7540_sum__count__set,axiom,
    ! [Xs: list_complex,X7: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ X7 )
     => ( ( finite3207457112153483333omplex @ X7 )
       => ( ( groups5693394587270226106ex_nat @ ( count_list_complex @ Xs ) @ X7 )
          = ( size_s3451745648224563538omplex @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_7541_sum__count__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ X7 )
     => ( ( finite6177210948735845034at_nat @ X7 )
       => ( ( groups977919841031483927at_nat @ ( count_4203492906077236349at_nat @ Xs ) @ X7 )
          = ( size_s5460976970255530739at_nat @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_7542_sum__count__set,axiom,
    ! [Xs: list_VEBT_VEBT,X7: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ X7 )
     => ( ( finite5795047828879050333T_VEBT @ X7 )
       => ( ( groups771621172384141258BT_nat @ ( count_list_VEBT_VEBT @ Xs ) @ X7 )
          = ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_7543_sum__count__set,axiom,
    ! [Xs: list_o,X7: set_o] :
      ( ( ord_less_eq_set_o @ ( set_o2 @ Xs ) @ X7 )
     => ( ( finite_finite_o @ X7 )
       => ( ( groups8507830703676809646_o_nat @ ( count_list_o @ Xs ) @ X7 )
          = ( size_size_list_o @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_7544_sum__count__set,axiom,
    ! [Xs: list_int,X7: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ X7 )
     => ( ( finite_finite_int @ X7 )
       => ( ( groups4541462559716669496nt_nat @ ( count_list_int @ Xs ) @ X7 )
          = ( size_size_list_int @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_7545_sum__count__set,axiom,
    ! [Xs: list_nat,X7: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ X7 )
     => ( ( finite_finite_nat @ X7 )
       => ( ( groups3542108847815614940at_nat @ ( count_list_nat @ Xs ) @ X7 )
          = ( size_size_list_nat @ Xs ) ) ) ) ).

% sum_count_set
thf(fact_7546_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > real] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X4: real] :
                  ( ( member_real @ X4 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
             => ( ord_less_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ ( groups8097168146408367636l_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7547_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ B ) )
           => ( ! [X4: complex] :
                  ( ( member_complex @ X4 @ B2 )
                 => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
             => ( ord_less_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ ( groups5808333547571424918x_real @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7548_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > rat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X4: real] :
                  ( ( member_real @ X4 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
             => ( ord_less_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ ( groups1300246762558778688al_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7549_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X4: complex] :
                  ( ( member_complex @ X4 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
             => ( ord_less_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ ( groups5058264527183730370ex_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7550_sum__strict__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,B: nat,F: nat > rat] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
         => ( ( ord_less_rat @ zero_zero_rat @ ( F @ B ) )
           => ( ! [X4: nat] :
                  ( ( member_nat @ X4 @ B2 )
                 => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
             => ( ord_less_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ ( groups2906978787729119204at_rat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7551_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > nat] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X4: real] :
                  ( ( member_real @ X4 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
             => ( ord_less_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ ( groups1935376822645274424al_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7552_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ B ) )
           => ( ! [X4: complex] :
                  ( ( member_complex @ X4 @ B2 )
                 => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
             => ( ord_less_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7553_sum__strict__mono2,axiom,
    ! [B2: set_real,A2: set_real,B: real,F: real > int] :
      ( ( finite_finite_real @ B2 )
     => ( ( ord_less_eq_set_real @ A2 @ B2 )
       => ( ( member_real @ B @ ( minus_minus_set_real @ B2 @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X4: real] :
                  ( ( member_real @ X4 @ B2 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
             => ( ord_less_int @ ( groups1932886352136224148al_int @ F @ A2 ) @ ( groups1932886352136224148al_int @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7554_sum__strict__mono2,axiom,
    ! [B2: set_complex,A2: set_complex,B: complex,F: complex > int] :
      ( ( finite3207457112153483333omplex @ B2 )
     => ( ( ord_le211207098394363844omplex @ A2 @ B2 )
       => ( ( member_complex @ B @ ( minus_811609699411566653omplex @ B2 @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X4: complex] :
                  ( ( member_complex @ X4 @ B2 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
             => ( ord_less_int @ ( groups5690904116761175830ex_int @ F @ A2 ) @ ( groups5690904116761175830ex_int @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7555_sum__strict__mono2,axiom,
    ! [B2: set_nat,A2: set_nat,B: nat,F: nat > int] :
      ( ( finite_finite_nat @ B2 )
     => ( ( ord_less_eq_set_nat @ A2 @ B2 )
       => ( ( member_nat @ B @ ( minus_minus_set_nat @ B2 @ A2 ) )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ B ) )
           => ( ! [X4: nat] :
                  ( ( member_nat @ X4 @ B2 )
                 => ( ord_less_eq_int @ zero_zero_int @ ( F @ X4 ) ) )
             => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ A2 ) @ ( groups3539618377306564664at_int @ F @ B2 ) ) ) ) ) ) ) ).

% sum_strict_mono2
thf(fact_7556_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > real] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7557_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > real] :
      ( ( member_real @ I @ A2 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7558_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > real] :
      ( ( member_int @ I @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_real @ ( F @ I ) @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7559_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > rat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7560_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > rat] :
      ( ( member_real @ I @ A2 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7561_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > rat] :
      ( ( member_int @ I @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7562_member__le__sum,axiom,
    ! [I: nat,A2: set_nat,F: nat > rat] :
      ( ( member_nat @ I @ A2 )
     => ( ! [X4: nat] :
            ( ( member_nat @ X4 @ ( minus_minus_set_nat @ A2 @ ( insert_nat @ I @ bot_bot_set_nat ) ) )
           => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X4 ) ) )
       => ( ( finite_finite_nat @ A2 )
         => ( ord_less_eq_rat @ ( F @ I ) @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7563_member__le__sum,axiom,
    ! [I: complex,A2: set_complex,F: complex > nat] :
      ( ( member_complex @ I @ A2 )
     => ( ! [X4: complex] :
            ( ( member_complex @ X4 @ ( minus_811609699411566653omplex @ A2 @ ( insert_complex @ I @ bot_bot_set_complex ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
       => ( ( finite3207457112153483333omplex @ A2 )
         => ( ord_less_eq_nat @ ( F @ I ) @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7564_member__le__sum,axiom,
    ! [I: real,A2: set_real,F: real > nat] :
      ( ( member_real @ I @ A2 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( minus_minus_set_real @ A2 @ ( insert_real @ I @ bot_bot_set_real ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
       => ( ( finite_finite_real @ A2 )
         => ( ord_less_eq_nat @ ( F @ I ) @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7565_member__le__sum,axiom,
    ! [I: int,A2: set_int,F: int > nat] :
      ( ( member_int @ I @ A2 )
     => ( ! [X4: int] :
            ( ( member_int @ X4 @ ( minus_minus_set_int @ A2 @ ( insert_int @ I @ bot_bot_set_int ) ) )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X4 ) ) )
       => ( ( finite_finite_int @ A2 )
         => ( ord_less_eq_nat @ ( F @ I ) @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ) ) ).

% member_le_sum
thf(fact_7566_power2__le__iff__abs__le,axiom,
    ! [Y3: code_integer,X3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y3 )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7567_power2__le__iff__abs__le,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7568_power2__le__iff__abs__le,axiom,
    ! [Y3: rat,X3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y3 )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7569_power2__le__iff__abs__le,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
     => ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ Y3 ) ) ) ).

% power2_le_iff_abs_le
thf(fact_7570_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X3: code_integer] :
      ( ! [X4: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
         => ( P @ X4 @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X3 ) @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7571_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X3: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
         => ( P @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X3 ) @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7572_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X3: rat] :
      ( ! [X4: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
         => ( P @ X4 @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X3 ) @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7573_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X3: int] :
      ( ! [X4: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X4 )
         => ( P @ X4 @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X3 ) @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_7574_abs__square__le__1,axiom,
    ! [X3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_7575_abs__square__le__1,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_7576_abs__square__le__1,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_7577_abs__square__le__1,axiom,
    ! [X3: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_7578_abs__square__less__1,axiom,
    ! [X3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X3 ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_7579_abs__square__less__1,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_7580_abs__square__less__1,axiom,
    ! [X3: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X3 ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_7581_abs__square__less__1,axiom,
    ! [X3: int] :
      ( ( ord_less_int @ ( power_power_int @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X3 ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_7582_power__mono__even,axiom,
    ! [N2: nat,A: code_integer,B: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7583_power__mono__even,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7584_power__mono__even,axiom,
    ! [N2: nat,A: rat,B: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7585_power__mono__even,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% power_mono_even
thf(fact_7586_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ( ord_less_eq_nat @ M @ I3 )
            & ( ord_less_nat @ I3 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I3: nat] :
                ( ( ord_less_eq_nat @ M @ I3 )
                & ( ord_less_eq_nat @ I3 @ N2 )
                & ( ( F @ I3 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_7587_incr__lemma,axiom,
    ! [D: int,Z2: int,X3: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z2 @ ( plus_plus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z2 ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_7588_decr__lemma,axiom,
    ! [D: int,X3: int,Z2: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X3 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X3 @ Z2 ) ) @ one_one_int ) @ D ) ) @ Z2 ) ) ).

% decr_lemma
thf(fact_7589_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > complex] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_complex @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [K3: nat] : ( minus_minus_complex @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) ) ) ).

% sum_natinterval_diff
thf(fact_7590_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_rat @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) ) ) ).

% sum_natinterval_diff
thf(fact_7591_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_int @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_int ) ) ) ).

% sum_natinterval_diff
thf(fact_7592_sum__natinterval__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = ( minus_minus_real @ ( F @ M ) @ ( F @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( plus_plus_nat @ K3 @ one_one_nat ) ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) ) ) ).

% sum_natinterval_diff
thf(fact_7593_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [K3: nat] : ( minus_minus_rat @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_rat @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_7594_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [K3: nat] : ( minus_minus_int @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_int @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_7595_sum__telescope_H_H,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [K3: nat] : ( minus_minus_real @ ( F @ K3 ) @ ( F @ ( minus_minus_nat @ K3 @ one_one_nat ) ) )
          @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ N2 ) )
        = ( minus_minus_real @ ( F @ N2 ) @ ( F @ M ) ) ) ) ).

% sum_telescope''
thf(fact_7596_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_7597_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_7598_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_7599_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M2: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M2 @ N ) @ ( modulo_modulo_nat @ M2 @ N ) ) ) ) ).

% divmod_nat_def
thf(fact_7600_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X3: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_7601_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X3: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_7602_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X3: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X3 @ M ) @ ( power_power_int @ X3 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_7603_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X3: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_7604_sum_Oin__pairs,axiom,
    ! [G: nat > rat,M: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I2: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7605_sum_Oin__pairs,axiom,
    ! [G: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7606_sum_Oin__pairs,axiom,
    ! [G: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7607_sum_Oin__pairs,axiom,
    ! [G: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_7608_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I3 @ one_one_nat ) ) @ ( F @ I3 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I3: nat] :
              ( ( ord_less_eq_nat @ I3 @ N2 )
              & ( ( F @ I3 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_7609_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_7610_gauss__sum__nat,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( 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_7611_of__int__round__abs__le,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X3 ) ) @ X3 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7612_of__int__round__abs__le,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X3 ) ) @ X3 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7613_listrel1p__def,axiom,
    ( listrel1p_int
    = ( ^ [R5: int > int > $o,Xs2: list_int,Ys3: list_int] : ( member6698963635872716290st_int @ ( produc364263696895485585st_int @ Xs2 @ Ys3 ) @ ( listrel1_int @ ( collec213857154873943460nt_int @ ( produc4947309494688390418_int_o @ R5 ) ) ) ) ) ) ).

% listrel1p_def
thf(fact_7614_lemma__interval,axiom,
    ! [A: real,X3: real,B: real] :
      ( ( ord_less_real @ A @ X3 )
     => ( ( ord_less_real @ X3 @ B )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [Y6: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) @ D4 )
               => ( ( ord_less_eq_real @ A @ Y6 )
                  & ( ord_less_eq_real @ Y6 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_7615_sum__gp,axiom,
    ! [N2: nat,M: nat,X3: complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X3 = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X3 != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X3 @ M ) @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7616_sum__gp,axiom,
    ! [N2: nat,M: nat,X3: rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X3 = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X3 != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X3 @ M ) @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7617_sum__gp,axiom,
    ! [N2: nat,M: nat,X3: real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X3 = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X3 != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X3 @ M ) @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_7618_gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum_from_Suc_0
thf(fact_7619_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_7620_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_7621_lemma__interval__lt,axiom,
    ! [A: real,X3: real,B: real] :
      ( ( ord_less_real @ A @ X3 )
     => ( ( ord_less_real @ X3 @ B )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [Y6: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y6 ) ) @ D4 )
               => ( ( ord_less_real @ A @ Y6 )
                  & ( ord_less_real @ Y6 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_7622_arctan__double,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X3 ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_7623_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_7624_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_7625_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_7626_split__part,axiom,
    ! [P: $o,Q: int > int > $o] :
      ( ( produc4947309494688390418_int_o
        @ ^ [A3: int,B3: int] :
            ( P
            & ( Q @ A3 @ B3 ) ) )
      = ( ^ [Ab: product_prod_int_int] :
            ( P
            & ( produc4947309494688390418_int_o @ Q @ Ab ) ) ) ) ).

% split_part
thf(fact_7627_int__eq__iff__numeral,axiom,
    ! [M: nat,V: num] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( numeral_numeral_int @ V ) )
      = ( M
        = ( numeral_numeral_nat @ V ) ) ) ).

% int_eq_iff_numeral
thf(fact_7628_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% abs_of_nat
thf(fact_7629_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( semiri4939895301339042750nteger @ N2 ) ) ).

% abs_of_nat
thf(fact_7630_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_7631_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% abs_of_nat
thf(fact_7632_negative__eq__positive,axiom,
    ! [N2: nat,M: nat] :
      ( ( ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) )
        = ( semiri1314217659103216013at_int @ M ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% negative_eq_positive
thf(fact_7633_of__int__of__nat__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_rat @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% of_int_of_nat_eq
thf(fact_7634_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_7635_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_7636_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_7637_int__dvd__int__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% int_dvd_int_iff
thf(fact_7638_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7639_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri681578069525770553at_rat @ M )
        = zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7640_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7641_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7642_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_7643_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_7644_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_7645_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_7646_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_7647_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_7648_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_7649_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_7650_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_7651_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_7652_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_7653_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_7654_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_7655_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_7656_of__nat__less__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_iff
thf(fact_7657_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_7658_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_7659_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_7660_of__nat__le__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% of_nat_le_iff
thf(fact_7661_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_add
thf(fact_7662_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_add
thf(fact_7663_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_add
thf(fact_7664_of__nat__add,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( plus_plus_nat @ M @ N2 ) )
      = ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_add
thf(fact_7665_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7666_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7667_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7668_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7669_of__nat__mult,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( times_times_nat @ M @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mult
thf(fact_7670_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_7671_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_7672_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_7673_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_7674_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_7675_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7676_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_rat
        = ( semiri681578069525770553at_rat @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_7677_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_7678_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_7679_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_7680_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ N2 )
        = one_one_complex )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7681_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri681578069525770553at_rat @ N2 )
        = one_one_rat )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_7682_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_7683_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_7684_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_7685_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_7686_arctan__less__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( arctan @ X3 ) @ zero_zero_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_7687_zero__less__arctan__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% zero_less_arctan_iff
thf(fact_7688_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_7689_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_7690_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_7691_of__nat__sum,axiom,
    ! [F: complex > nat,A2: set_complex] :
      ( ( semiri8010041392384452111omplex @ ( groups5693394587270226106ex_nat @ F @ A2 ) )
      = ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( semiri8010041392384452111omplex @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7692_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7693_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7694_of__nat__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_sum
thf(fact_7695_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7696_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7697_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7698_of__nat__le__0__iff,axiom,
    ! [M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int )
      = ( M = zero_zero_nat ) ) ).

% of_nat_le_0_iff
thf(fact_7699_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_7700_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7701_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_7702_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_7703_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_7704_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_7705_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_7706_numeral__le__real__of__nat__iff,axiom,
    ! [N2: num,M: nat] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( semiri5074537144036343181t_real @ M ) )
      = ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ M ) ) ).

% numeral_le_real_of_nat_iff
thf(fact_7707_of__nat__0__less__iff,axiom,
    ! [N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% of_nat_0_less_iff
thf(fact_7708_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_7709_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_7710_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_7711_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7712_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7713_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7714_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_7715_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7716_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7717_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7718_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_7719_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7720_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7721_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7722_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X3: nat,B: nat,W2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ B @ W2 ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_7723_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7724_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7725_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7726_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W2: nat,X3: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W2 ) @ X3 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_7727_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X3 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7728_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X3 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7729_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X3 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7730_of__nat__zero__less__power__iff,axiom,
    ! [X3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X3 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_7731_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X3: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7732_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X3: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7733_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X3: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7734_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X3: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_7735_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7736_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7737_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7738_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_7739_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7740_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X3 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7741_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7742_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X3: nat,I: num,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) )
      = ( ord_less_eq_nat @ X3 @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_7743_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X3: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I ) @ N2 ) @ ( semiri5074537144036343181t_real @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7744_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X3: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I ) @ N2 ) @ ( semiri681578069525770553at_rat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7745_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X3: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7746_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I: num,N2: nat,X3: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I ) @ N2 ) @ ( semiri1314217659103216013at_int @ X3 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I ) @ N2 ) @ X3 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_7747_prod_Odisc__eq__case,axiom,
    ! [Prod: product_prod_int_int] :
      ( produc4947309494688390418_int_o
      @ ^ [Uu3: int,Uv3: int] : $true
      @ Prod ) ).

% prod.disc_eq_case
thf(fact_7748_int__sum,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups3542108847815614940at_nat @ F @ A2 ) )
      = ( groups3539618377306564664at_int
        @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% int_sum
thf(fact_7749_real__arch__simple,axiom,
    ! [X3: real] :
    ? [N3: nat] : ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% real_arch_simple
thf(fact_7750_real__arch__simple,axiom,
    ! [X3: rat] :
    ? [N3: nat] : ( ord_less_eq_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% real_arch_simple
thf(fact_7751_reals__Archimedean2,axiom,
    ! [X3: rat] :
    ? [N3: nat] : ( ord_less_rat @ X3 @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% reals_Archimedean2
thf(fact_7752_reals__Archimedean2,axiom,
    ! [X3: real] :
    ? [N3: nat] : ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% reals_Archimedean2
thf(fact_7753_mult__of__nat__commute,axiom,
    ! [X3: nat,Y3: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X3 ) @ Y3 )
      = ( times_times_complex @ Y3 @ ( semiri8010041392384452111omplex @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7754_mult__of__nat__commute,axiom,
    ! [X3: nat,Y3: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ X3 ) @ Y3 )
      = ( times_times_rat @ Y3 @ ( semiri681578069525770553at_rat @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7755_mult__of__nat__commute,axiom,
    ! [X3: nat,Y3: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X3 ) @ Y3 )
      = ( times_times_int @ Y3 @ ( semiri1314217659103216013at_int @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7756_mult__of__nat__commute,axiom,
    ! [X3: nat,Y3: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X3 ) @ Y3 )
      = ( times_times_real @ Y3 @ ( semiri5074537144036343181t_real @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7757_mult__of__nat__commute,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X3 ) @ Y3 )
      = ( times_times_nat @ Y3 @ ( semiri1316708129612266289at_nat @ X3 ) ) ) ).

% mult_of_nat_commute
thf(fact_7758_int__cases2,axiom,
    ! [Z2: int] :
      ( ! [N3: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ).

% int_cases2
thf(fact_7759_arctan__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% arctan_less_iff
thf(fact_7760_arctan__monotone,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) ) ) ).

% arctan_monotone
thf(fact_7761_int__diff__cases,axiom,
    ! [Z2: int] :
      ~ ! [M3: nat,N3: nat] :
          ( Z2
         != ( minus_minus_int @ ( semiri1314217659103216013at_int @ M3 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_diff_cases
thf(fact_7762_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X3: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X3 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X3 ) ) ).

% of_nat_less_of_int_iff
thf(fact_7763_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X3: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X3 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X3 ) ) ).

% of_nat_less_of_int_iff
thf(fact_7764_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X3: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X3 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X3 ) ) ).

% of_nat_less_of_int_iff
thf(fact_7765_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_7766_of__nat__0__le__iff,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ).

% of_nat_0_le_iff
thf(fact_7767_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_7768_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_7769_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_7770_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_7771_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_7772_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_7773_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_7774_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_7775_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_7776_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_7777_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_7778_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_7779_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_7780_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_7781_of__nat__less__imp__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
     => ( ord_less_nat @ M @ N2 ) ) ).

% of_nat_less_imp_less
thf(fact_7782_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_7783_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_7784_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_7785_less__imp__of__nat__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% less_imp_of_nat_less
thf(fact_7786_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_7787_of__nat__mono,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_eq_nat @ I @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_7788_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_7789_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_7790_int__ops_I1_J,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% int_ops(1)
thf(fact_7791_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_7792_int__of__nat__induct,axiom,
    ! [P: int > $o,Z2: int] :
      ( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
     => ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
       => ( P @ Z2 ) ) ) ).

% int_of_nat_induct
thf(fact_7793_int__cases,axiom,
    ! [Z2: int] :
      ( ! [N3: nat] :
          ( Z2
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( Z2
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% int_cases
thf(fact_7794_zle__int,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% zle_int
thf(fact_7795_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_7796_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_7797_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_7798_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z2: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z2 ) ) ).

% zadd_int_left
thf(fact_7799_int__ops_I5_J,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% int_ops(5)
thf(fact_7800_int__plus,axiom,
    ! [N2: nat,M: nat] :
      ( ( semiri1314217659103216013at_int @ ( plus_plus_nat @ N2 @ M ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% int_plus
thf(fact_7801_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_7802_not__int__zless__negative,axiom,
    ! [N2: nat,M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% not_int_zless_negative
thf(fact_7803_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_7804_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_7805_ex__less__of__nat__mult,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ? [N3: nat] : ( ord_less_rat @ Y3 @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X3 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_7806_ex__less__of__nat__mult,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ? [N3: nat] : ( ord_less_real @ Y3 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_7807_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri681578069525770553at_rat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_7808_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_7809_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri5074537144036343181t_real @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_7810_of__nat__diff,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1316708129612266289at_nat @ ( minus_minus_nat @ M @ N2 ) )
        = ( minus_minus_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ) ).

% of_nat_diff
thf(fact_7811_reals__Archimedean3,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ! [Y6: real] :
        ? [N3: nat] : ( ord_less_real @ Y6 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X3 ) ) ) ).

% reals_Archimedean3
thf(fact_7812_int__cases4,axiom,
    ! [M: int] :
      ( ! [N3: nat] :
          ( M
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( M
             != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N3 ) ) ) ) ) ).

% int_cases4
thf(fact_7813_int__zle__neg,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ M ) ) )
      = ( ( N2 = zero_zero_nat )
        & ( M = zero_zero_nat ) ) ) ).

% int_zle_neg
thf(fact_7814_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_7815_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_7816_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_7817_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_7818_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_7819_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_7820_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z6: complex] :
                ( ( power_power_complex @ Z6 @ N2 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_7821_mod__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( modulo364778990260209775nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( modulo364778990260209775nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) @ ( modulo364778990260209775nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7822_mod__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( modulo_modulo_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( modulo_modulo_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) @ ( modulo_modulo_int @ A @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7823_mod__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( modulo_modulo_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( modulo_modulo_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) @ ( modulo_modulo_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) ) ) ).

% mod_mult2_eq'
thf(fact_7824_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_7825_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_7826_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_7827_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat,M2: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M2 ) ) ) ) ).

% nat_less_real_le
thf(fact_7828_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N: nat,M2: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M2 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_7829_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_7830_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_7831_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_7832_negD,axiom,
    ! [X3: int] :
      ( ( ord_less_int @ X3 @ zero_zero_int )
     => ? [N3: nat] :
          ( X3
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% negD
thf(fact_7833_int__ops_I6_J,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = zero_zero_int ) )
      & ( ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) )
       => ( ( semiri1314217659103216013at_int @ ( minus_minus_nat @ A @ B ) )
          = ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ) ) ).

% int_ops(6)
thf(fact_7834_nat__approx__posE,axiom,
    ! [E2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ E2 )
     => ~ ! [N3: nat] :
            ~ ( ord_less_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ E2 ) ) ).

% nat_approx_posE
thf(fact_7835_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_7836_of__nat__less__two__power,axiom,
    ! [N2: nat] : ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ).

% of_nat_less_two_power
thf(fact_7837_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_7838_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_7839_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_7840_inverse__of__nat__le,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( N2 != zero_zero_nat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) @ ( divide_divide_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ) ).

% inverse_of_nat_le
thf(fact_7841_real__archimedian__rdiv__eq__0,axiom,
    ! [X3: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( 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 ) @ X3 ) @ C ) )
         => ( X3 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_7842_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_7843_zdiff__int__split,axiom,
    ! [P: int > $o,X3: nat,Y3: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X3 @ Y3 ) ) )
      = ( ( ( ord_less_eq_nat @ Y3 @ X3 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X3 ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) )
        & ( ( ord_less_nat @ X3 @ Y3 )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_7844_ln__realpow,axiom,
    ! [X3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ln_ln_real @ ( power_power_real @ X3 @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X3 ) ) ) ) ).

% ln_realpow
thf(fact_7845_arctan__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X3 ) @ ( arctan @ Y3 ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X3 @ Y3 ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X3 @ Y3 ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_7846_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_7847_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_7848_double__gauss__sum,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum
thf(fact_7849_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_7850_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_7851_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_7852_double__arith__series,axiom,
    ! [A: complex,D: complex,N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I2: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I2 ) @ 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_7853_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
        @ ( groups7108830773950497114d_enat
          @ ^ [I2: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I2 ) @ 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_7854_double__arith__series,axiom,
    ! [A: rat,D: rat,N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I2 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ A ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_7855_double__arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ 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_7856_double__arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ 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_7857_double__arith__series,axiom,
    ! [A: real,D: real,N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I2 ) @ 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_7858_gauss__sum,axiom,
    ! [N2: nat] :
      ( ( groups7501900531339628137nteger @ semiri4939895301339042750nteger @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% gauss_sum
thf(fact_7859_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_7860_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_7861_arith__series,axiom,
    ! [A: code_integer,D: code_integer,N2: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I2: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I2 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide6298287555418463151nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( semiri4939895301339042750nteger @ N2 ) @ one_one_Code_integer ) @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ N2 ) @ D ) ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_7862_arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I2: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I2 ) @ 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_7863_arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ 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_7864_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_7865_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_7866_double__gauss__sum__from__Suc__0,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( groups2906978787729119204at_rat @ semiri681578069525770553at_rat @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) ) ).

% double_gauss_sum_from_Suc_0
thf(fact_7867_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_7868_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_7869_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_7870_sum__gp__offset,axiom,
    ! [X3: complex,M: nat,N2: nat] :
      ( ( ( X3 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
      & ( ( X3 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X3 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X3 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7871_sum__gp__offset,axiom,
    ! [X3: rat,M: nat,N2: nat] :
      ( ( ( X3 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
      & ( ( X3 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X3 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X3 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7872_sum__gp__offset,axiom,
    ! [X3: real,M: nat,N2: nat] :
      ( ( ( X3 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
      & ( ( X3 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X3 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X3 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X3 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_7873_of__nat__code__if,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] :
          ( if_complex @ ( N = zero_zero_nat ) @ zero_zero_complex
          @ ( produc1917071388513777916omplex
            @ ^ [M2: nat,Q5: nat] : ( if_complex @ ( Q5 = zero_zero_nat ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M2 ) ) @ one_one_complex ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7874_of__nat__code__if,axiom,
    ( semiri4216267220026989637d_enat
    = ( ^ [N: nat] :
          ( if_Extended_enat @ ( N = zero_zero_nat ) @ zero_z5237406670263579293d_enat
          @ ( produc2676513652042109336d_enat
            @ ^ [M2: nat,Q5: nat] : ( if_Extended_enat @ ( Q5 = zero_zero_nat ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M2 ) ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ ( semiri4216267220026989637d_enat @ M2 ) ) @ one_on7984719198319812577d_enat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7875_of__nat__code__if,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( if_rat @ ( N = zero_zero_nat ) @ zero_zero_rat
          @ ( produc6207742614233964070at_rat
            @ ^ [M2: nat,Q5: nat] : ( if_rat @ ( Q5 = zero_zero_nat ) @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M2 ) ) @ one_one_rat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7876_of__nat__code__if,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( if_int @ ( N = zero_zero_nat ) @ zero_zero_int
          @ ( produc6840382203811409530at_int
            @ ^ [M2: nat,Q5: nat] : ( if_int @ ( Q5 = zero_zero_nat ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ M2 ) ) @ one_one_int ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7877_of__nat__code__if,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( produc1703576794950452218t_real
            @ ^ [M2: nat,Q5: nat] : ( if_real @ ( Q5 = zero_zero_nat ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M2 ) ) @ one_one_real ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7878_of__nat__code__if,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( produc6842872674320459806at_nat
            @ ^ [M2: nat,Q5: nat] : ( if_nat @ ( Q5 = zero_zero_nat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ M2 ) ) @ one_one_nat ) )
            @ ( divmod_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% of_nat_code_if
thf(fact_7879_monoseq__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ 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 @ X3 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_7880_lemma__termdiff3,axiom,
    ! [H2: real,Z2: real,K5: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z2 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z2 @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z2 @ ( 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_7881_lemma__termdiff3,axiom,
    ! [H2: complex,Z2: complex,K5: real,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z2 ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z2 @ H2 ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z2 @ ( 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_7882_ln__series,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X3 )
          = ( 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 @ X3 @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).

% ln_series
thf(fact_7883_lemma__termdiff2,axiom,
    ! [H2: complex,Z2: complex,N2: nat] :
      ( ( H2 != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ N2 ) @ ( power_power_complex @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z2 @ ( 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 @ Z2 @ H2 ) @ Q5 ) @ ( power_power_complex @ Z2 @ ( 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_7884_lemma__termdiff2,axiom,
    ! [H2: rat,Z2: rat,N2: nat] :
      ( ( H2 != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ N2 ) @ ( power_power_rat @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H2
          @ ( groups2906978787729119204at_rat
            @ ^ [P6: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ Q5 ) @ ( power_power_rat @ Z2 @ ( 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_7885_lemma__termdiff2,axiom,
    ! [H2: real,Z2: real,N2: nat] :
      ( ( H2 != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ N2 ) @ ( power_power_real @ Z2 @ N2 ) ) @ H2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z2 @ ( 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 @ Z2 @ H2 ) @ Q5 ) @ ( power_power_real @ Z2 @ ( 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_7886_arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( arctan @ X3 )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_7887_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_7888_lessThan__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I @ K ) ) ).

% lessThan_iff
thf(fact_7889_lessThan__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I @ K ) ) ).

% lessThan_iff
thf(fact_7890_lessThan__iff,axiom,
    ! [I: nat,K: nat] :
      ( ( member_nat @ I @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I @ K ) ) ).

% lessThan_iff
thf(fact_7891_lessThan__iff,axiom,
    ! [I: int,K: int] :
      ( ( member_int @ I @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I @ K ) ) ).

% lessThan_iff
thf(fact_7892_lessThan__iff,axiom,
    ! [I: real,K: real] :
      ( ( member_real @ I @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I @ K ) ) ).

% lessThan_iff
thf(fact_7893_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_7894_lessThan__subset__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X3 ) @ ( set_ord_lessThan_rat @ Y3 ) )
      = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7895_lessThan__subset__iff,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X3 ) @ ( set_ord_lessThan_num @ Y3 ) )
      = ( ord_less_eq_num @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7896_lessThan__subset__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X3 ) @ ( set_ord_lessThan_nat @ Y3 ) )
      = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7897_lessThan__subset__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X3 ) @ ( set_ord_lessThan_int @ Y3 ) )
      = ( ord_less_eq_int @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7898_lessThan__subset__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X3 ) @ ( set_or5984915006950818249n_real @ Y3 ) )
      = ( ord_less_eq_real @ X3 @ Y3 ) ) ).

% lessThan_subset_iff
thf(fact_7899_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_7900_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_7901_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_7902_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_7903_sum_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% sum.lessThan_Suc
thf(fact_7904_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_7905_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_7906_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_7907_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_7908_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_7909_int__int__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% int_int_eq
thf(fact_7910_lessThan__non__empty,axiom,
    ! [X3: int] :
      ( ( set_ord_lessThan_int @ X3 )
     != bot_bot_set_int ) ).

% lessThan_non_empty
thf(fact_7911_lessThan__non__empty,axiom,
    ! [X3: real] :
      ( ( set_or5984915006950818249n_real @ X3 )
     != bot_bot_set_real ) ).

% lessThan_non_empty
thf(fact_7912_infinite__Iio,axiom,
    ! [A: int] :
      ~ ( finite_finite_int @ ( set_ord_lessThan_int @ A ) ) ).

% infinite_Iio
thf(fact_7913_infinite__Iio,axiom,
    ! [A: real] :
      ~ ( finite_finite_real @ ( set_or5984915006950818249n_real @ A ) ) ).

% infinite_Iio
thf(fact_7914_lessThan__def,axiom,
    ( set_or890127255671739683et_nat
    = ( ^ [U2: set_nat] :
          ( collect_set_nat
          @ ^ [X: set_nat] : ( ord_less_set_nat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7915_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7916_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7917_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7918_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7919_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_7920_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_7921_lessThan__strict__subset__iff,axiom,
    ! [M: rat,N2: rat] :
      ( ( ord_less_set_rat @ ( set_ord_lessThan_rat @ M ) @ ( set_ord_lessThan_rat @ N2 ) )
      = ( ord_less_rat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7922_lessThan__strict__subset__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_set_num @ ( set_ord_lessThan_num @ M ) @ ( set_ord_lessThan_num @ N2 ) )
      = ( ord_less_num @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7923_lessThan__strict__subset__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_set_nat @ ( set_ord_lessThan_nat @ M ) @ ( set_ord_lessThan_nat @ N2 ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7924_lessThan__strict__subset__iff,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_set_int @ ( set_ord_lessThan_int @ M ) @ ( set_ord_lessThan_int @ N2 ) )
      = ( ord_less_int @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7925_lessThan__strict__subset__iff,axiom,
    ! [M: real,N2: real] :
      ( ( ord_less_set_real @ ( set_or5984915006950818249n_real @ M ) @ ( set_or5984915006950818249n_real @ N2 ) )
      = ( ord_less_real @ M @ N2 ) ) ).

% lessThan_strict_subset_iff
thf(fact_7926_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_7927_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_7928_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S7: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_7929_finite__nat__bounded,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [K2: nat] : ( ord_less_eq_set_nat @ S3 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).

% finite_nat_bounded
thf(fact_7930_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_7931_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_7932_sum__diff__distrib,axiom,
    ! [Q: int > nat,P: int > nat,N2: int] :
      ( ! [X4: int] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N2 ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N2 ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_int @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_7933_sum__diff__distrib,axiom,
    ! [Q: real > nat,P: real > nat,N2: real] :
      ( ! [X4: real] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
     => ( ( minus_minus_nat @ ( groups1935376822645274424al_nat @ P @ ( set_or5984915006950818249n_real @ N2 ) ) @ ( groups1935376822645274424al_nat @ Q @ ( set_or5984915006950818249n_real @ N2 ) ) )
        = ( groups1935376822645274424al_nat
          @ ^ [X: real] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_or5984915006950818249n_real @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_7934_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N2: nat] :
      ( ! [X4: nat] : ( ord_less_eq_nat @ ( Q @ X4 ) @ ( P @ X4 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_7935_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7936_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
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7937_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
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7938_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
          @ ^ [I2: nat] : ( G @ ( suc @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_7939_sum__lessThan__telescope,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N: nat] : ( minus_minus_rat @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7940_sum__lessThan__telescope,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7941_sum__lessThan__telescope,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ ( suc @ N ) ) @ ( F @ N ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ M ) @ ( F @ zero_zero_nat ) ) ) ).

% sum_lessThan_telescope
thf(fact_7942_sum__lessThan__telescope_H,axiom,
    ! [F: nat > rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [N: nat] : ( minus_minus_rat @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7943_sum__lessThan__telescope_H,axiom,
    ! [F: nat > int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [N: nat] : ( minus_minus_int @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7944_sum__lessThan__telescope_H,axiom,
    ! [F: nat > real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( minus_minus_real @ ( F @ N ) @ ( F @ ( suc @ N ) ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ M ) ) ) ).

% sum_lessThan_telescope'
thf(fact_7945_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_7946_sum_OatLeast1__atMost__eq,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ ( suc @ zero_zero_nat ) @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( G @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.atLeast1_atMost_eq
thf(fact_7947_lemma__termdiff1,axiom,
    ! [Z2: complex,H2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P6: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_complex @ Z2 @ P6 ) ) @ ( power_power_complex @ Z2 @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ Z2 @ P6 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_complex @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7948_lemma__termdiff1,axiom,
    ! [Z2: rat,H2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P6: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_rat @ Z2 @ P6 ) ) @ ( power_power_rat @ Z2 @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ Z2 @ P6 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_rat @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7949_lemma__termdiff1,axiom,
    ! [Z2: int,H2: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P6: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_int @ Z2 @ P6 ) ) @ ( power_power_int @ Z2 @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ Z2 @ P6 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_int @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7950_lemma__termdiff1,axiom,
    ! [Z2: real,H2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P6: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_real @ Z2 @ P6 ) ) @ ( power_power_real @ Z2 @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ Z2 @ P6 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z2 @ H2 ) @ ( minus_minus_nat @ M @ P6 ) ) @ ( power_power_real @ Z2 @ ( minus_minus_nat @ M @ P6 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_7951_diff__power__eq__sum,axiom,
    ! [X3: complex,N2: nat,Y3: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X3 @ ( suc @ N2 ) ) @ ( power_power_complex @ Y3 @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
        @ ( groups2073611262835488442omplex
          @ ^ [P6: nat] : ( times_times_complex @ ( power_power_complex @ X3 @ P6 ) @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7952_diff__power__eq__sum,axiom,
    ! [X3: rat,N2: nat,Y3: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X3 @ ( suc @ N2 ) ) @ ( power_power_rat @ Y3 @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
        @ ( groups2906978787729119204at_rat
          @ ^ [P6: nat] : ( times_times_rat @ ( power_power_rat @ X3 @ P6 ) @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7953_diff__power__eq__sum,axiom,
    ! [X3: int,N2: nat,Y3: int] :
      ( ( minus_minus_int @ ( power_power_int @ X3 @ ( suc @ N2 ) ) @ ( power_power_int @ Y3 @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
        @ ( groups3539618377306564664at_int
          @ ^ [P6: nat] : ( times_times_int @ ( power_power_int @ X3 @ P6 ) @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7954_diff__power__eq__sum,axiom,
    ! [X3: real,N2: nat,Y3: real] :
      ( ( minus_minus_real @ ( power_power_real @ X3 @ ( suc @ N2 ) ) @ ( power_power_real @ Y3 @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
        @ ( groups6591440286371151544t_real
          @ ^ [P6: nat] : ( times_times_real @ ( power_power_real @ X3 @ P6 ) @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N2 @ P6 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_7955_power__diff__sumr2,axiom,
    ! [X3: complex,N2: nat,Y3: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X3 @ N2 ) @ ( power_power_complex @ Y3 @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ X3 @ Y3 )
        @ ( groups2073611262835488442omplex
          @ ^ [I2: nat] : ( times_times_complex @ ( power_power_complex @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_complex @ X3 @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7956_power__diff__sumr2,axiom,
    ! [X3: rat,N2: nat,Y3: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X3 @ N2 ) @ ( power_power_rat @ Y3 @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ X3 @ Y3 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( times_times_rat @ ( power_power_rat @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_rat @ X3 @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7957_power__diff__sumr2,axiom,
    ! [X3: int,N2: nat,Y3: int] :
      ( ( minus_minus_int @ ( power_power_int @ X3 @ N2 ) @ ( power_power_int @ Y3 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X3 @ Y3 )
        @ ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( times_times_int @ ( power_power_int @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_int @ X3 @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7958_power__diff__sumr2,axiom,
    ! [X3: real,N2: nat,Y3: real] :
      ( ( minus_minus_real @ ( power_power_real @ X3 @ N2 ) @ ( power_power_real @ Y3 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ X3 @ Y3 )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ Y3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) ) @ ( power_power_real @ X3 @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_7959_real__sum__nat__ivl__bounded2,axiom,
    ! [N2: nat,F: nat > rat,K5: rat,K: nat] :
      ( ! [P7: nat] :
          ( ( ord_less_nat @ P7 @ N2 )
         => ( ord_less_eq_rat @ ( F @ P7 ) @ K5 ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ K5 )
       => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ K5 ) ) ) ) ).

% real_sum_nat_ivl_bounded2
thf(fact_7960_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_7961_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_7962_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_7963_one__diff__power__eq_H,axiom,
    ! [X3: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X3 @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X3 )
        @ ( groups2073611262835488442omplex
          @ ^ [I2: nat] : ( power_power_complex @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7964_one__diff__power__eq_H,axiom,
    ! [X3: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X3 @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X3 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I2: nat] : ( power_power_rat @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7965_one__diff__power__eq_H,axiom,
    ! [X3: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X3 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X3 )
        @ ( groups3539618377306564664at_int
          @ ^ [I2: nat] : ( power_power_int @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7966_one__diff__power__eq_H,axiom,
    ! [X3: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X3 )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( power_power_real @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_7967_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I2: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ ( F @ I2 ) @ ( G @ I2 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I2: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I2 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_split_even_odd
thf(fact_7968_norm__le__zero__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X3 ) @ zero_zero_real )
      = ( X3 = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_7969_norm__le__zero__iff,axiom,
    ! [X3: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real )
      = ( X3 = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_7970_zero__less__norm__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X3 ) )
      = ( X3 != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_7971_zero__less__norm__iff,axiom,
    ! [X3: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X3 ) )
      = ( X3 != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_7972_suminf__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( ( suminf_real @ ( power_power_real @ C ) )
        = ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7973_suminf__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( ( suminf_complex @ ( power_power_complex @ C ) )
        = ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% suminf_geometric
thf(fact_7974_norm__zero,axiom,
    ( ( real_V7735802525324610683m_real @ zero_zero_real )
    = zero_zero_real ) ).

% norm_zero
thf(fact_7975_norm__zero,axiom,
    ( ( real_V1022390504157884413omplex @ zero_zero_complex )
    = zero_zero_real ) ).

% norm_zero
thf(fact_7976_norm__eq__zero,axiom,
    ! [X3: real] :
      ( ( ( real_V7735802525324610683m_real @ X3 )
        = zero_zero_real )
      = ( X3 = zero_zero_real ) ) ).

% norm_eq_zero
thf(fact_7977_norm__eq__zero,axiom,
    ! [X3: complex] :
      ( ( ( real_V1022390504157884413omplex @ X3 )
        = zero_zero_real )
      = ( X3 = zero_zero_complex ) ) ).

% norm_eq_zero
thf(fact_7978_suminf__zero,axiom,
    ( ( suminf_complex
      @ ^ [N: nat] : zero_zero_complex )
    = zero_zero_complex ) ).

% suminf_zero
thf(fact_7979_suminf__zero,axiom,
    ( ( suminf_real
      @ ^ [N: nat] : zero_zero_real )
    = zero_zero_real ) ).

% suminf_zero
thf(fact_7980_suminf__zero,axiom,
    ( ( suminf_nat
      @ ^ [N: nat] : zero_zero_nat )
    = zero_zero_nat ) ).

% suminf_zero
thf(fact_7981_suminf__zero,axiom,
    ( ( suminf_int
      @ ^ [N: nat] : zero_zero_int )
    = zero_zero_int ) ).

% suminf_zero
thf(fact_7982_norm__not__less__zero,axiom,
    ! [X3: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_7983_norm__uminus__minus,axiom,
    ! [X3: real,Y3: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X3 ) @ Y3 ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).

% norm_uminus_minus
thf(fact_7984_norm__uminus__minus,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X3 ) @ Y3 ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).

% norm_uminus_minus
thf(fact_7985_nonzero__norm__divide,axiom,
    ! [B: real,A: real] :
      ( ( B != zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
        = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_7986_nonzero__norm__divide,axiom,
    ! [B: complex,A: complex] :
      ( ( B != zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
        = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ) ).

% nonzero_norm_divide
thf(fact_7987_power__eq__imp__eq__norm,axiom,
    ! [W2: real,N2: nat,Z2: real] :
      ( ( ( power_power_real @ W2 @ N2 )
        = ( power_power_real @ Z2 @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V7735802525324610683m_real @ W2 )
          = ( real_V7735802525324610683m_real @ Z2 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_7988_power__eq__imp__eq__norm,axiom,
    ! [W2: complex,N2: nat,Z2: complex] :
      ( ( ( power_power_complex @ W2 @ N2 )
        = ( power_power_complex @ Z2 @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V1022390504157884413omplex @ W2 )
          = ( real_V1022390504157884413omplex @ Z2 ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_7989_norm__mult__less,axiom,
    ! [X3: real,R2: real,Y3: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X3 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_7990_norm__mult__less,axiom,
    ! [X3: complex,R2: real,Y3: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X3 @ Y3 ) ) @ ( times_times_real @ R2 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_7991_norm__triangle__lt,axiom,
    ! [X3: real,Y3: real,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_7992_norm__triangle__lt,axiom,
    ! [X3: complex,Y3: complex,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_7993_norm__add__less,axiom,
    ! [X3: real,R2: real,Y3: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ R2 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_7994_norm__add__less,axiom,
    ! [X3: complex,R2: real,Y3: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ R2 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y3 ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_add_less
thf(fact_7995_norm__triangle__mono,axiom,
    ! [A: real,R2: real,B: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_7996_norm__triangle__mono,axiom,
    ! [A: complex,R2: real,B: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R2 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R2 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_7997_norm__triangle__ineq,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) ) ).

% norm_triangle_ineq
thf(fact_7998_norm__triangle__ineq,axiom,
    ! [X3: complex,Y3: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) ) ).

% norm_triangle_ineq
thf(fact_7999_norm__triangle__le,axiom,
    ! [X3: real,Y3: real,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ Y3 ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_8000_norm__triangle__le,axiom,
    ! [X3: complex,Y3: complex,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ Y3 ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_8001_norm__add__leD,axiom,
    ! [A: real,B: real,C: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_8002_norm__add__leD,axiom,
    ! [A: complex,B: complex,C: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ C )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ C ) ) ) ).

% norm_add_leD
thf(fact_8003_norm__diff__triangle__less,axiom,
    ! [X3: real,Y3: real,E1: real,Z2: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Y3 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y3 @ Z2 ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_8004_norm__diff__triangle__less,axiom,
    ! [X3: complex,Y3: complex,E1: real,Z2: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Y3 ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y3 @ Z2 ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X3 @ Z2 ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_8005_norm__diff__ineq,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_8006_norm__diff__ineq,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) ) ).

% norm_diff_ineq
thf(fact_8007_suminf__finite,axiom,
    ! [N7: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_complex ) )
       => ( ( suminf_complex @ F )
          = ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).

% suminf_finite
thf(fact_8008_suminf__finite,axiom,
    ! [N7: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( ( suminf_int @ F )
          = ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).

% suminf_finite
thf(fact_8009_suminf__finite,axiom,
    ! [N7: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( ( suminf_nat @ F )
          = ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).

% suminf_finite
thf(fact_8010_suminf__finite,axiom,
    ! [N7: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( ( suminf_real @ F )
          = ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).

% suminf_finite
thf(fact_8011_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_8012_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_8013_norm__diff__triangle__ineq,axiom,
    ! [A: real,B: real,C: real,D: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ ( plus_plus_real @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ C ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_8014_norm__diff__triangle__ineq,axiom,
    ! [A: complex,B: complex,C: complex,D: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( plus_plus_complex @ A @ B ) @ ( plus_plus_complex @ C @ D ) ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ C ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ B @ D ) ) ) ) ).

% norm_diff_triangle_ineq
thf(fact_8015_sum__bounds__lt__plus1,axiom,
    ! [F: nat > nat,Mm: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_8016_sum__bounds__lt__plus1,axiom,
    ! [F: nat > real,Mm: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( F @ ( suc @ K3 ) )
        @ ( set_ord_lessThan_nat @ Mm ) )
      = ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ one_one_nat @ Mm ) ) ) ).

% sum_bounds_lt_plus1
thf(fact_8017_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ zero_zero_real @ M2 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_8018_pi__series,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( suminf_real
      @ ^ [K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% pi_series
thf(fact_8019_summable__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X3 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_8020_ceiling__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) )
          = ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% ceiling_log_nat_eq_powr_iff
thf(fact_8021_geometric__deriv__sums,axiom,
    ! [Z2: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ one_one_real )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8022_geometric__deriv__sums,axiom,
    ! [Z2: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ one_one_real )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8023_summable__zero,axiom,
    ( summable_complex
    @ ^ [N: nat] : zero_zero_complex ) ).

% summable_zero
thf(fact_8024_summable__zero,axiom,
    ( summable_real
    @ ^ [N: nat] : zero_zero_real ) ).

% summable_zero
thf(fact_8025_summable__zero,axiom,
    ( summable_nat
    @ ^ [N: nat] : zero_zero_nat ) ).

% summable_zero
thf(fact_8026_summable__zero,axiom,
    ( summable_int
    @ ^ [N: nat] : zero_zero_int ) ).

% summable_zero
thf(fact_8027_summable__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( summable_complex
      @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex ) ) ).

% summable_single
thf(fact_8028_summable__single,axiom,
    ! [I: nat,F: nat > real] :
      ( summable_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real ) ) ).

% summable_single
thf(fact_8029_summable__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( summable_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat ) ) ).

% summable_single
thf(fact_8030_summable__single,axiom,
    ! [I: nat,F: nat > int] :
      ( summable_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int ) ) ).

% summable_single
thf(fact_8031_sums__zero,axiom,
    ( sums_complex
    @ ^ [N: nat] : zero_zero_complex
    @ zero_zero_complex ) ).

% sums_zero
thf(fact_8032_sums__zero,axiom,
    ( sums_real
    @ ^ [N: nat] : zero_zero_real
    @ zero_zero_real ) ).

% sums_zero
thf(fact_8033_sums__zero,axiom,
    ( sums_nat
    @ ^ [N: nat] : zero_zero_nat
    @ zero_zero_nat ) ).

% sums_zero
thf(fact_8034_sums__zero,axiom,
    ( sums_int
    @ ^ [N: nat] : zero_zero_int
    @ zero_zero_int ) ).

% sums_zero
thf(fact_8035_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_8036_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_8037_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_8038_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_8039_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_8040_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_8041_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_complex
        @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite
thf(fact_8042_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite
thf(fact_8043_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite
thf(fact_8044_summable__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite
thf(fact_8045_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_complex
        @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex ) ) ) ).

% summable_If_finite_set
thf(fact_8046_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real ) ) ) ).

% summable_If_finite_set
thf(fact_8047_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat ) ) ) ).

% summable_If_finite_set
thf(fact_8048_summable__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( summable_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int ) ) ) ).

% summable_If_finite_set
thf(fact_8049_log__eq__one,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ A )
          = one_one_real ) ) ) ).

% log_eq_one
thf(fact_8050_log__less__cancel__iff,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
            = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_8051_log__less__one__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ ( log @ A @ X3 ) @ one_one_real )
          = ( ord_less_real @ X3 @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_8052_one__less__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X3 ) )
          = ( ord_less_real @ A @ X3 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_8053_log__less__zero__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ ( log @ A @ X3 ) @ zero_zero_real )
          = ( ord_less_real @ X3 @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_8054_zero__less__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X3 ) )
          = ( ord_less_real @ one_one_real @ X3 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_8055_log__le__cancel__iff,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) )
            = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_8056_log__le__one__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ one_one_real )
          = ( ord_less_eq_real @ X3 @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_8057_one__le__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X3 ) )
          = ( ord_less_eq_real @ A @ X3 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_8058_log__le__zero__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ ( log @ A @ X3 ) @ zero_zero_real )
          = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_8059_zero__le__log__cancel__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X3 ) )
          = ( ord_less_eq_real @ one_one_real @ X3 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_8060_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X3: complex] :
      ( ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
        @ X3 )
      = ( ( A @ zero_zero_nat )
        = X3 ) ) ).

% powser_sums_zero_iff
thf(fact_8061_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X3: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
        @ X3 )
      = ( ( A @ zero_zero_nat )
        = X3 ) ) ).

% powser_sums_zero_iff
thf(fact_8062_summable__geometric__iff,axiom,
    ! [C: real] :
      ( ( summable_real @ ( power_power_real @ C ) )
      = ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_8063_summable__geometric__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex @ ( power_power_complex @ C ) )
      = ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real ) ) ).

% summable_geometric_iff
thf(fact_8064_log__pow__cancel,axiom,
    ! [A: real,B: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( power_power_real @ A @ B ) )
          = ( semiri5074537144036343181t_real @ B ) ) ) ) ).

% log_pow_cancel
thf(fact_8065_sums__0,axiom,
    ! [F: nat > complex] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_complex )
     => ( sums_complex @ F @ zero_zero_complex ) ) ).

% sums_0
thf(fact_8066_sums__0,axiom,
    ! [F: nat > real] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_real )
     => ( sums_real @ F @ zero_zero_real ) ) ).

% sums_0
thf(fact_8067_sums__0,axiom,
    ! [F: nat > nat] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_nat )
     => ( sums_nat @ F @ zero_zero_nat ) ) ).

% sums_0
thf(fact_8068_sums__0,axiom,
    ! [F: nat > int] :
      ( ! [N3: nat] :
          ( ( F @ N3 )
          = zero_zero_int )
     => ( sums_int @ F @ zero_zero_int ) ) ).

% sums_0
thf(fact_8069_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_8070_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_8071_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_8072_sums__single,axiom,
    ! [I: nat,F: nat > complex] :
      ( sums_complex
      @ ^ [R5: nat] : ( if_complex @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_complex )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8073_sums__single,axiom,
    ! [I: nat,F: nat > real] :
      ( sums_real
      @ ^ [R5: nat] : ( if_real @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_real )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8074_sums__single,axiom,
    ! [I: nat,F: nat > nat] :
      ( sums_nat
      @ ^ [R5: nat] : ( if_nat @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_nat )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8075_sums__single,axiom,
    ! [I: nat,F: nat > int] :
      ( sums_int
      @ ^ [R5: nat] : ( if_int @ ( R5 = I ) @ ( F @ R5 ) @ zero_zero_int )
      @ ( F @ I ) ) ).

% sums_single
thf(fact_8076_sums__add,axiom,
    ! [F: nat > real,A: real,G: nat > real,B: real] :
      ( ( sums_real @ F @ A )
     => ( ( sums_real @ G @ B )
       => ( sums_real
          @ ^ [N: nat] : ( plus_plus_real @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_real @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8077_sums__add,axiom,
    ! [F: nat > nat,A: nat,G: nat > nat,B: nat] :
      ( ( sums_nat @ F @ A )
     => ( ( sums_nat @ G @ B )
       => ( sums_nat
          @ ^ [N: nat] : ( plus_plus_nat @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8078_sums__add,axiom,
    ! [F: nat > int,A: int,G: nat > int,B: int] :
      ( ( sums_int @ F @ A )
     => ( ( sums_int @ G @ B )
       => ( sums_int
          @ ^ [N: nat] : ( plus_plus_int @ ( F @ N ) @ ( G @ N ) )
          @ ( plus_plus_int @ A @ B ) ) ) ) ).

% sums_add
thf(fact_8079_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_8080_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_8081_summable__comparison__test_H,axiom,
    ! [G: nat > real,N7: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N7 @ N3 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_8082_summable__comparison__test_H,axiom,
    ! [G: nat > real,N7: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N7 @ N3 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_8083_summable__const__iff,axiom,
    ! [C: complex] :
      ( ( summable_complex
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_complex ) ) ).

% summable_const_iff
thf(fact_8084_summable__const__iff,axiom,
    ! [C: real] :
      ( ( summable_real
        @ ^ [Uu3: nat] : C )
      = ( C = zero_zero_real ) ) ).

% summable_const_iff
thf(fact_8085_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_8086_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_8087_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_8088_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_8089_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_8090_powser__insidea,axiom,
    ! [F: nat > real,X3: real,Z2: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X3 @ N ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X3 ) )
       => ( summable_real
          @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ) ) ).

% powser_insidea
thf(fact_8091_powser__insidea,axiom,
    ! [F: nat > complex,X3: complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X3 @ N ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X3 ) )
       => ( summable_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ) ) ).

% powser_insidea
thf(fact_8092_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_8093_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_8094_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_8095_summable__finite,axiom,
    ! [N7: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_complex ) )
       => ( summable_complex @ F ) ) ) ).

% summable_finite
thf(fact_8096_summable__finite,axiom,
    ! [N7: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( summable_real @ F ) ) ) ).

% summable_finite
thf(fact_8097_summable__finite,axiom,
    ! [N7: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( summable_nat @ F ) ) ) ).

% summable_finite
thf(fact_8098_summable__finite,axiom,
    ! [N7: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( summable_int @ F ) ) ) ).

% summable_finite
thf(fact_8099_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_8100_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_8101_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_8102_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_8103_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_8104_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_8105_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_8106_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_8107_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_8108_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_8109_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_8110_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_8111_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_8112_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_8113_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_8114_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_8115_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_8116_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_8117_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_8118_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_8119_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_8120_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_8121_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_8122_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_8123_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_8124_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_8125_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_8126_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_8127_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_8128_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_8129_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_8130_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > complex,S: complex] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ( F @ I3 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
          @ S )
        = ( sums_complex @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_8131_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > real,S: real] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ I3 @ N2 )
         => ( ( F @ I3 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
          @ S )
        = ( sums_real @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_8132_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_8133_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_8134_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_8135_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_8136_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_8137_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_8138_summable__powser__split__head,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_8139_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_8140_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_8141_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z2: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_complex @ Z2 @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_8142_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z2: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_real @ Z2 @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_8143_sums__finite,axiom,
    ! [N7: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_complex ) )
       => ( sums_complex @ F @ ( groups2073611262835488442omplex @ F @ N7 ) ) ) ) ).

% sums_finite
thf(fact_8144_sums__finite,axiom,
    ! [N7: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_int ) )
       => ( sums_int @ F @ ( groups3539618377306564664at_int @ F @ N7 ) ) ) ) ).

% sums_finite
thf(fact_8145_sums__finite,axiom,
    ! [N7: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_nat ) )
       => ( sums_nat @ F @ ( groups3542108847815614940at_nat @ F @ N7 ) ) ) ) ).

% sums_finite
thf(fact_8146_sums__finite,axiom,
    ! [N7: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ N7 )
     => ( ! [N3: nat] :
            ( ~ ( member_nat @ N3 @ N7 )
           => ( ( F @ N3 )
              = zero_zero_real ) )
       => ( sums_real @ F @ ( groups6591440286371151544t_real @ F @ N7 ) ) ) ) ).

% sums_finite
thf(fact_8147_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > complex] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_complex
        @ ^ [R5: nat] : ( if_complex @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8148_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > int] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_int
        @ ^ [R5: nat] : ( if_int @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8149_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > nat] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_nat
        @ ^ [R5: nat] : ( if_nat @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8150_sums__If__finite,axiom,
    ! [P: nat > $o,F: nat > real] :
      ( ( finite_finite_nat @ ( collect_nat @ P ) )
     => ( sums_real
        @ ^ [R5: nat] : ( if_real @ ( P @ R5 ) @ ( F @ R5 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ ( collect_nat @ P ) ) ) ) ).

% sums_If_finite
thf(fact_8151_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_complex
        @ ^ [R5: nat] : ( if_complex @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_complex )
        @ ( groups2073611262835488442omplex @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8152_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_int
        @ ^ [R5: nat] : ( if_int @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_int )
        @ ( groups3539618377306564664at_int @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8153_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_nat
        @ ^ [R5: nat] : ( if_nat @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8154_sums__If__finite__set,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( sums_real
        @ ^ [R5: nat] : ( if_real @ ( member_nat @ R5 @ A2 ) @ ( F @ R5 ) @ zero_zero_real )
        @ ( groups6591440286371151544t_real @ F @ A2 ) ) ) ).

% sums_If_finite_set
thf(fact_8155_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_8156_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_8157_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 ) )
          = ( ? [I2: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I2 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_8158_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 ) )
          = ( ? [I2: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_8159_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 ) )
          = ( ? [I2: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I2 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_8160_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_8161_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_8162_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_8163_suminf__le__const,axiom,
    ! [F: nat > int,X3: int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X3 ) ) ) ).

% suminf_le_const
thf(fact_8164_suminf__le__const,axiom,
    ! [F: nat > nat,X3: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X3 ) ) ) ).

% suminf_le_const
thf(fact_8165_suminf__le__const,axiom,
    ! [F: nat > real,X3: real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X3 )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X3 ) ) ) ).

% suminf_le_const
thf(fact_8166_powser__sums__if,axiom,
    ! [M: nat,Z2: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( times_times_complex @ ( if_complex @ ( N = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z2 @ N ) )
      @ ( power_power_complex @ Z2 @ M ) ) ).

% powser_sums_if
thf(fact_8167_powser__sums__if,axiom,
    ! [M: nat,Z2: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z2 @ N ) )
      @ ( power_power_real @ Z2 @ M ) ) ).

% powser_sums_if
thf(fact_8168_powser__sums__if,axiom,
    ! [M: nat,Z2: int] :
      ( sums_int
      @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z2 @ N ) )
      @ ( power_power_int @ Z2 @ M ) ) ).

% powser_sums_if
thf(fact_8169_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_8170_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_8171_log__base__change,axiom,
    ! [A: real,B: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X3 )
          = ( divide_divide_real @ ( log @ A @ X3 ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_8172_log__of__power__eq,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( semiri5074537144036343181t_real @ N2 )
          = ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) ) ) ) ).

% log_of_power_eq
thf(fact_8173_less__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( ( ord_less_real @ ( power_power_real @ B @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).

% less_log_of_power
thf(fact_8174_powser__inside,axiom,
    ! [F: nat > real,X3: real,Z2: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X3 @ N ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z2 ) @ ( real_V7735802525324610683m_real @ X3 ) )
       => ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) ) ) ) ).

% powser_inside
thf(fact_8175_powser__inside,axiom,
    ! [F: nat > complex,X3: complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X3 @ N ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z2 ) @ ( real_V1022390504157884413omplex @ X3 ) )
       => ( summable_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) ) ) ) ).

% powser_inside
thf(fact_8176_sums__iff__shift,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_8177_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X3: 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 ) ) @ X3 )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_8178_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X3: 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 ) ) @ X3 )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_8179_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X3: 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 ) ) @ X3 )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_8180_sums__split__initial__segment,axiom,
    ! [F: nat > real,S: real,N2: nat] :
      ( ( sums_real @ F @ S )
     => ( sums_real
        @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_8181_sums__iff__shift_H,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_real @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_8182_complete__algebra__summable__geometric,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X3 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_8183_complete__algebra__summable__geometric,axiom,
    ! [X3: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X3 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_8184_summable__geometric,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ C ) ) ) ).

% summable_geometric
thf(fact_8185_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_8186_sums__If__finite__set_H,axiom,
    ! [G: nat > real,S3: real,A2: set_nat,S6: real,F: nat > real] :
      ( ( sums_real @ G @ S3 )
     => ( ( finite_finite_nat @ A2 )
       => ( ( S6
            = ( plus_plus_real @ S3
              @ ( 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 ) )
            @ S6 ) ) ) ) ).

% sums_If_finite_set'
thf(fact_8187_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_8188_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_8189_sum__le__suminf,axiom,
    ! [F: nat > int,I5: set_nat] :
      ( ( summable_int @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) ) )
         => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ I5 ) @ ( suminf_int @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8190_sum__le__suminf,axiom,
    ! [F: nat > nat,I5: set_nat] :
      ( ( summable_nat @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) ) )
         => ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ I5 ) @ ( suminf_nat @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8191_sum__le__suminf,axiom,
    ! [F: nat > real,I5: set_nat] :
      ( ( summable_real @ F )
     => ( ( finite_finite_nat @ I5 )
       => ( ! [N3: nat] :
              ( ( member_nat @ N3 @ ( uminus5710092332889474511et_nat @ I5 ) )
             => ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) ) )
         => ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ I5 ) @ ( suminf_real @ F ) ) ) ) ) ).

% sum_le_suminf
thf(fact_8192_log__mult,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( ord_less_real @ zero_zero_real @ Y3 )
           => ( ( log @ A @ ( times_times_real @ X3 @ Y3 ) )
              = ( plus_plus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).

% log_mult
thf(fact_8193_log__divide,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( ord_less_real @ zero_zero_real @ Y3 )
           => ( ( log @ A @ ( divide_divide_real @ X3 @ Y3 ) )
              = ( minus_minus_real @ ( log @ A @ X3 ) @ ( log @ A @ Y3 ) ) ) ) ) ) ) ).

% log_divide
thf(fact_8194_le__log__of__power,axiom,
    ! [B: real,N2: nat,M: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ B @ N2 ) @ M )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ M ) ) ) ) ).

% le_log_of_power
thf(fact_8195_log__base__pow,axiom,
    ! [A: real,N2: nat,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N2 ) @ X3 )
        = ( divide_divide_real @ ( log @ A @ X3 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log_base_pow
thf(fact_8196_log__nat__power,axiom,
    ! [X3: real,B: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( log @ B @ ( power_power_real @ X3 @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X3 ) ) ) ) ).

% log_nat_power
thf(fact_8197_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_8198_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_8199_pi__half__less__two,axiom,
    ord_less_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_less_two
thf(fact_8200_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_8201_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_8202_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_8203_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) )
            @ Z2 ) ) ) ) ).

% powser_split_head(1)
thf(fact_8204_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) )
            @ Z2 ) ) ) ) ).

% powser_split_head(1)
thf(fact_8205_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z2: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z2 @ N ) ) )
          @ Z2 )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z2 @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8206_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z2: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z2 @ N ) ) )
          @ Z2 )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z2 @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8207_log__of__power__less,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_less
thf(fact_8208_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ B )
         => ( ( B != one_one_real )
           => ( ( ord_less_real @ zero_zero_real @ X3 )
             => ( ( log @ A @ X3 )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X3 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_8209_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E2: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M6: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M6 )
                 => ! [N6: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M6 @ N6 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_8210_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E2: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N9: nat] :
              ~ ! [M6: nat] :
                  ( ( ord_less_eq_nat @ N9 @ M6 )
                 => ! [N6: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M6 @ N6 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_8211_suminf__exist__split,axiom,
    ! [R2: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_real @ F )
       => ? [N9: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ N9 @ N6 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N6 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_8212_suminf__exist__split,axiom,
    ! [R2: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R2 )
     => ( ( summable_complex @ F )
       => ? [N9: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ N9 @ N6 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I2: nat] : ( F @ ( plus_plus_nat @ I2 @ N6 ) ) ) )
              @ R2 ) ) ) ) ).

% suminf_exist_split
thf(fact_8213_summable__power__series,axiom,
    ! [F: nat > real,Z2: real] :
      ( ! [I3: nat] : ( ord_less_eq_real @ ( F @ I3 ) @ one_one_real )
     => ( ! [I3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I3 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( summable_real
              @ ^ [I2: nat] : ( times_times_real @ ( F @ I2 ) @ ( power_power_real @ Z2 @ I2 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8214_Abel__lemma,axiom,
    ! [R2: real,R0: real,A: nat > complex,M7: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R2 )
     => ( ( ord_less_real @ R2 @ R0 )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N3 ) ) @ ( power_power_real @ R0 @ N3 ) ) @ M7 )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( real_V1022390504157884413omplex @ ( A @ N ) ) @ ( power_power_real @ R2 @ N ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_8215_pi__half__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_gt_zero
thf(fact_8216_m2pi__less__pi,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) @ pi ).

% m2pi_less_pi
thf(fact_8217_summable__ratio__test,axiom,
    ! [C: real,N7: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N7 @ 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_8218_summable__ratio__test,axiom,
    ! [C: real,N7: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N7 @ 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_8219_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_8220_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_8221_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_8222_arctan__ubound,axiom,
    ! [Y3: real] : ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_8223_geometric__sums,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ C ) @ one_one_real )
     => ( sums_real @ ( power_power_real @ C ) @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ C ) ) ) ) ).

% geometric_sums
thf(fact_8224_geometric__sums,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( sums_complex @ ( power_power_complex @ C ) @ ( divide1717551699836669952omplex @ one_one_complex @ ( minus_minus_complex @ one_one_complex @ C ) ) ) ) ).

% geometric_sums
thf(fact_8225_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_8226_log__of__power__le,axiom,
    ! [M: nat,B: real,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ M ) @ ( power_power_real @ B @ N2 ) )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ord_less_nat @ zero_zero_nat @ M )
         => ( ord_less_eq_real @ ( log @ B @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_of_power_le
thf(fact_8227_minus__pi__half__less__zero,axiom,
    ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ zero_zero_real ).

% minus_pi_half_less_zero
thf(fact_8228_arctan__lbound,axiom,
    ! [Y3: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) ) ).

% arctan_lbound
thf(fact_8229_arctan__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
      & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_8230_less__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% less_log2_of_power
thf(fact_8231_le__log2__of__power,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ M )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% le_log2_of_power
thf(fact_8232_log2__of__power__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_less
thf(fact_8233_log2__of__power__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ M )
       => ( ord_less_eq_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log2_of_power_le
thf(fact_8234_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D4: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D4 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_8235_ceiling__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_eq_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim7802044766580827645g_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ) ) ) ).

% ceiling_log_nat_eq_if
thf(fact_8236_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_8237_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_8238_diffs__equiv,axiom,
    ! [C: nat > complex,X3: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X3 @ N ) ) )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( C @ N ) ) @ ( power_power_complex @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X3 @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_8239_diffs__equiv,axiom,
    ! [C: nat > real,X3: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X3 @ N ) ) )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( C @ N ) ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_8240_cos__pi__eq__zero,axiom,
    ! [M: nat] :
      ( ( cos_real @ ( divide_divide_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = zero_zero_real ) ).

% cos_pi_eq_zero
thf(fact_8241_ceiling__log__eq__powr__iff,axiom,
    ! [X3: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X3 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X3 )
            & ( ord_less_eq_real @ X3 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_8242_floor__log__nat__eq__powr__iff,axiom,
    ! [B: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) )
          = ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
            & ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ) ) ) ) ).

% floor_log_nat_eq_powr_iff
thf(fact_8243_sin__zero,axiom,
    ( ( sin_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sin_zero
thf(fact_8244_sin__zero,axiom,
    ( ( sin_real @ zero_zero_real )
    = zero_zero_real ) ).

% sin_zero
thf(fact_8245_powr__eq__0__iff,axiom,
    ! [W2: real,Z2: real] :
      ( ( ( powr_real @ W2 @ Z2 )
        = zero_zero_real )
      = ( W2 = zero_zero_real ) ) ).

% powr_eq_0_iff
thf(fact_8246_powr__0,axiom,
    ! [Z2: real] :
      ( ( powr_real @ zero_zero_real @ Z2 )
      = zero_zero_real ) ).

% powr_0
thf(fact_8247_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_8248_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_8249_powr__zero__eq__one,axiom,
    ! [X3: real] :
      ( ( ( X3 = zero_zero_real )
       => ( ( powr_real @ X3 @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X3 != zero_zero_real )
       => ( ( powr_real @ X3 @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_8250_floor__zero,axiom,
    ( ( archim6058952711729229775r_real @ zero_zero_real )
    = zero_zero_int ) ).

% floor_zero
thf(fact_8251_floor__zero,axiom,
    ( ( archim3151403230148437115or_rat @ zero_zero_rat )
    = zero_zero_int ) ).

% floor_zero
thf(fact_8252_powr__gt__zero,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X3 @ A ) )
      = ( X3 != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_8253_powr__less__cancel__iff,axiom,
    ! [X3: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel_iff
thf(fact_8254_powr__eq__one__iff,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X3 )
          = one_one_real )
        = ( X3 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_8255_powr__le__cancel__iff,axiom,
    ! [X3: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ( ord_less_eq_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_8256_zero__le__floor,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X3 ) ) ).

% zero_le_floor
thf(fact_8257_zero__le__floor,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X3 ) ) ).

% zero_le_floor
thf(fact_8258_sin__cos__squared__add3,axiom,
    ! [X3: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ X3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ X3 ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_8259_sin__cos__squared__add3,axiom,
    ! [X3: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ X3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ X3 ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_8260_floor__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ zero_zero_int )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% floor_less_zero
thf(fact_8261_floor__less__zero,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ zero_zero_int )
      = ( ord_less_rat @ X3 @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_8262_numeral__le__floor,axiom,
    ! [V: num,X3: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X3 ) ) ).

% numeral_le_floor
thf(fact_8263_numeral__le__floor,axiom,
    ! [V: num,X3: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X3 ) ) ).

% numeral_le_floor
thf(fact_8264_zero__less__floor,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ one_one_real @ X3 ) ) ).

% zero_less_floor
thf(fact_8265_zero__less__floor,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ).

% zero_less_floor
thf(fact_8266_floor__le__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ zero_zero_int )
      = ( ord_less_real @ X3 @ one_one_real ) ) ).

% floor_le_zero
thf(fact_8267_floor__le__zero,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ zero_zero_int )
      = ( ord_less_rat @ X3 @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_8268_floor__less__numeral,axiom,
    ! [X3: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X3 @ ( numeral_numeral_real @ V ) ) ) ).

% floor_less_numeral
thf(fact_8269_floor__less__numeral,axiom,
    ! [X3: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X3 @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_8270_one__le__floor,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ one_one_real @ X3 ) ) ).

% one_le_floor
thf(fact_8271_one__le__floor,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X3 ) ) ).

% one_le_floor
thf(fact_8272_floor__less__one,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
      = ( ord_less_real @ X3 @ one_one_real ) ) ).

% floor_less_one
thf(fact_8273_floor__less__one,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
      = ( ord_less_rat @ X3 @ one_one_rat ) ) ).

% floor_less_one
thf(fact_8274_powr__log__cancel,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( powr_real @ A @ ( log @ A @ X3 ) )
            = X3 ) ) ) ) ).

% powr_log_cancel
thf(fact_8275_log__powr__cancel,axiom,
    ! [A: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y3 ) )
          = Y3 ) ) ) ).

% log_powr_cancel
thf(fact_8276_numeral__less__floor,axiom,
    ! [V: num,X3: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X3 ) ) ).

% numeral_less_floor
thf(fact_8277_numeral__less__floor,axiom,
    ! [V: num,X3: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X3 ) ) ).

% numeral_less_floor
thf(fact_8278_floor__le__numeral,axiom,
    ! [X3: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X3 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_8279_floor__le__numeral,axiom,
    ! [X3: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_8280_one__less__floor,axiom,
    ! [X3: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) ).

% one_less_floor
thf(fact_8281_one__less__floor,axiom,
    ! [X3: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X3 ) ) ).

% one_less_floor
thf(fact_8282_floor__le__one,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
      = ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_8283_floor__le__one,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
      = ( ord_less_rat @ X3 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_8284_neg__numeral__le__floor,axiom,
    ! [V: num,X3: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X3 ) ) ).

% neg_numeral_le_floor
thf(fact_8285_neg__numeral__le__floor,axiom,
    ! [V: num,X3: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X3 ) ) ).

% neg_numeral_le_floor
thf(fact_8286_floor__less__neg__numeral,axiom,
    ! [X3: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_8287_floor__less__neg__numeral,axiom,
    ! [X3: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X3 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_8288_sin__cos__squared__add2,axiom,
    ! [X3: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_8289_sin__cos__squared__add2,axiom,
    ! [X3: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_8290_sin__cos__squared__add,axiom,
    ! [X3: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_8291_sin__cos__squared__add,axiom,
    ! [X3: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_8292_neg__numeral__less__floor,axiom,
    ! [V: num,X3: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X3 ) ) ).

% neg_numeral_less_floor
thf(fact_8293_neg__numeral__less__floor,axiom,
    ! [V: num,X3: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X3 ) ) ).

% neg_numeral_less_floor
thf(fact_8294_floor__le__neg__numeral,axiom,
    ! [X3: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X3 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% floor_le_neg_numeral
thf(fact_8295_floor__le__neg__numeral,axiom,
    ! [X3: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_8296_cos__one__sin__zero,axiom,
    ! [X3: complex] :
      ( ( ( cos_complex @ X3 )
        = one_one_complex )
     => ( ( sin_complex @ X3 )
        = zero_zero_complex ) ) ).

% cos_one_sin_zero
thf(fact_8297_cos__one__sin__zero,axiom,
    ! [X3: real] :
      ( ( ( cos_real @ X3 )
        = one_one_real )
     => ( ( sin_real @ X3 )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_8298_sin__add,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( sin_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).

% sin_add
thf(fact_8299_sin__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).

% sin_add
thf(fact_8300_cos__diff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).

% cos_diff
thf(fact_8301_cos__diff,axiom,
    ! [X3: real,Y3: real] :
      ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).

% cos_diff
thf(fact_8302_cos__add,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) @ ( times_times_complex @ ( sin_complex @ X3 ) @ ( sin_complex @ Y3 ) ) ) ) ).

% cos_add
thf(fact_8303_cos__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) @ ( times_times_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) ) ) ) ).

% cos_add
thf(fact_8304_sin__zero__norm__cos__one,axiom,
    ! [X3: real] :
      ( ( ( sin_real @ X3 )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X3 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_8305_sin__zero__norm__cos__one,axiom,
    ! [X3: complex] :
      ( ( ( sin_complex @ X3 )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X3 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_8306_sincos__principal__value,axiom,
    ! [X3: real] :
    ? [Y5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y5 )
      & ( ord_less_eq_real @ Y5 @ pi )
      & ( ( sin_real @ Y5 )
        = ( sin_real @ X3 ) )
      & ( ( cos_real @ Y5 )
        = ( cos_real @ X3 ) ) ) ).

% sincos_principal_value
thf(fact_8307_floor__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ Y3 )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) ).

% floor_mono
thf(fact_8308_floor__mono,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_rat @ X3 @ Y3 )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) ).

% floor_mono
thf(fact_8309_of__int__floor__le,axiom,
    ! [X3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) @ X3 ) ).

% of_int_floor_le
thf(fact_8310_of__int__floor__le,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) @ X3 ) ).

% of_int_floor_le
thf(fact_8311_floor__less__cancel,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) )
     => ( ord_less_real @ X3 @ Y3 ) ) ).

% floor_less_cancel
thf(fact_8312_floor__less__cancel,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) )
     => ( ord_less_rat @ X3 @ Y3 ) ) ).

% floor_less_cancel
thf(fact_8313_powr__non__neg,axiom,
    ! [A: real,X3: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X3 ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_8314_powr__less__mono2__neg,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ X3 @ Y3 )
         => ( ord_less_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_8315_powr__less__cancel,axiom,
    ! [X3: real,A: real,B: real] :
      ( ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) )
     => ( ( ord_less_real @ one_one_real @ X3 )
       => ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel
thf(fact_8316_powr__less__mono,axiom,
    ! [A: real,B: real,X3: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ one_one_real @ X3 )
       => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ) ).

% powr_less_mono
thf(fact_8317_le__floor__iff,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_eq_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ).

% le_floor_iff
thf(fact_8318_le__floor__iff,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_eq_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ).

% le_floor_iff
thf(fact_8319_floor__less__iff,axiom,
    ! [X3: real,Z2: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
      = ( ord_less_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ).

% floor_less_iff
thf(fact_8320_floor__less__iff,axiom,
    ! [X3: rat,Z2: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
      = ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ).

% floor_less_iff
thf(fact_8321_powr__mono2_H,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ X3 @ Y3 )
         => ( ord_less_eq_real @ ( powr_real @ Y3 @ A ) @ ( powr_real @ X3 @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_8322_powr__less__mono2,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ X3 @ Y3 )
         => ( ord_less_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ Y3 @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_8323_le__floor__add,axiom,
    ! [X3: real,Y3: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).

% le_floor_add
thf(fact_8324_le__floor__add,axiom,
    ! [X3: rat,Y3: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y3 ) ) ) ).

% le_floor_add
thf(fact_8325_gr__one__powr,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ Y3 )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X3 @ Y3 ) ) ) ) ).

% gr_one_powr
thf(fact_8326_powr__inj,axiom,
    ! [A: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X3 )
            = ( powr_real @ A @ Y3 ) )
          = ( X3 = Y3 ) ) ) ) ).

% powr_inj
thf(fact_8327_int__add__floor,axiom,
    ! [Z2: int,X3: real] :
      ( ( plus_plus_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ X3 ) ) ) ).

% int_add_floor
thf(fact_8328_int__add__floor,axiom,
    ! [Z2: int,X3: rat] :
      ( ( plus_plus_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 ) ) ) ).

% int_add_floor
thf(fact_8329_floor__add__int,axiom,
    ! [X3: real,Z2: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ ( ring_1_of_int_real @ Z2 ) ) ) ) ).

% floor_add_int
thf(fact_8330_floor__add__int,axiom,
    ! [X3: rat,Z2: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ ( ring_1_of_int_rat @ Z2 ) ) ) ) ).

% floor_add_int
thf(fact_8331_sin__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).

% sin_gt_zero
thf(fact_8332_sin__times__sin,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_8333_sin__times__sin,axiom,
    ! [W2: real,Z2: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_8334_sin__times__cos,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( times_times_complex @ ( sin_complex @ W2 ) @ ( cos_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_8335_sin__times__cos,axiom,
    ! [W2: real,Z2: real] :
      ( ( times_times_real @ ( sin_real @ W2 ) @ ( cos_real @ Z2 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_8336_cos__times__sin,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( sin_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) @ ( sin_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_8337_cos__times__sin,axiom,
    ! [W2: real,Z2: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( sin_real @ Z2 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W2 @ Z2 ) ) @ ( sin_real @ ( minus_minus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_8338_sin__plus__sin,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_8339_sin__plus__sin,axiom,
    ! [W2: real,Z2: real] :
      ( ( plus_plus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_8340_sin__diff__sin,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W2 ) @ ( sin_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_8341_sin__diff__sin,axiom,
    ! [W2: real,Z2: real] :
      ( ( minus_minus_real @ ( sin_real @ W2 ) @ ( sin_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_8342_cos__diff__cos,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z2 @ W2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_8343_cos__diff__cos,axiom,
    ! [W2: real,Z2: real] :
      ( ( minus_minus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z2 @ W2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_8344_powr__add,axiom,
    ! [X3: real,A: real,B: real] :
      ( ( powr_real @ X3 @ ( plus_plus_real @ A @ B ) )
      = ( times_times_real @ ( powr_real @ X3 @ A ) @ ( powr_real @ X3 @ B ) ) ) ).

% powr_add
thf(fact_8345_one__add__floor,axiom,
    ! [X3: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_8346_one__add__floor,axiom,
    ! [X3: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_8347_floor__log__eq__powr__iff,axiom,
    ! [X3: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X3 ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X3 )
            & ( ord_less_real @ X3 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_8348_powr__realpow,axiom,
    ! [X3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) )
        = ( power_power_real @ X3 @ N2 ) ) ) ).

% powr_realpow
thf(fact_8349_less__log__iff,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ Y3 @ ( log @ B @ X3 ) )
          = ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ).

% less_log_iff
thf(fact_8350_log__less__iff,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ ( log @ B @ X3 ) @ Y3 )
          = ( ord_less_real @ X3 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).

% log_less_iff
thf(fact_8351_less__powr__iff,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ X3 @ ( powr_real @ B @ Y3 ) )
          = ( ord_less_real @ ( log @ B @ X3 ) @ Y3 ) ) ) ) ).

% less_powr_iff
thf(fact_8352_powr__less__iff,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_real @ ( powr_real @ B @ Y3 ) @ X3 )
          = ( ord_less_real @ Y3 @ ( log @ B @ X3 ) ) ) ) ) ).

% powr_less_iff
thf(fact_8353_cos__mono__less__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ X3 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ pi )
           => ( ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) )
              = ( ord_less_real @ Y3 @ X3 ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_8354_cos__monotone__0__pi,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => ( ( ord_less_eq_real @ X3 @ pi )
         => ( ord_less_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_8355_sin__eq__0__pi,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
     => ( ( ord_less_real @ X3 @ pi )
       => ( ( ( sin_real @ X3 )
            = zero_zero_real )
         => ( X3 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_8356_real__of__int__floor__add__one__gt,axiom,
    ! [R2: real] : ( ord_less_real @ R2 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_8357_floor__eq,axiom,
    ! [N2: int,X3: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X3 )
          = N2 ) ) ) ).

% floor_eq
thf(fact_8358_sin__zero__pi__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ pi )
     => ( ( ( sin_real @ X3 )
          = zero_zero_real )
        = ( X3 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_8359_real__of__int__floor__gt__diff__one,axiom,
    ! [R2: real] : ( ord_less_real @ ( minus_minus_real @ R2 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R2 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_8360_diffs__def,axiom,
    ( diffs_complex
    = ( ^ [C3: nat > complex,N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_8361_diffs__def,axiom,
    ( diffs_rat
    = ( ^ [C3: nat > rat,N: nat] : ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_8362_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C3: nat > int,N: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_8363_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C3: nat > real,N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( C3 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_8364_sin__expansion__lemma,axiom,
    ! [X3: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_8365_floor__unique,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X3 )
          = Z2 ) ) ) ).

% floor_unique
thf(fact_8366_floor__unique,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X3 )
     => ( ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X3 )
          = Z2 ) ) ) ).

% floor_unique
thf(fact_8367_floor__eq__iff,axiom,
    ! [X3: real,A: int] :
      ( ( ( archim6058952711729229775r_real @ X3 )
        = A )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X3 )
        & ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_8368_floor__eq__iff,axiom,
    ! [X3: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X3 )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X3 )
        & ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_8369_floor__split,axiom,
    ! [P: int > $o,T: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I2 ) @ T )
              & ( ord_less_real @ T @ ( plus_plus_real @ ( ring_1_of_int_real @ I2 ) @ one_one_real ) ) )
           => ( P @ I2 ) ) ) ) ).

% floor_split
thf(fact_8370_floor__split,axiom,
    ! [P: int > $o,T: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T ) )
      = ( ! [I2: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I2 ) @ T )
              & ( ord_less_rat @ T @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I2 ) @ one_one_rat ) ) )
           => ( P @ I2 ) ) ) ) ).

% floor_split
thf(fact_8371_le__mult__floor,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_8372_le__mult__floor,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ B )
       => ( ord_less_eq_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ).

% le_mult_floor
thf(fact_8373_less__floor__iff,axiom,
    ! [Z2: int,X3: real] :
      ( ( ord_less_int @ Z2 @ ( archim6058952711729229775r_real @ X3 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) @ X3 ) ) ).

% less_floor_iff
thf(fact_8374_less__floor__iff,axiom,
    ! [Z2: int,X3: rat] :
      ( ( ord_less_int @ Z2 @ ( archim3151403230148437115or_rat @ X3 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) @ X3 ) ) ).

% less_floor_iff
thf(fact_8375_floor__le__iff,axiom,
    ! [X3: real,Z2: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X3 ) @ Z2 )
      = ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z2 ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_8376_floor__le__iff,axiom,
    ! [X3: rat,Z2: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X3 ) @ Z2 )
      = ( ord_less_rat @ X3 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z2 ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_8377_floor__correct,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X3 ) ) @ X3 )
      & ( ord_less_real @ X3 @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_8378_floor__correct,axiom,
    ! [X3: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X3 ) ) @ X3 )
      & ( ord_less_rat @ X3 @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_8379_cos__expansion__lemma,axiom,
    ! [X3: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_8380_powr__neg__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X3 ) ) ) ).

% powr_neg_one
thf(fact_8381_sin__gt__zero__02,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).

% sin_gt_zero_02
thf(fact_8382_le__log__iff,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ Y3 @ ( log @ B @ X3 ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ).

% le_log_iff
thf(fact_8383_log__le__iff,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ ( log @ B @ X3 ) @ Y3 )
          = ( ord_less_eq_real @ X3 @ ( powr_real @ B @ Y3 ) ) ) ) ) ).

% log_le_iff
thf(fact_8384_le__powr__iff,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ X3 @ ( powr_real @ B @ Y3 ) )
          = ( ord_less_eq_real @ ( log @ B @ X3 ) @ Y3 ) ) ) ) ).

% le_powr_iff
thf(fact_8385_powr__le__iff,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y3 ) @ X3 )
          = ( ord_less_eq_real @ Y3 @ ( log @ B @ X3 ) ) ) ) ) ).

% powr_le_iff
thf(fact_8386_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_8387_floor__eq2,axiom,
    ! [N2: int,X3: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X3 )
          = N2 ) ) ) ).

% floor_eq2
thf(fact_8388_cos__monotone__minus__pi__0,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => ( ( ord_less_eq_real @ X3 @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y3 ) @ ( cos_real @ X3 ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_8389_floor__divide__lower,axiom,
    ! [Q4: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q4 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).

% floor_divide_lower
thf(fact_8390_floor__divide__lower,axiom,
    ! [Q4: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ Q4 ) @ P4 ) ) ).

% floor_divide_lower
thf(fact_8391_sincos__total__2pi,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X3
                  = ( cos_real @ T6 ) )
               => ( Y3
                 != ( sin_real @ T6 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_8392_ln__powr__bound,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X3 ) @ ( divide_divide_real @ ( powr_real @ X3 @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_8393_ln__powr__bound2,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X3 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X3 ) ) ) ) ).

% ln_powr_bound2
thf(fact_8394_log__add__eq__powr,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( plus_plus_real @ ( log @ B @ X3 ) @ Y3 )
            = ( log @ B @ ( times_times_real @ X3 @ ( powr_real @ B @ Y3 ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_8395_add__log__eq__powr,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( plus_plus_real @ Y3 @ ( log @ B @ X3 ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_8396_minus__log__eq__powr,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( minus_minus_real @ Y3 @ ( log @ B @ X3 ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y3 ) @ X3 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_8397_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_8398_cos__times__cos,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( times_times_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W2 @ Z2 ) ) @ ( cos_complex @ ( plus_plus_complex @ W2 @ Z2 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_8399_cos__times__cos,axiom,
    ! [W2: real,Z2: real] :
      ( ( times_times_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W2 @ Z2 ) ) @ ( cos_real @ ( plus_plus_real @ W2 @ Z2 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_8400_cos__plus__cos,axiom,
    ! [W2: complex,Z2: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W2 ) @ ( cos_complex @ Z2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W2 @ Z2 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_8401_cos__plus__cos,axiom,
    ! [W2: real,Z2: real] :
      ( ( plus_plus_real @ ( cos_real @ W2 ) @ ( cos_real @ Z2 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W2 @ Z2 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_8402_floor__divide__upper,axiom,
    ! [Q4: real,P4: real] :
      ( ( ord_less_real @ zero_zero_real @ Q4 )
     => ( ord_less_real @ P4 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P4 @ Q4 ) ) ) @ one_one_real ) @ Q4 ) ) ) ).

% floor_divide_upper
thf(fact_8403_floor__divide__upper,axiom,
    ! [Q4: rat,P4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q4 )
     => ( ord_less_rat @ P4 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P4 @ Q4 ) ) ) @ one_one_rat ) @ Q4 ) ) ) ).

% floor_divide_upper
thf(fact_8404_round__def,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X: real] : ( archim6058952711729229775r_real @ ( plus_plus_real @ X @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8405_round__def,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X: rat] : ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% round_def
thf(fact_8406_sin__gt__zero2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X3 ) ) ) ) ).

% sin_gt_zero2
thf(fact_8407_sin__lt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ pi @ X3 )
     => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_8408_cos__double__less__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_8409_cos__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).

% cos_gt_zero
thf(fact_8410_log__minus__eq__powr,axiom,
    ! [B: real,X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( minus_minus_real @ ( log @ B @ X3 ) @ Y3 )
            = ( log @ B @ ( times_times_real @ X3 @ ( powr_real @ B @ ( uminus_uminus_real @ Y3 ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_8411_termdiff__converges,axiom,
    ! [X3: real,K5: real,C: nat > real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ K5 )
     => ( ! [X4: real] :
            ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ K5 )
           => ( summable_real
              @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X4 @ N ) ) ) )
       => ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X3 @ N ) ) ) ) ) ).

% termdiff_converges
thf(fact_8412_termdiff__converges,axiom,
    ! [X3: complex,K5: real,C: nat > complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ K5 )
     => ( ! [X4: complex] :
            ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ K5 )
           => ( summable_complex
              @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X4 @ N ) ) ) )
       => ( summable_complex
          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X3 @ N ) ) ) ) ) ).

% termdiff_converges
thf(fact_8413_powr__neg__numeral,axiom,
    ! [X3: real,N2: num] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( powr_real @ X3 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_8414_sin__le__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ pi @ X3 )
     => ( ( ord_less_real @ X3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_8415_sin__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X3 ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_8416_sin__mono__less__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_eq_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X3 ) @ ( sin_real @ Y3 ) )
              = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_8417_sin__monotone__2pi,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => ( ( ord_less_eq_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y3 ) @ ( sin_real @ X3 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_8418_cos__gt__zero__pi,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X3 ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_8419_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_8420_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_8421_floor__log__nat__eq__if,axiom,
    ! [B: nat,N2: nat,K: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ B @ N2 ) @ K )
     => ( ( ord_less_nat @ K @ ( power_power_nat @ B @ ( plus_plus_nat @ N2 @ one_one_nat ) ) )
       => ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B )
         => ( ( archim6058952711729229775r_real @ ( log @ ( semiri5074537144036343181t_real @ B ) @ ( semiri5074537144036343181t_real @ K ) ) )
            = ( semiri1314217659103216013at_int @ N2 ) ) ) ) ) ).

% floor_log_nat_eq_if
thf(fact_8422_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
       => ? [T6: real] :
            ( ( ord_less_real @ X3 @ T6 )
            & ( ord_less_real @ T6 @ zero_zero_real )
            & ( ( cos_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
                  @ ( 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 @ X3 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_8423_Maclaurin__cos__expansion2,axiom,
    ! [X3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X3 )
            & ( ( cos_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( cos_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
                  @ ( 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 @ X3 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_8424_sin__paired,axiom,
    ! [X3: 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 @ X3 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
      @ ( sin_real @ X3 ) ) ).

% sin_paired
thf(fact_8425_tan__double,axiom,
    ! [X3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X3 ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_8426_tan__double,axiom,
    ! [X3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X3 ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X3 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_8427_arcosh__def,axiom,
    ( arcosh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( minus_minus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arcosh_def
thf(fact_8428_tan__zero,axiom,
    ( ( tan_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% tan_zero
thf(fact_8429_tan__zero,axiom,
    ( ( tan_real @ zero_zero_real )
    = zero_zero_real ) ).

% tan_zero
thf(fact_8430_of__real__eq__0__iff,axiom,
    ! [X3: real] :
      ( ( ( real_V1803761363581548252l_real @ X3 )
        = zero_zero_real )
      = ( X3 = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_8431_of__real__eq__0__iff,axiom,
    ! [X3: real] :
      ( ( ( real_V4546457046886955230omplex @ X3 )
        = zero_zero_complex )
      = ( X3 = zero_zero_real ) ) ).

% of_real_eq_0_iff
thf(fact_8432_of__real__0,axiom,
    ( ( real_V1803761363581548252l_real @ zero_zero_real )
    = zero_zero_real ) ).

% of_real_0
thf(fact_8433_of__real__0,axiom,
    ( ( real_V4546457046886955230omplex @ zero_zero_real )
    = zero_zero_complex ) ).

% of_real_0
thf(fact_8434_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_8435_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_8436_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_8437_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_8438_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_8439_of__real__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X3 @ Y3 ) )
      = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ ( real_V1803761363581548252l_real @ Y3 ) ) ) ).

% of_real_add
thf(fact_8440_of__real__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X3 @ Y3 ) )
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ ( real_V4546457046886955230omplex @ Y3 ) ) ) ).

% of_real_add
thf(fact_8441_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_8442_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_8443_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_8444_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_8445_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_8446_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_Suc
thf(fact_8447_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( suc @ N2 ) )
      = ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_Suc
thf(fact_8448_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_Suc
thf(fact_8449_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_Suc
thf(fact_8450_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_Suc
thf(fact_8451_sin__of__real__pi,axiom,
    ( ( sin_real @ ( real_V1803761363581548252l_real @ pi ) )
    = zero_zero_real ) ).

% sin_of_real_pi
thf(fact_8452_sin__of__real__pi,axiom,
    ( ( sin_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = zero_zero_complex ) ).

% sin_of_real_pi
thf(fact_8453_norm__of__real__add1,axiom,
    ! [X3: real] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ one_one_real ) )
      = ( abs_abs_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8454_norm__of__real__add1,axiom,
    ! [X3: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ one_one_complex ) )
      = ( abs_abs_real @ ( plus_plus_real @ X3 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_8455_norm__of__real__addn,axiom,
    ! [X3: real,B: num] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X3 ) @ ( numeral_numeral_real @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_8456_norm__of__real__addn,axiom,
    ! [X3: real,B: num] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X3 ) @ ( numera6690914467698888265omplex @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X3 @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_8457_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_8458_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_8459_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ N2 )
     != zero_zero_complex ) ).

% fact_nonzero
thf(fact_8460_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ N2 )
     != zero_zero_rat ) ).

% fact_nonzero
thf(fact_8461_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ N2 )
     != zero_zero_int ) ).

% fact_nonzero
thf(fact_8462_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ N2 )
     != zero_zero_real ) ).

% fact_nonzero
thf(fact_8463_fact__nonzero,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ N2 )
     != zero_zero_nat ) ).

% fact_nonzero
thf(fact_8464_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_zero
thf(fact_8465_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_zero
thf(fact_8466_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_zero
thf(fact_8467_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_zero
thf(fact_8468_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_8469_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_8470_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_8471_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_8472_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_gt_zero
thf(fact_8473_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_gt_zero
thf(fact_8474_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_gt_zero
thf(fact_8475_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_gt_zero
thf(fact_8476_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_1
thf(fact_8477_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_8478_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_1
thf(fact_8479_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_8480_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_mono
thf(fact_8481_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_mono
thf(fact_8482_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_mono
thf(fact_8483_fact__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono
thf(fact_8484_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) ) ) ).

% fact_dvd
thf(fact_8485_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_Code_integer @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) ) ) ).

% fact_dvd
thf(fact_8486_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ M ) ) ) ).

% fact_dvd
thf(fact_8487_fact__dvd,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( dvd_dvd_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) ) ) ).

% fact_dvd
thf(fact_8488_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_rat @ ( semiri773545260158071498ct_rat @ M ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_8489_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_int @ ( semiri1406184849735516958ct_int @ M ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_8490_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_real @ ( semiri2265585572941072030t_real @ M ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_8491_fact__less__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono
thf(fact_8492_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ N2 ) ) @ ( semiri3624122377584611663nteger @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8493_fact__fact__dvd__fact,axiom,
    ! [K: nat,N2: nat] : ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ N2 ) ) @ ( semiri773545260158071498ct_rat @ ( plus_plus_nat @ K @ N2 ) ) ) ).

% fact_fact_dvd_fact
thf(fact_8494_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_8495_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_8496_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_8497_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_int @ ( semiri1406184849735516958ct_int @ N2 ) @ ( semiri1406184849735516958ct_int @ M ) )
        = zero_zero_int ) ) ).

% fact_mod
thf(fact_8498_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_8499_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ M ) )
        = zero_zero_nat ) ) ).

% fact_mod
thf(fact_8500_fact__le__power,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri681578069525770553at_rat @ ( power_power_nat @ N2 @ N2 ) ) ) ).

% fact_le_power
thf(fact_8501_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_8502_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_8503_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_8504_norm__less__p1,axiom,
    ! [X3: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X3 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X3 ) ) @ one_one_real ) ) ) ).

% norm_less_p1
thf(fact_8505_norm__less__p1,axiom,
    ! [X3: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X3 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X3 ) ) @ one_one_complex ) ) ) ).

% norm_less_p1
thf(fact_8506_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_Code_integer @ ( times_3573771949741848930nteger @ ( semiri3624122377584611663nteger @ K ) @ ( semiri3624122377584611663nteger @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).

% choose_dvd
thf(fact_8507_choose__dvd,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( dvd_dvd_rat @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% choose_dvd
thf(fact_8508_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_8509_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_8510_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_8511_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M2: nat] : ( if_complex @ ( M2 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M2 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8512_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M2: nat] : ( if_rat @ ( M2 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8513_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M2: nat] : ( if_int @ ( M2 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M2 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8514_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M2: nat] : ( if_real @ ( M2 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M2 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8515_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M2: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M2 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_8516_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_8517_fact__reduce,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( semiri773545260158071498ct_rat @ N2 )
        = ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ) ).

% fact_reduce
thf(fact_8518_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_8519_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_8520_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_8521_tan__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).

% tan_gt_zero
thf(fact_8522_lemma__tan__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ? [X4: real] :
          ( ( ord_less_real @ zero_zero_real @ X4 )
          & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y3 @ ( tan_real @ X4 ) ) ) ) ).

% lemma_tan_total
thf(fact_8523_lemma__tan__total1,axiom,
    ! [Y3: real] :
    ? [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
      & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X4 )
        = Y3 ) ) ).

% lemma_tan_total1
thf(fact_8524_tan__mono__lt__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
              = ( ord_less_real @ X3 @ Y3 ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_8525_tan__monotone_H,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
         => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y3 @ X3 )
              = ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_8526_tan__monotone,axiom,
    ! [Y3: real,X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ X3 )
       => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y3 ) @ ( tan_real @ X3 ) ) ) ) ) ).

% tan_monotone
thf(fact_8527_tan__total,axiom,
    ! [Y3: real] :
    ? [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
      & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X4 )
        = Y3 )
      & ! [Y6: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y6 )
            & ( ord_less_real @ Y6 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y6 )
              = Y3 ) )
         => ( Y6 = X4 ) ) ) ).

% tan_total
thf(fact_8528_add__tan__eq,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) )
          = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_8529_add__tan__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_8530_tan__total__pos,axiom,
    ! [Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
     => ? [X4: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X4 )
          & ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X4 )
            = Y3 ) ) ) ).

% tan_total_pos
thf(fact_8531_tan__pos__pi2__le,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X3 ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_8532_tan__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X3 ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_8533_tan__mono__le__eq,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y3 )
         => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) )
              = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_8534_tan__mono__le,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ).

% tan_mono_le
thf(fact_8535_tan__bound__pi2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X3 ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_8536_arctan,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y3 ) )
      & ( ord_less_real @ ( arctan @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y3 ) )
        = Y3 ) ) ).

% arctan
thf(fact_8537_arctan__tan,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X3 ) )
          = X3 ) ) ) ).

% arctan_tan
thf(fact_8538_arctan__unique,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ( tan_real @ X3 )
            = Y3 )
         => ( ( arctan @ Y3 )
            = X3 ) ) ) ) ).

% arctan_unique
thf(fact_8539_Maclaurin__zero,axiom,
    ! [X3: real,N2: nat,Diff: nat > complex > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_8540_Maclaurin__zero,axiom,
    ! [X3: real,N2: nat,Diff: nat > real > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_8541_Maclaurin__zero,axiom,
    ! [X3: real,N2: nat,Diff: nat > rat > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8542_Maclaurin__zero,axiom,
    ! [X3: real,N2: nat,Diff: nat > nat > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_8543_Maclaurin__zero,axiom,
    ! [X3: real,N2: nat,Diff: nat > int > real] :
      ( ( X3 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_8544_Maclaurin__lemma,axiom,
    ! [H2: real,F: real > real,J: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H2 )
     => ? [B9: real] :
          ( ( F @ H2 )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
              @ ( set_ord_lessThan_nat @ N2 ) )
            @ ( times_times_real @ B9 @ ( divide_divide_real @ ( power_power_real @ H2 @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_8545_tan__add,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_8546_tan__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X3 @ Y3 ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_8547_tan__diff,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( minus_minus_complex @ X3 @ Y3 ) )
            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_8548_tan__diff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X3 @ Y3 ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X3 @ Y3 ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_8549_lemma__tan__add1,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( cos_complex @ X3 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y3 )
         != zero_zero_complex )
       => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X3 ) @ ( tan_complex @ Y3 ) ) )
          = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) @ ( times_times_complex @ ( cos_complex @ X3 ) @ ( cos_complex @ Y3 ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_8550_lemma__tan__add1,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( cos_real @ X3 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y3 )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X3 ) @ ( tan_real @ Y3 ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X3 @ Y3 ) ) @ ( times_times_real @ ( cos_real @ X3 ) @ ( cos_real @ Y3 ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_8551_minus__sin__cos__eq,axiom,
    ! [X3: real] :
      ( ( uminus_uminus_real @ ( sin_real @ X3 ) )
      = ( cos_real @ ( plus_plus_real @ X3 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8552_minus__sin__cos__eq,axiom,
    ! [X3: complex] :
      ( ( uminus1482373934393186551omplex @ ( sin_complex @ X3 ) )
      = ( cos_complex @ ( plus_plus_complex @ X3 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_8553_tan__total__pi4,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ? [Z3: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z3 )
          & ( ord_less_real @ Z3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z3 )
            = X3 ) ) ) ).

% tan_total_pi4
thf(fact_8554_tan__half,axiom,
    ( tan_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_complex @ ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X ) ) @ one_one_complex ) ) ) ) ).

% tan_half
thf(fact_8555_tan__half,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ ( plus_plus_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X ) ) @ one_one_real ) ) ) ) ).

% tan_half
thf(fact_8556_arsinh__def,axiom,
    ( arsinh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( powr_real @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( real_V1803761363581548252l_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% arsinh_def
thf(fact_8557_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ? [T6: real] :
            ( ( ord_less_real @ zero_zero_real @ T6 )
            & ( ord_less_real @ T6 @ X3 )
            & ( ( sin_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
                  @ ( 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 @ X3 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_8558_Maclaurin__sin__expansion4,axiom,
    ! [X3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ? [T6: real] :
          ( ( ord_less_real @ zero_zero_real @ T6 )
          & ( ord_less_eq_real @ T6 @ X3 )
          & ( ( sin_real @ X3 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M2: nat] : ( times_times_real @ ( sin_coeff @ M2 ) @ ( power_power_real @ X3 @ M2 ) )
                @ ( 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 @ X3 @ N2 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_8559_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_8560_Maclaurin__exp__lt,axiom,
    ! [X3: real,N2: nat] :
      ( ( X3 != 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 @ X3 ) )
            & ( ( exp_real @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( divide_divide_real @ ( power_power_real @ X3 @ M2 ) @ ( semiri2265585572941072030t_real @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_8561_round__altdef,axiom,
    ( archim8280529875227126926d_real
    = ( ^ [X: real] : ( if_int @ ( ord_less_eq_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( archim2898591450579166408c_real @ X ) ) @ ( archim7802044766580827645g_real @ X ) @ ( archim6058952711729229775r_real @ X ) ) ) ) ).

% round_altdef
thf(fact_8562_round__altdef,axiom,
    ( archim7778729529865785530nd_rat
    = ( ^ [X: rat] : ( if_int @ ( ord_less_eq_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( archimedean_frac_rat @ X ) ) @ ( archim2889992004027027881ng_rat @ X ) @ ( archim3151403230148437115or_rat @ X ) ) ) ) ).

% round_altdef
thf(fact_8563_exp__less__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ).

% exp_less_mono
thf(fact_8564_exp__less__cancel__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% exp_less_cancel_iff
thf(fact_8565_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_8566_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_8567_frac__of__int,axiom,
    ! [Z2: int] :
      ( ( archim2898591450579166408c_real @ ( ring_1_of_int_real @ Z2 ) )
      = zero_zero_real ) ).

% frac_of_int
thf(fact_8568_frac__of__int,axiom,
    ! [Z2: int] :
      ( ( archimedean_frac_rat @ ( ring_1_of_int_rat @ Z2 ) )
      = zero_zero_rat ) ).

% frac_of_int
thf(fact_8569_sin__coeff__0,axiom,
    ( ( sin_coeff @ zero_zero_nat )
    = zero_zero_real ) ).

% sin_coeff_0
thf(fact_8570_one__less__exp__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% one_less_exp_iff
thf(fact_8571_exp__less__one__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( exp_real @ X3 ) @ one_one_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_8572_exp__ln__iff,axiom,
    ! [X3: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X3 ) )
        = X3 )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% exp_ln_iff
thf(fact_8573_exp__ln,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( exp_real @ ( ln_ln_real @ X3 ) )
        = X3 ) ) ).

% exp_ln
thf(fact_8574_fact__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_mono_nat
thf(fact_8575_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_8576_exp__less__cancel,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
     => ( ord_less_real @ X3 @ Y3 ) ) ).

% exp_less_cancel
thf(fact_8577_exp__not__eq__zero,axiom,
    ! [X3: complex] :
      ( ( exp_complex @ X3 )
     != zero_zero_complex ) ).

% exp_not_eq_zero
thf(fact_8578_exp__not__eq__zero,axiom,
    ! [X3: real] :
      ( ( exp_real @ X3 )
     != zero_zero_real ) ).

% exp_not_eq_zero
thf(fact_8579_fact__less__mono__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ M @ N2 )
       => ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% fact_less_mono_nat
thf(fact_8580_exp__total,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ Y3 )
     => ? [X4: real] :
          ( ( exp_real @ X4 )
          = Y3 ) ) ).

% exp_total
thf(fact_8581_exp__gt__zero,axiom,
    ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X3 ) ) ).

% exp_gt_zero
thf(fact_8582_not__exp__less__zero,axiom,
    ! [X3: real] :
      ~ ( ord_less_real @ ( exp_real @ X3 ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_8583_mult__exp__exp,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) )
      = ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) ) ) ).

% mult_exp_exp
thf(fact_8584_mult__exp__exp,axiom,
    ! [X3: real,Y3: real] :
      ( ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) )
      = ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) ) ) ).

% mult_exp_exp
thf(fact_8585_exp__add__commuting,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( ( times_times_complex @ X3 @ Y3 )
        = ( times_times_complex @ Y3 @ X3 ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X3 @ Y3 ) )
        = ( times_times_complex @ ( exp_complex @ X3 ) @ ( exp_complex @ Y3 ) ) ) ) ).

% exp_add_commuting
thf(fact_8586_exp__add__commuting,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( times_times_real @ X3 @ Y3 )
        = ( times_times_real @ Y3 @ X3 ) )
     => ( ( exp_real @ ( plus_plus_real @ X3 @ Y3 ) )
        = ( times_times_real @ ( exp_real @ X3 ) @ ( exp_real @ Y3 ) ) ) ) ).

% exp_add_commuting
thf(fact_8587_frac__ge__0,axiom,
    ! [X3: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X3 ) ) ).

% frac_ge_0
thf(fact_8588_frac__ge__0,axiom,
    ! [X3: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X3 ) ) ).

% frac_ge_0
thf(fact_8589_frac__lt__1,axiom,
    ! [X3: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X3 ) @ one_one_real ) ).

% frac_lt_1
thf(fact_8590_frac__lt__1,axiom,
    ! [X3: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X3 ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_8591_frac__1__eq,axiom,
    ! [X3: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X3 @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X3 ) ) ).

% frac_1_eq
thf(fact_8592_frac__1__eq,axiom,
    ! [X3: rat] :
      ( ( archimedean_frac_rat @ ( plus_plus_rat @ X3 @ one_one_rat ) )
      = ( archimedean_frac_rat @ X3 ) ) ).

% frac_1_eq
thf(fact_8593_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_8594_exp__gt__one,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X3 ) ) ) ).

% exp_gt_one
thf(fact_8595_dvd__fact,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( dvd_dvd_nat @ M @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ) ).

% dvd_fact
thf(fact_8596_fact__diff__Suc,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ M ) )
     => ( ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) )
        = ( times_times_nat @ ( minus_minus_nat @ ( suc @ M ) @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M @ N2 ) ) ) ) ) ).

% fact_diff_Suc
thf(fact_8597_ln__ge__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ Y3 @ ( ln_ln_real @ X3 ) )
        = ( ord_less_eq_real @ ( exp_real @ Y3 ) @ X3 ) ) ) ).

% ln_ge_iff
thf(fact_8598_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_8599_powr__def,axiom,
    ( powr_real
    = ( ^ [X: real,A3: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( exp_real @ ( times_times_real @ A3 @ ( ln_ln_real @ X ) ) ) ) ) ) ).

% powr_def
thf(fact_8600_exp__divide__power__eq,axiom,
    ! [N2: nat,X3: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X3 @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
        = ( exp_complex @ X3 ) ) ) ).

% exp_divide_power_eq
thf(fact_8601_exp__divide__power__eq,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
        = ( exp_real @ X3 ) ) ) ).

% exp_divide_power_eq
thf(fact_8602_frac__eq,axiom,
    ! [X3: real] :
      ( ( ( archim2898591450579166408c_real @ X3 )
        = X3 )
      = ( ( ord_less_eq_real @ zero_zero_real @ X3 )
        & ( ord_less_real @ X3 @ one_one_real ) ) ) ).

% frac_eq
thf(fact_8603_frac__eq,axiom,
    ! [X3: rat] :
      ( ( ( archimedean_frac_rat @ X3 )
        = X3 )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X3 )
        & ( ord_less_rat @ X3 @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_8604_frac__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X3 @ Y3 ) )
          = ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X3 @ Y3 ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real ) ) ) ) ).

% frac_add
thf(fact_8605_frac__add,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_8606_tanh__altdef,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_8607_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_8608_floor__add,axiom,
    ! [X3: real,Y3: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y3 ) )
          = ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X3 ) @ ( archim2898591450579166408c_real @ Y3 ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X3 @ Y3 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X3 ) @ ( archim6058952711729229775r_real @ Y3 ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_8609_floor__add,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X3 ) @ ( archimedean_frac_rat @ Y3 ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X3 @ Y3 ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X3 ) @ ( archim3151403230148437115or_rat @ Y3 ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_8610_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_8611_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X3 ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_8612_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X3: real,N2: nat] :
      ( ( ord_less_eq_real @ X3 @ ( 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 @ X3 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X3 ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_8613_log__base__10__eq2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X3 ) ) ) ) ).

% log_base_10_eq2
thf(fact_8614_log__base__10__eq1,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X3 )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X3 ) ) ) ) ).

% log_base_10_eq1
thf(fact_8615_sin__tan,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X3 )
        = ( divide_divide_real @ ( tan_real @ X3 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_8616_cos__tan,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X3 )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_8617_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X5: nat > real] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_real @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_real @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8618_monoseq__def,axiom,
    ( topolo3100542954746470799et_int
    = ( ^ [X5: nat > set_int] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_set_int @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_set_int @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8619_monoseq__def,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X5: nat > rat] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_rat @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_rat @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8620_monoseq__def,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X5: nat > num] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_num @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_num @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8621_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X5: nat > nat] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_nat @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_nat @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8622_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X5: nat > int] :
          ( ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_int @ ( X5 @ M2 ) @ ( X5 @ N ) ) )
          | ! [M2: nat,N: nat] :
              ( ( ord_less_eq_nat @ M2 @ N )
             => ( ord_less_eq_int @ ( X5 @ N ) @ ( X5 @ M2 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8623_monoI2,axiom,
    ! [X7: nat > real] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo6980174941875973593q_real @ X7 ) ) ).

% monoI2
thf(fact_8624_monoI2,axiom,
    ! [X7: nat > set_int] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_set_int @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo3100542954746470799et_int @ X7 ) ) ).

% monoI2
thf(fact_8625_monoI2,axiom,
    ! [X7: nat > rat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X7 ) ) ).

% monoI2
thf(fact_8626_monoI2,axiom,
    ! [X7: nat > num] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo1459490580787246023eq_num @ X7 ) ) ).

% monoI2
thf(fact_8627_monoI2,axiom,
    ! [X7: nat > nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X7 ) ) ).

% monoI2
thf(fact_8628_monoI2,axiom,
    ! [X7: nat > int] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ M3 ) ) )
     => ( topolo4899668324122417113eq_int @ X7 ) ) ).

% monoI2
thf(fact_8629_monoI1,axiom,
    ! [X7: nat > real] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_real @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X7 ) ) ).

% monoI1
thf(fact_8630_monoI1,axiom,
    ! [X7: nat > set_int] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_set_int @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo3100542954746470799et_int @ X7 ) ) ).

% monoI1
thf(fact_8631_monoI1,axiom,
    ! [X7: nat > rat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_rat @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X7 ) ) ).

% monoI1
thf(fact_8632_monoI1,axiom,
    ! [X7: nat > num] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_num @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo1459490580787246023eq_num @ X7 ) ) ).

% monoI1
thf(fact_8633_monoI1,axiom,
    ! [X7: nat > nat] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_nat @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X7 ) ) ).

% monoI1
thf(fact_8634_monoI1,axiom,
    ! [X7: nat > int] :
      ( ! [M3: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M3 @ N3 )
         => ( ord_less_eq_int @ ( X7 @ M3 ) @ ( X7 @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X7 ) ) ).

% monoI1
thf(fact_8635_real__sqrt__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% real_sqrt_less_iff
thf(fact_8636_real__sqrt__lt__0__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( sqrt @ X3 ) @ zero_zero_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_8637_real__sqrt__gt__0__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ zero_zero_real @ Y3 ) ) ).

% real_sqrt_gt_0_iff
thf(fact_8638_real__sqrt__lt__1__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( sqrt @ X3 ) @ one_one_real )
      = ( ord_less_real @ X3 @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_8639_real__sqrt__gt__1__iff,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y3 ) )
      = ( ord_less_real @ one_one_real @ Y3 ) ) ).

% real_sqrt_gt_1_iff
thf(fact_8640_real__sqrt__less__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ( ord_less_real @ ( sqrt @ X3 ) @ ( sqrt @ Y3 ) ) ) ).

% real_sqrt_less_mono
thf(fact_8641_real__sqrt__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X3 ) ) ) ).

% real_sqrt_gt_zero
thf(fact_8642_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_8643_real__less__rsqrt,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y3 )
     => ( ord_less_real @ X3 @ ( sqrt @ Y3 ) ) ) ).

% real_less_rsqrt
thf(fact_8644_lemma__real__divide__sqrt__less,axiom,
    ! [U: real] :
      ( ( ord_less_real @ zero_zero_real @ U )
     => ( ord_less_real @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ U ) ) ).

% lemma_real_divide_sqrt_less
thf(fact_8645_real__less__lsqrt,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ X3 @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X3 ) @ Y3 ) ) ) ) ).

% real_less_lsqrt
thf(fact_8646_ln__sqrt,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ln_ln_real @ ( sqrt @ X3 ) )
        = ( divide_divide_real @ ( ln_ln_real @ X3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_8647_arsinh__real__aux,axiom,
    ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X3 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_8648_real__sqrt__sum__squares__less,axiom,
    ! [X3: real,U: real,Y3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y3 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_8649_sqrt__sum__squares__half__less,axiom,
    ! [X3: real,U: real,Y3: real] :
      ( ( ord_less_real @ X3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y3 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_8650_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X5: nat > real] :
          ( ! [N: nat] : ( ord_less_eq_real @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_real @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8651_monoseq__Suc,axiom,
    ( topolo3100542954746470799et_int
    = ( ^ [X5: nat > set_int] :
          ( ! [N: nat] : ( ord_less_eq_set_int @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_set_int @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8652_monoseq__Suc,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X5: nat > rat] :
          ( ! [N: nat] : ( ord_less_eq_rat @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_rat @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8653_monoseq__Suc,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X5: nat > num] :
          ( ! [N: nat] : ( ord_less_eq_num @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_num @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8654_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X5: nat > nat] :
          ( ! [N: nat] : ( ord_less_eq_nat @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_nat @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8655_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X5: nat > int] :
          ( ! [N: nat] : ( ord_less_eq_int @ ( X5 @ N ) @ ( X5 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_int @ ( X5 @ ( suc @ N ) ) @ ( X5 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8656_mono__SucI2,axiom,
    ! [X7: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo6980174941875973593q_real @ X7 ) ) ).

% mono_SucI2
thf(fact_8657_mono__SucI2,axiom,
    ! [X7: nat > set_int] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo3100542954746470799et_int @ X7 ) ) ).

% mono_SucI2
thf(fact_8658_mono__SucI2,axiom,
    ! [X7: nat > rat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo4267028734544971653eq_rat @ X7 ) ) ).

% mono_SucI2
thf(fact_8659_mono__SucI2,axiom,
    ! [X7: nat > num] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo1459490580787246023eq_num @ X7 ) ) ).

% mono_SucI2
thf(fact_8660_mono__SucI2,axiom,
    ! [X7: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo4902158794631467389eq_nat @ X7 ) ) ).

% mono_SucI2
thf(fact_8661_mono__SucI2,axiom,
    ! [X7: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ ( suc @ N3 ) ) @ ( X7 @ N3 ) )
     => ( topolo4899668324122417113eq_int @ X7 ) ) ).

% mono_SucI2
thf(fact_8662_mono__SucI1,axiom,
    ! [X7: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X7 ) ) ).

% mono_SucI1
thf(fact_8663_mono__SucI1,axiom,
    ! [X7: nat > set_int] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo3100542954746470799et_int @ X7 ) ) ).

% mono_SucI1
thf(fact_8664_mono__SucI1,axiom,
    ! [X7: nat > rat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X7 ) ) ).

% mono_SucI1
thf(fact_8665_mono__SucI1,axiom,
    ! [X7: nat > num] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo1459490580787246023eq_num @ X7 ) ) ).

% mono_SucI1
thf(fact_8666_mono__SucI1,axiom,
    ! [X7: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X7 ) ) ).

% mono_SucI1
thf(fact_8667_mono__SucI1,axiom,
    ! [X7: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X7 @ N3 ) @ ( X7 @ ( suc @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X7 ) ) ).

% mono_SucI1
thf(fact_8668_pochhammer__double,axiom,
    ! [Z2: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z2 ) @ ( 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 @ Z2 @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_8669_pochhammer__double,axiom,
    ! [Z2: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z2 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_8670_pochhammer__double,axiom,
    ! [Z2: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z2 ) @ ( 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 @ Z2 @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_8671_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I2: complex] : ( plus_plus_complex @ I2 @ one_one_complex )
          @ N
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_8672_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I2: rat] : ( plus_plus_rat @ I2 @ one_one_rat )
          @ N
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_8673_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I2: int] : ( plus_plus_int @ I2 @ one_one_int )
          @ N
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_8674_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I2: real] : ( plus_plus_real @ I2 @ one_one_real )
          @ N
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_8675_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ one_one_nat )
          @ N
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_8676_gchoose__row__sum__weighted,axiom,
    ! [R2: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R2 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R2 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8677_gchoose__row__sum__weighted,axiom,
    ! [R2: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R2 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R2 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ ( suc @ M ) ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8678_gchoose__row__sum__weighted,axiom,
    ! [R2: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R2 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R2 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ R2 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_8679_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_8680_complex__unimodular__polar,axiom,
    ! [Z2: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z2 )
        = one_one_real )
     => ~ ! [T6: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T6 )
           => ( ( ord_less_real @ T6 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z2
               != ( complex2 @ ( cos_real @ T6 ) @ ( sin_real @ T6 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_8681_binomial__Suc__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ N2 )
      = ( suc @ N2 ) ) ).

% binomial_Suc_n
thf(fact_8682_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
      = zero_zero_complex ) ).

% gbinomial_0(2)
thf(fact_8683_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_8684_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_8685_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_8686_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_8687_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
      = N2 ) ).

% binomial_1
thf(fact_8688_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_8689_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_8690_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_8691_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_8692_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_8693_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_8694_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_8695_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_8696_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_8697_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_8698_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_8699_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_8700_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_8701_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_8702_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_8703_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_8704_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_8705_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_8706_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_8707_choose__mult__lemma,axiom,
    ! [M: nat,R2: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R2 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R2 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_8708_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_8709_pochhammer__pos,axiom,
    ! [X3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8710_pochhammer__pos,axiom,
    ! [X3: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8711_pochhammer__pos,axiom,
    ! [X3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X3 )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8712_pochhammer__pos,axiom,
    ! [X3: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X3 )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_8713_pochhammer__neq__0__mono,axiom,
    ! [A: complex,M: nat,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ M )
       != zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ N2 )
         != zero_zero_complex ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8714_pochhammer__neq__0__mono,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ M )
       != zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ N2 )
         != zero_zero_real ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8715_pochhammer__neq__0__mono,axiom,
    ! [A: rat,M: nat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ M )
       != zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ N2 )
         != zero_zero_rat ) ) ) ).

% pochhammer_neq_0_mono
thf(fact_8716_pochhammer__eq__0__mono,axiom,
    ! [A: complex,N2: nat,M: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s2602460028002588243omplex @ A @ M )
          = zero_zero_complex ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8717_pochhammer__eq__0__mono,axiom,
    ! [A: real,N2: nat,M: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s7457072308508201937r_real @ A @ M )
          = zero_zero_real ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8718_pochhammer__eq__0__mono,axiom,
    ! [A: rat,N2: nat,M: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( ( comm_s4028243227959126397er_rat @ A @ M )
          = zero_zero_rat ) ) ) ).

% pochhammer_eq_0_mono
thf(fact_8719_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_8720_Suc__times__binomial__add,axiom,
    ! [A: nat,B: nat] :
      ( ( times_times_nat @ ( suc @ A ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ ( suc @ A ) ) )
      = ( times_times_nat @ ( suc @ B ) @ ( binomial @ ( suc @ ( plus_plus_nat @ A @ B ) ) @ A ) ) ) ).

% Suc_times_binomial_add
thf(fact_8721_choose__mult,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( times_times_nat @ ( binomial @ N2 @ M ) @ ( binomial @ M @ K ) )
          = ( times_times_nat @ ( binomial @ N2 @ K ) @ ( binomial @ ( minus_minus_nat @ N2 @ K ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ) ).

% choose_mult
thf(fact_8722_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_8723_gbinomial__pochhammer_H,axiom,
    ( gbinomial_complex
    = ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ A3 @ ( semiri8010041392384452111omplex @ K3 ) ) @ one_one_complex ) @ K3 ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8724_gbinomial__pochhammer_H,axiom,
    ( gbinomial_rat
    = ( ^ [A3: rat,K3: nat] : ( divide_divide_rat @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ K3 ) ) @ one_one_rat ) @ K3 ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8725_gbinomial__pochhammer_H,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K3: nat] : ( divide_divide_real @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ A3 @ ( semiri5074537144036343181t_real @ K3 ) ) @ one_one_real ) @ K3 ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer'
thf(fact_8726_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_8727_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_8728_gbinomial__Suc__Suc,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ).

% gbinomial_Suc_Suc
thf(fact_8729_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_8730_pochhammer__nonneg,axiom,
    ! [X3: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X3 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8731_pochhammer__nonneg,axiom,
    ! [X3: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X3 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X3 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8732_pochhammer__nonneg,axiom,
    ! [X3: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X3 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X3 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8733_pochhammer__nonneg,axiom,
    ! [X3: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X3 )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X3 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_8734_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_8735_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_8736_pochhammer__0__left,axiom,
    ! [N2: nat] :
      ( ( ( N2 = zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = one_one_rat ) )
      & ( ( N2 != zero_zero_nat )
       => ( ( comm_s4028243227959126397er_rat @ zero_zero_rat @ N2 )
          = zero_zero_rat ) ) ) ).

% pochhammer_0_left
thf(fact_8737_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_8738_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_8739_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_8740_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_8741_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_8742_gbinomial__addition__formula,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ A @ ( suc @ K ) )
      = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_addition_formula
thf(fact_8743_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_8744_gbinomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ K ) @ A )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_ge_n_over_k_pow_k
thf(fact_8745_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_8746_gbinomial__mult__1_H,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ A )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1'
thf(fact_8747_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_8748_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_8749_gbinomial__mult__1,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ A @ ( gbinomial_rat @ A @ K ) )
      = ( plus_plus_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ K ) @ ( gbinomial_rat @ A @ K ) ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) ) ) ) ).

% gbinomial_mult_1
thf(fact_8750_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_8751_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_8752_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_8753_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_8754_pochhammer__rec,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ A @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ N2 ) ) ) ).

% pochhammer_rec
thf(fact_8755_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_8756_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_8757_pochhammer__rec_H,axiom,
    ! [Z2: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N2 ) )
      = ( times_times_complex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8758_pochhammer__rec_H,axiom,
    ! [Z2: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8759_pochhammer__rec_H,axiom,
    ! [Z2: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z2 @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8760_pochhammer__rec_H,axiom,
    ! [Z2: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N2 ) )
      = ( times_times_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8761_pochhammer__rec_H,axiom,
    ! [Z2: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z2 @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z2 @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_8762_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_8763_pochhammer__Suc,axiom,
    ! [A: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ A @ ( suc @ N2 ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ A @ N2 ) @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) ) ) ).

% pochhammer_Suc
thf(fact_8764_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_8765_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_8766_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_8767_pochhammer__eq__0__iff,axiom,
    ! [A: complex,N2: nat] :
      ( ( ( comm_s2602460028002588243omplex @ A @ N2 )
        = zero_zero_complex )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8768_pochhammer__eq__0__iff,axiom,
    ! [A: rat,N2: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ A @ N2 )
        = zero_zero_rat )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8769_pochhammer__eq__0__iff,axiom,
    ! [A: real,N2: nat] :
      ( ( ( comm_s7457072308508201937r_real @ A @ N2 )
        = zero_zero_real )
      = ( ? [K3: nat] :
            ( ( ord_less_nat @ K3 @ N2 )
            & ( A
              = ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K3 ) ) ) ) ) ) ).

% pochhammer_eq_0_iff
thf(fact_8770_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_8771_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
        = zero_zero_rat )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8772_pochhammer__of__nat__eq__0__iff,axiom,
    ! [N2: nat,K: nat] :
      ( ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
        = zero_z3403309356797280102nteger )
      = ( ord_less_nat @ N2 @ K ) ) ).

% pochhammer_of_nat_eq_0_iff
thf(fact_8773_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_8774_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_8775_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_8776_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
        = zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8777_pochhammer__of__nat__eq__0__lemma,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
        = zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma
thf(fact_8778_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_8779_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_8780_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_8781_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ K )
       != zero_zero_rat ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8782_pochhammer__of__nat__eq__0__lemma_H,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ K )
       != zero_z3403309356797280102nteger ) ) ).

% pochhammer_of_nat_eq_0_lemma'
thf(fact_8783_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_8784_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_8785_pochhammer__product_H,axiom,
    ! [Z2: complex,N2: nat,M: nat] :
      ( ( comm_s2602460028002588243omplex @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8786_pochhammer__product_H,axiom,
    ! [Z2: rat,N2: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8787_pochhammer__product_H,axiom,
    ! [Z2: int,N2: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8788_pochhammer__product_H,axiom,
    ! [Z2: real,N2: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8789_pochhammer__product_H,axiom,
    ! [Z2: nat,N2: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z2 @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_8790_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_8791_binomial__ge__n__over__k__pow__k,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ord_less_eq_rat @ ( power_power_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( semiri681578069525770553at_rat @ K ) ) @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) ) ) ).

% binomial_ge_n_over_k_pow_k
thf(fact_8792_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_8793_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_8794_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_8795_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_8796_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_8797_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_8798_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_8799_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_8800_Suc__times__gbinomial,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) ) )
      = ( times_times_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% Suc_times_gbinomial
thf(fact_8801_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_8802_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_8803_gbinomial__absorption,axiom,
    ! [K: nat,A: rat] :
      ( ( times_times_rat @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) @ ( gbinomial_rat @ A @ ( suc @ K ) ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorption
thf(fact_8804_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_8805_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_8806_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: complex] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_complex @ ( gbinomial_complex @ A @ M ) @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ M ) @ K ) )
        = ( times_times_complex @ ( gbinomial_complex @ A @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8807_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: rat] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_rat @ ( gbinomial_rat @ A @ M ) @ ( gbinomial_rat @ ( semiri681578069525770553at_rat @ M ) @ K ) )
        = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8808_gbinomial__trinomial__revision,axiom,
    ! [K: nat,M: nat,A: real] :
      ( ( ord_less_eq_nat @ K @ M )
     => ( ( times_times_real @ ( gbinomial_real @ A @ M ) @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ M ) @ K ) )
        = ( times_times_real @ ( gbinomial_real @ A @ K ) @ ( gbinomial_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( minus_minus_nat @ M @ K ) ) ) ) ) ).

% gbinomial_trinomial_revision
thf(fact_8809_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s2602460028002588243omplex @ Z2 @ N2 )
        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8810_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ Z2 @ N2 )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8811_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z2 @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z2 @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8812_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s7457072308508201937r_real @ Z2 @ N2 )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8813_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z2 @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z2 @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z2 @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_8814_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_8815_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_8816_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_8817_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_8818_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_8819_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_8820_binomial__fact,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ) ).

% binomial_fact
thf(fact_8821_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_8822_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_8823_fact__binomial,axiom,
    ! [K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ K @ N2 )
     => ( ( times_times_rat @ ( semiri773545260158071498ct_rat @ K ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K ) ) )
        = ( divide_divide_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ N2 @ K ) ) ) ) ) ).

% fact_binomial
thf(fact_8824_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_8825_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_8826_gbinomial__rec,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( gbinomial_rat @ A @ K ) @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) ) ) ).

% gbinomial_rec
thf(fact_8827_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_8828_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_8829_gbinomial__factors,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( suc @ K ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ A @ one_one_rat ) @ ( semiri681578069525770553at_rat @ ( suc @ K ) ) ) @ ( gbinomial_rat @ A @ K ) ) ) ).

% gbinomial_factors
thf(fact_8830_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_8831_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_8832_pochhammer__absorb__comp,axiom,
    ! [R2: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R2 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R2 ) @ K ) )
      = ( times_times_rat @ R2 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R2 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8833_pochhammer__absorb__comp,axiom,
    ! [R2: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R2 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R2 ) @ K ) )
      = ( times_3573771949741848930nteger @ R2 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R2 ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_8834_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_8835_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_8836_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_8837_gbinomial__minus,axiom,
    ! [A: rat,K: nat] :
      ( ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% gbinomial_minus
thf(fact_8838_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_8839_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_8840_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_8841_gbinomial__reduce__nat,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( plus_plus_rat @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ) ).

% gbinomial_reduce_nat
thf(fact_8842_pochhammer__minus,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8843_pochhammer__minus,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8844_pochhammer__minus,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8845_pochhammer__minus,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8846_pochhammer__minus,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_minus
thf(fact_8847_pochhammer__minus_H,axiom,
    ! [B: complex,K: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( minus_minus_complex @ B @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ K )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8848_pochhammer__minus_H,axiom,
    ! [B: rat,K: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( minus_minus_rat @ B @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ K )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8849_pochhammer__minus_H,axiom,
    ! [B: code_integer,K: nat] :
      ( ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ B @ ( semiri4939895301339042750nteger @ K ) ) @ one_one_Code_integer ) @ K )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ K ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8850_pochhammer__minus_H,axiom,
    ! [B: int,K: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( minus_minus_int @ B @ ( semiri1314217659103216013at_int @ K ) ) @ one_one_int ) @ K )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ K ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8851_pochhammer__minus_H,axiom,
    ! [B: real,K: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( minus_minus_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ K )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ B ) @ K ) ) ) ).

% pochhammer_minus'
thf(fact_8852_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [J3: nat] : ( gbinomial_complex @ ( semiri8010041392384452111omplex @ J3 ) @ 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_8853_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [J3: nat] : ( gbinomial_rat @ ( semiri681578069525770553at_rat @ J3 ) @ K )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ).

% gbinomial_sum_up_index
thf(fact_8854_gbinomial__sum__up__index,axiom,
    ! [K: nat,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [J3: nat] : ( gbinomial_real @ ( semiri5074537144036343181t_real @ J3 ) @ 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_8855_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_8856_gbinomial__absorption_H,axiom,
    ! [K: nat,A: rat] :
      ( ( ord_less_nat @ zero_zero_nat @ K )
     => ( ( gbinomial_rat @ A @ K )
        = ( times_times_rat @ ( divide_divide_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ ( minus_minus_nat @ K @ one_one_nat ) ) ) ) ) ).

% gbinomial_absorption'
thf(fact_8857_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_8858_binomial__code,axiom,
    ( binomial
    = ( ^ [N: nat,K3: nat] : ( if_nat @ ( ord_less_nat @ N @ K3 ) @ zero_zero_nat @ ( if_nat @ ( ord_less_nat @ N @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K3 ) ) @ ( binomial @ N @ ( minus_minus_nat @ N @ K3 ) ) @ ( divide_divide_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( plus_plus_nat @ ( minus_minus_nat @ N @ K3 ) @ one_one_nat ) @ N @ one_one_nat ) @ ( semiri1408675320244567234ct_nat @ K3 ) ) ) ) ) ) ).

% binomial_code
thf(fact_8859_gbinomial__code,axiom,
    ( gbinomial_complex
    = ( ^ [A3: complex,K3: nat] :
          ( if_complex @ ( K3 = 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 @ K3 @ one_one_nat )
              @ one_one_complex )
            @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8860_gbinomial__code,axiom,
    ( gbinomial_rat
    = ( ^ [A3: rat,K3: nat] :
          ( if_rat @ ( K3 = zero_zero_nat ) @ one_one_rat
          @ ( divide_divide_rat
            @ ( set_fo1949268297981939178at_rat
              @ ^ [L2: nat] : ( times_times_rat @ ( minus_minus_rat @ A3 @ ( semiri681578069525770553at_rat @ L2 ) ) )
              @ zero_zero_nat
              @ ( minus_minus_nat @ K3 @ one_one_nat )
              @ one_one_rat )
            @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8861_gbinomial__code,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K3: nat] :
          ( if_real @ ( K3 = 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 @ K3 @ one_one_nat )
              @ one_one_real )
            @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ) ).

% gbinomial_code
thf(fact_8862_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: complex,N2: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z2 @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K3: nat] : ( plus_plus_complex @ Z2 @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( 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_8863_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: rat,N2: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z2 @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z2 @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8864_pochhammer__times__pochhammer__half,axiom,
    ! [Z2: real,N2: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z2 @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z2 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K3: nat] : ( plus_plus_real @ Z2 @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_8865_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_8866_pochhammer__code,axiom,
    ( comm_s4028243227959126397er_rat
    = ( ^ [A3: rat,N: nat] :
          ( if_rat @ ( N = zero_zero_nat ) @ one_one_rat
          @ ( set_fo1949268297981939178at_rat
            @ ^ [O: nat] : ( times_times_rat @ ( plus_plus_rat @ A3 @ ( semiri681578069525770553at_rat @ O ) ) )
            @ zero_zero_nat
            @ ( minus_minus_nat @ N @ one_one_nat )
            @ one_one_rat ) ) ) ) ).

% pochhammer_code
thf(fact_8867_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_8868_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_8869_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_8870_gbinomial__partial__row__sum,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ one_one_complex ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8871_gbinomial__partial__row__sum,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ A @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( semiri681578069525770553at_rat @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( divide_divide_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ one_one_rat ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( gbinomial_rat @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8872_gbinomial__partial__row__sum,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( semiri5074537144036343181t_real @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( divide_divide_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ one_one_real ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( gbinomial_real @ A @ ( plus_plus_nat @ M @ one_one_nat ) ) ) ) ).

% gbinomial_partial_row_sum
thf(fact_8873_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_8874_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_8875_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_8876_atMost__iff,axiom,
    ! [I: rat,K: rat] :
      ( ( member_rat @ I @ ( set_ord_atMost_rat @ K ) )
      = ( ord_less_eq_rat @ I @ K ) ) ).

% atMost_iff
thf(fact_8877_atMost__iff,axiom,
    ! [I: num,K: num] :
      ( ( member_num @ I @ ( set_ord_atMost_num @ K ) )
      = ( ord_less_eq_num @ I @ K ) ) ).

% atMost_iff
thf(fact_8878_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_8879_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_8880_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_8881_of__nat__prod,axiom,
    ! [F: int > nat,A2: set_int] :
      ( ( semiri1314217659103216013at_int @ ( groups1707563613775114915nt_nat @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8882_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri5074537144036343181t_real @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X: nat] : ( semiri5074537144036343181t_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8883_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1316708129612266289at_nat @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( semiri1316708129612266289at_nat @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8884_of__nat__prod,axiom,
    ! [F: nat > nat,A2: set_nat] :
      ( ( semiri1314217659103216013at_int @ ( groups708209901874060359at_nat @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X: nat] : ( semiri1314217659103216013at_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_nat_prod
thf(fact_8885_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_real @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups129246275422532515t_real
        @ ^ [X: nat] : ( ring_1_of_int_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8886_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_rat @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups73079841787564623at_rat
        @ ^ [X: nat] : ( ring_1_of_int_rat @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8887_of__int__prod,axiom,
    ! [F: nat > int,A2: set_nat] :
      ( ( ring_1_of_int_int @ ( groups705719431365010083at_int @ F @ A2 ) )
      = ( groups705719431365010083at_int
        @ ^ [X: nat] : ( ring_1_of_int_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8888_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_real @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups2316167850115554303t_real
        @ ^ [X: int] : ( ring_1_of_int_real @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8889_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_rat @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups1072433553688619179nt_rat
        @ ^ [X: int] : ( ring_1_of_int_rat @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8890_of__int__prod,axiom,
    ! [F: int > int,A2: set_int] :
      ( ( ring_1_of_int_int @ ( groups1705073143266064639nt_int @ F @ A2 ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( ring_1_of_int_int @ ( F @ X ) )
        @ A2 ) ) ).

% of_int_prod
thf(fact_8891_prod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups6464643781859351333omplex @ F @ A2 )
          = zero_zero_complex )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_complex ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8892_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups7440179247065528705omplex @ F @ A2 )
          = zero_zero_complex )
        = ( ? [X: int] :
              ( ( member_int @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_complex ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8893_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups3708469109370488835omplex @ F @ A2 )
          = zero_zero_complex )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_complex ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8894_prod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups129246275422532515t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8895_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups2316167850115554303t_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X: int] :
              ( ( member_int @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8896_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups766887009212190081x_real @ F @ A2 )
          = zero_zero_real )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_real ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8897_prod__zero__iff,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( ( groups73079841787564623at_rat @ F @ A2 )
          = zero_zero_rat )
        = ( ? [X: nat] :
              ( ( member_nat @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8898_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1072433553688619179nt_rat @ F @ A2 )
          = zero_zero_rat )
        = ( ? [X: int] :
              ( ( member_int @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8899_prod__zero__iff,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( ( groups225925009352817453ex_rat @ F @ A2 )
          = zero_zero_rat )
        = ( ? [X: complex] :
              ( ( member_complex @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_rat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8900_prod__zero__iff,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( ( groups1707563613775114915nt_nat @ F @ A2 )
          = zero_zero_nat )
        = ( ? [X: int] :
              ( ( member_int @ X @ A2 )
              & ( ( F @ X )
                = zero_zero_nat ) ) ) ) ) ).

% prod_zero_iff
thf(fact_8901_prod_Oempty,axiom,
    ! [G: real > complex] :
      ( ( groups713298508707869441omplex @ G @ bot_bot_set_real )
      = one_one_complex ) ).

% prod.empty
thf(fact_8902_prod_Oempty,axiom,
    ! [G: real > real] :
      ( ( groups1681761925125756287l_real @ G @ bot_bot_set_real )
      = one_one_real ) ).

% prod.empty
thf(fact_8903_prod_Oempty,axiom,
    ! [G: real > rat] :
      ( ( groups4061424788464935467al_rat @ G @ bot_bot_set_real )
      = one_one_rat ) ).

% prod.empty
thf(fact_8904_prod_Oempty,axiom,
    ! [G: real > nat] :
      ( ( groups4696554848551431203al_nat @ G @ bot_bot_set_real )
      = one_one_nat ) ).

% prod.empty
thf(fact_8905_prod_Oempty,axiom,
    ! [G: real > int] :
      ( ( groups4694064378042380927al_int @ G @ bot_bot_set_real )
      = one_one_int ) ).

% prod.empty
thf(fact_8906_prod_Oempty,axiom,
    ! [G: nat > complex] :
      ( ( groups6464643781859351333omplex @ G @ bot_bot_set_nat )
      = one_one_complex ) ).

% prod.empty
thf(fact_8907_prod_Oempty,axiom,
    ! [G: nat > real] :
      ( ( groups129246275422532515t_real @ G @ bot_bot_set_nat )
      = one_one_real ) ).

% prod.empty
thf(fact_8908_prod_Oempty,axiom,
    ! [G: nat > rat] :
      ( ( groups73079841787564623at_rat @ G @ bot_bot_set_nat )
      = one_one_rat ) ).

% prod.empty
thf(fact_8909_prod_Oempty,axiom,
    ! [G: int > complex] :
      ( ( groups7440179247065528705omplex @ G @ bot_bot_set_int )
      = one_one_complex ) ).

% prod.empty
thf(fact_8910_prod_Oempty,axiom,
    ! [G: int > real] :
      ( ( groups2316167850115554303t_real @ G @ bot_bot_set_int )
      = one_one_real ) ).

% prod.empty
thf(fact_8911_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > complex] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups6464643781859351333omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_8912_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > complex] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups7440179247065528705omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_8913_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > complex] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups3708469109370488835omplex @ G @ A2 )
        = one_one_complex ) ) ).

% prod.infinite
thf(fact_8914_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > real] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups129246275422532515t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8915_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > real] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups2316167850115554303t_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8916_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > real] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups766887009212190081x_real @ G @ A2 )
        = one_one_real ) ) ).

% prod.infinite
thf(fact_8917_prod_Oinfinite,axiom,
    ! [A2: set_nat,G: nat > rat] :
      ( ~ ( finite_finite_nat @ A2 )
     => ( ( groups73079841787564623at_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8918_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > rat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1072433553688619179nt_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8919_prod_Oinfinite,axiom,
    ! [A2: set_complex,G: complex > rat] :
      ( ~ ( finite3207457112153483333omplex @ A2 )
     => ( ( groups225925009352817453ex_rat @ G @ A2 )
        = one_one_rat ) ) ).

% prod.infinite
thf(fact_8920_prod_Oinfinite,axiom,
    ! [A2: set_int,G: int > nat] :
      ( ~ ( finite_finite_int @ A2 )
     => ( ( groups1707563613775114915nt_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.infinite
thf(fact_8921_atMost__subset__iff,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X3 ) @ ( set_or58775011639299419et_int @ Y3 ) )
      = ( ord_less_eq_set_int @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8922_atMost__subset__iff,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X3 ) @ ( set_ord_atMost_rat @ Y3 ) )
      = ( ord_less_eq_rat @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8923_atMost__subset__iff,axiom,
    ! [X3: num,Y3: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X3 ) @ ( set_ord_atMost_num @ Y3 ) )
      = ( ord_less_eq_num @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8924_atMost__subset__iff,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X3 ) @ ( set_ord_atMost_nat @ Y3 ) )
      = ( ord_less_eq_nat @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8925_atMost__subset__iff,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X3 ) @ ( set_ord_atMost_int @ Y3 ) )
      = ( ord_less_eq_int @ X3 @ Y3 ) ) ).

% atMost_subset_iff
thf(fact_8926_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: nat,F: real > nat] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_nat @ B @ ( groups4696554848551431203al_nat @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8927_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: nat,F: int > nat] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_nat @ B @ ( groups1707563613775114915nt_nat @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8928_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: nat,F: complex > nat] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_nat @ B @ ( groups861055069439313189ex_nat @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8929_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: int,F: real > int] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_int @ B @ ( groups4694064378042380927al_int @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8930_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: int,F: complex > int] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_int @ B @ ( groups858564598930262913ex_int @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8931_dvd__prod__eqI,axiom,
    ! [A2: set_real,A: real,B: code_integer,F: real > code_integer] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_Code_integer @ B @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8932_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B: code_integer,F: nat > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_Code_integer @ B @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8933_dvd__prod__eqI,axiom,
    ! [A2: set_int,A: int,B: code_integer,F: int > code_integer] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_Code_integer @ B @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8934_dvd__prod__eqI,axiom,
    ! [A2: set_complex,A: complex,B: code_integer,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_Code_integer @ B @ ( groups8682486955453173170nteger @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8935_dvd__prod__eqI,axiom,
    ! [A2: set_nat,A: nat,B: nat,F: nat > nat] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( ( B
            = ( F @ A ) )
         => ( dvd_dvd_nat @ B @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ) ) ).

% dvd_prod_eqI
thf(fact_8936_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_8937_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_8938_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_8939_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_8940_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_8941_dvd__prodI,axiom,
    ! [A2: set_real,A: real,F: real > code_integer] :
      ( ( finite_finite_real @ A2 )
     => ( ( member_real @ A @ A2 )
       => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups6225526099057966256nteger @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_8942_dvd__prodI,axiom,
    ! [A2: set_nat,A: nat,F: nat > code_integer] :
      ( ( finite_finite_nat @ A2 )
     => ( ( member_nat @ A @ A2 )
       => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3455450783089532116nteger @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_8943_dvd__prodI,axiom,
    ! [A2: set_int,A: int,F: int > code_integer] :
      ( ( finite_finite_int @ A2 )
     => ( ( member_int @ A @ A2 )
       => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups3827104343326376752nteger @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_8944_dvd__prodI,axiom,
    ! [A2: set_complex,A: complex,F: complex > code_integer] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ( member_complex @ A @ A2 )
       => ( dvd_dvd_Code_integer @ ( F @ A ) @ ( groups8682486955453173170nteger @ F @ A2 ) ) ) ) ).

% dvd_prodI
thf(fact_8945_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_8946_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_8947_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_8948_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_8949_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( K3 = A ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta
thf(fact_8950_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8951_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8952_prod_Odelta,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8953_prod_Odelta,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( K3 = A ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta
thf(fact_8954_prod_Odelta,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_8955_prod_Odelta,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( K3 = A ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta
thf(fact_8956_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > complex] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups713298508707869441omplex
              @ ^ [K3: real] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_8957_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > complex] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups6464643781859351333omplex
              @ ^ [K3: nat] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_8958_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > complex] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups7440179247065528705omplex
              @ ^ [K3: int] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_8959_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > complex] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups3708469109370488835omplex
              @ ^ [K3: complex] : ( if_complex @ ( A = K3 ) @ ( B @ K3 ) @ one_one_complex )
              @ S3 )
            = one_one_complex ) ) ) ) ).

% prod.delta'
thf(fact_8960_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > real] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups1681761925125756287l_real
              @ ^ [K3: real] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8961_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > real] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups129246275422532515t_real
              @ ^ [K3: nat] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8962_prod_Odelta_H,axiom,
    ! [S3: set_int,A: int,B: int > real] :
      ( ( finite_finite_int @ S3 )
     => ( ( ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_int @ A @ S3 )
         => ( ( groups2316167850115554303t_real
              @ ^ [K3: int] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8963_prod_Odelta_H,axiom,
    ! [S3: set_complex,A: complex,B: complex > real] :
      ( ( finite3207457112153483333omplex @ S3 )
     => ( ( ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_complex @ A @ S3 )
         => ( ( groups766887009212190081x_real
              @ ^ [K3: complex] : ( if_real @ ( A = K3 ) @ ( B @ K3 ) @ one_one_real )
              @ S3 )
            = one_one_real ) ) ) ) ).

% prod.delta'
thf(fact_8964_prod_Odelta_H,axiom,
    ! [S3: set_real,A: real,B: real > rat] :
      ( ( finite_finite_real @ S3 )
     => ( ( ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_real @ A @ S3 )
         => ( ( groups4061424788464935467al_rat
              @ ^ [K3: real] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_8965_prod_Odelta_H,axiom,
    ! [S3: set_nat,A: nat,B: nat > rat] :
      ( ( finite_finite_nat @ S3 )
     => ( ( ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = ( B @ A ) ) )
        & ( ~ ( member_nat @ A @ S3 )
         => ( ( groups73079841787564623at_rat
              @ ^ [K3: nat] : ( if_rat @ ( A = K3 ) @ ( B @ K3 ) @ one_one_rat )
              @ S3 )
            = one_one_rat ) ) ) ) ).

% prod.delta'
thf(fact_8966_prod_Oinsert,axiom,
    ! [A2: set_real,X3: real,G: real > complex] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X3 @ A2 )
       => ( ( groups713298508707869441omplex @ G @ ( insert_real @ X3 @ A2 ) )
          = ( times_times_complex @ ( G @ X3 ) @ ( groups713298508707869441omplex @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8967_prod_Oinsert,axiom,
    ! [A2: set_nat,X3: nat,G: nat > complex] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X3 @ A2 )
       => ( ( groups6464643781859351333omplex @ G @ ( insert_nat @ X3 @ A2 ) )
          = ( times_times_complex @ ( G @ X3 ) @ ( groups6464643781859351333omplex @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8968_prod_Oinsert,axiom,
    ! [A2: set_int,X3: int,G: int > complex] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X3 @ A2 )
       => ( ( groups7440179247065528705omplex @ G @ ( insert_int @ X3 @ A2 ) )
          = ( times_times_complex @ ( G @ X3 ) @ ( groups7440179247065528705omplex @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8969_prod_Oinsert,axiom,
    ! [A2: set_complex,X3: complex,G: complex > complex] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X3 @ A2 )
       => ( ( groups3708469109370488835omplex @ G @ ( insert_complex @ X3 @ A2 ) )
          = ( times_times_complex @ ( G @ X3 ) @ ( groups3708469109370488835omplex @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8970_prod_Oinsert,axiom,
    ! [A2: set_real,X3: real,G: real > real] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X3 @ A2 )
       => ( ( groups1681761925125756287l_real @ G @ ( insert_real @ X3 @ A2 ) )
          = ( times_times_real @ ( G @ X3 ) @ ( groups1681761925125756287l_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8971_prod_Oinsert,axiom,
    ! [A2: set_nat,X3: nat,G: nat > real] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X3 @ A2 )
       => ( ( groups129246275422532515t_real @ G @ ( insert_nat @ X3 @ A2 ) )
          = ( times_times_real @ ( G @ X3 ) @ ( groups129246275422532515t_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8972_prod_Oinsert,axiom,
    ! [A2: set_int,X3: int,G: int > real] :
      ( ( finite_finite_int @ A2 )
     => ( ~ ( member_int @ X3 @ A2 )
       => ( ( groups2316167850115554303t_real @ G @ ( insert_int @ X3 @ A2 ) )
          = ( times_times_real @ ( G @ X3 ) @ ( groups2316167850115554303t_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8973_prod_Oinsert,axiom,
    ! [A2: set_complex,X3: complex,G: complex > real] :
      ( ( finite3207457112153483333omplex @ A2 )
     => ( ~ ( member_complex @ X3 @ A2 )
       => ( ( groups766887009212190081x_real @ G @ ( insert_complex @ X3 @ A2 ) )
          = ( times_times_real @ ( G @ X3 ) @ ( groups766887009212190081x_real @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8974_prod_Oinsert,axiom,
    ! [A2: set_real,X3: real,G: real > rat] :
      ( ( finite_finite_real @ A2 )
     => ( ~ ( member_real @ X3 @ A2 )
       => ( ( groups4061424788464935467al_rat @ G @ ( insert_real @ X3 @ A2 ) )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups4061424788464935467al_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8975_prod_Oinsert,axiom,
    ! [A2: set_nat,X3: nat,G: nat > rat] :
      ( ( finite_finite_nat @ A2 )
     => ( ~ ( member_nat @ X3 @ A2 )
       => ( ( groups73079841787564623at_rat @ G @ ( insert_nat @ X3 @ A2 ) )
          = ( times_times_rat @ ( G @ X3 ) @ ( groups73079841787564623at_rat @ G @ A2 ) ) ) ) ) ).

% prod.insert
thf(fact_8976_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_8977_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H2: rat,H3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H2 ) @ ( set_ord_atMost_rat @ H3 ) )
      = ( ~ ( ord_less_eq_rat @ L @ H2 )
        | ( ord_less_eq_rat @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8978_Icc__subset__Iic__iff,axiom,
    ! [L: num,H2: num,H3: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H2 ) @ ( set_ord_atMost_num @ H3 ) )
      = ( ~ ( ord_less_eq_num @ L @ H2 )
        | ( ord_less_eq_num @ H2 @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_8979_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_8980_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_8981_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_8982_sum_OatMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% sum.atMost_Suc
thf(fact_8983_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_8984_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_8985_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_8986_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_8987_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_8988_prod_OlessThan__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_lessThan_nat @ N2 ) ) @ ( G @ N2 ) ) ) ).

% prod.lessThan_Suc
thf(fact_8989_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_8990_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_8991_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_8992_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_8993_prod_OatMost__Suc,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_ord_atMost_nat @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ).

% prod.atMost_Suc
thf(fact_8994_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_8995_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_8996_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_8997_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > complex] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_complex ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_complex @ ( groups6464643781859351333omplex @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8998_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > real] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_real @ ( groups129246275422532515t_real @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_8999_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > rat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_rat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_rat @ ( groups73079841787564623at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_9000_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > nat] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_nat ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_nat @ ( groups708209901874060359at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_9001_prod_Ocl__ivl__Suc,axiom,
    ! [N2: nat,M: nat,G: nat > int] :
      ( ( ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = one_one_int ) )
      & ( ~ ( ord_less_nat @ ( suc @ N2 ) @ M )
       => ( ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( suc @ N2 ) ) )
          = ( times_times_int @ ( groups705719431365010083at_int @ G @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) @ ( G @ ( suc @ N2 ) ) ) ) ) ) ).

% prod.cl_ivl_Suc
thf(fact_9002_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_9003_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_9004_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9005_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S7: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S7 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_9006_sum__choose__upper,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ K3 @ M )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ N2 ) @ ( suc @ M ) ) ) ).

% sum_choose_upper
thf(fact_9007_sum__choose__lower,axiom,
    ! [R2: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R2 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R2 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_9008_choose__rising__sum_I1_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) ).

% choose_rising_sum(1)
thf(fact_9009_choose__rising__sum_I2_J,axiom,
    ! [N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [J3: nat] : ( binomial @ ( plus_plus_nat @ N2 @ J3 ) @ N2 )
        @ ( set_ord_atMost_nat @ M ) )
      = ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ N2 @ M ) @ one_one_nat ) @ M ) ) ).

% choose_rising_sum(2)
thf(fact_9010_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_9011_fact__eq__fact__times,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( semiri1408675320244567234ct_nat @ M )
        = ( times_times_nat @ ( semiri1408675320244567234ct_nat @ N2 )
          @ ( groups708209901874060359at_nat
            @ ^ [X: nat] : X
            @ ( set_or1269000886237332187st_nat @ ( suc @ N2 ) @ M ) ) ) ) ) ).

% fact_eq_fact_times
thf(fact_9012_sum__choose__diagonal,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3542108847815614940at_nat
          @ ^ [K3: nat] : ( binomial @ ( minus_minus_nat @ N2 @ K3 ) @ ( minus_minus_nat @ M @ K3 ) )
          @ ( set_ord_atMost_nat @ M ) )
        = ( binomial @ ( suc @ N2 ) @ M ) ) ) ).

% sum_choose_diagonal
thf(fact_9013_vandermonde,axiom,
    ! [M: nat,N2: nat,R2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R2 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R2 ) ) ).

% vandermonde
thf(fact_9014_fact__div__fact,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ M ) @ ( semiri1408675320244567234ct_nat @ N2 ) )
        = ( groups708209901874060359at_nat
          @ ^ [X: nat] : X
          @ ( set_or1269000886237332187st_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) ) ).

% fact_div_fact
thf(fact_9015_binomial,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ A @ B ) @ N2 )
      = ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_nat @ A @ K3 ) ) @ ( power_power_nat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial
thf(fact_9016_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X3: nat] :
      ( ! [I3: nat] :
          ( ( ord_less_nat @ M @ I3 )
         => ( ( A @ I3 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( power_power_nat @ X3 @ I2 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X3 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3542108847815614940at_nat
            @ ^ [R5: nat] :
                ( times_times_nat
                @ ( groups3542108847815614940at_nat
                  @ ^ [K3: nat] : ( times_times_nat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_nat @ X3 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_9017_binomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) @ one_one_nat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% binomial_r_part_sum
thf(fact_9018_cot__less__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X3 )
     => ( ( ord_less_real @ X3 @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X3 ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_9019_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z6: int] :
                ( ( ord_less_eq_int @ D5 @ Z6 )
                & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9020_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_9021_prod__int__plus__eq,axiom,
    ! [I: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I @ ( plus_plus_nat @ I @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9022_cot__gt__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X3 ) ) ) ) ).

% cot_gt_zero
thf(fact_9023_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z6: int] :
                ( ( ord_less_eq_int @ D5 @ Z7 )
                & ( ord_less_int @ Z7 @ Z6 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9024_arctan__def,axiom,
    ( arctan
    = ( ^ [Y: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( tan_real @ X )
                = Y ) ) ) ) ) ).

% arctan_def
thf(fact_9025_modulo__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: 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 @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
          = ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) ) )
      & ( ~ ( ( ( 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 @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( modulo_modulo_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( times_times_int @ ( sgn_sgn_int @ L )
                @ ( minus_minus_int
                  @ ( semiri1314217659103216013at_int
                    @ ( times_times_nat @ N2
                      @ ( zero_n2687167440665602831ol_nat
                        @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) )
                  @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ) ) ) ) ) ).

% modulo_int_unfold
thf(fact_9026_powr__int,axiom,
    ! [X3: real,I: int] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
            = ( power_power_real @ X3 @ ( nat2 @ I ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I )
         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ I ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ I ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_9027_nat__int,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( semiri1314217659103216013at_int @ N2 ) )
      = N2 ) ).

% nat_int
thf(fact_9028_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_9029_nat__of__bool,axiom,
    ! [P: $o] :
      ( ( nat2 @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% nat_of_bool
thf(fact_9030_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_9031_nat__0__iff,axiom,
    ! [I: int] :
      ( ( ( nat2 @ I )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_9032_nat__le__0,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
     => ( ( nat2 @ Z2 )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_9033_zless__nat__conj,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
      = ( ( ord_less_int @ zero_zero_int @ Z2 )
        & ( ord_less_int @ W2 @ Z2 ) ) ) ).

% zless_nat_conj
thf(fact_9034_nat__neg__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = zero_zero_nat ) ).

% nat_neg_numeral
thf(fact_9035_nat__zminus__int,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = zero_zero_nat ) ).

% nat_zminus_int
thf(fact_9036_int__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = Z2 ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_9037_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_9038_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_9039_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_9040_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_9041_zero__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ zero_zero_int @ Z2 ) ) ).

% zero_less_nat_eq
thf(fact_9042_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_9043_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X3: num,N2: nat,Y3: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 )
        = ( nat2 @ Y3 ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 )
        = Y3 ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_9044_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y3: int,X3: num,N2: nat] :
      ( ( ( nat2 @ Y3 )
        = ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) )
      = ( Y3
        = ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_9045_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_9046_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_9047_nat__ceiling__le__eq,axiom,
    ! [X3: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X3 ) ) @ A )
      = ( ord_less_eq_real @ X3 @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_9048_one__less__nat__eq,axiom,
    ! [Z2: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ one_one_int @ Z2 ) ) ).

% one_less_nat_eq
thf(fact_9049_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_9050_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_9051_numeral__power__less__nat__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_9052_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X3: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_9053_numeral__power__le__nat__cancel__iff,axiom,
    ! [X3: num,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X3 ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X3 ) @ N2 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_9054_nat__zero__as__int,axiom,
    ( zero_zero_nat
    = ( nat2 @ zero_zero_int ) ) ).

% nat_zero_as_int
thf(fact_9055_nat__mono,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ X3 @ Y3 )
     => ( ord_less_eq_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ).

% nat_mono
thf(fact_9056_int__sgnE,axiom,
    ! [K: int] :
      ~ ! [N3: nat,L4: int] :
          ( K
         != ( times_times_int @ ( sgn_sgn_int @ L4 ) @ ( semiri1314217659103216013at_int @ N3 ) ) ) ).

% int_sgnE
thf(fact_9057_ex__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ? [X6: nat] : ( P2 @ X6 ) )
    = ( ^ [P3: nat > $o] :
        ? [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
          & ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% ex_nat
thf(fact_9058_all__nat,axiom,
    ( ( ^ [P2: nat > $o] :
        ! [X6: nat] : ( P2 @ X6 ) )
    = ( ^ [P3: nat > $o] :
        ! [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( P3 @ ( nat2 @ X ) ) ) ) ) ).

% all_nat
thf(fact_9059_eq__nat__nat__iff,axiom,
    ! [Z2: int,Z8: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
       => ( ( ( nat2 @ Z2 )
            = ( nat2 @ Z8 ) )
          = ( Z2 = Z8 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_9060_nat__mono__iff,axiom,
    ! [Z2: int,W2: int] :
      ( ( ord_less_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W2 @ Z2 ) ) ) ).

% nat_mono_iff
thf(fact_9061_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z2: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z2 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z2 ) ) ).

% zless_nat_eq_int_zless
thf(fact_9062_nat__le__iff,axiom,
    ! [X3: int,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X3 ) @ N2 )
      = ( ord_less_eq_int @ X3 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_9063_nat__0__le,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z2 ) )
        = Z2 ) ) ).

% nat_0_le
thf(fact_9064_int__eq__iff,axiom,
    ! [M: nat,Z2: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z2 )
      = ( ( M
          = ( nat2 @ Z2 ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z2 ) ) ) ).

% int_eq_iff
thf(fact_9065_nat__int__add,axiom,
    ! [A: nat,B: nat] :
      ( ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) )
      = ( plus_plus_nat @ A @ B ) ) ).

% nat_int_add
thf(fact_9066_nat__abs__mult__distrib,axiom,
    ! [W2: int,Z2: int] :
      ( ( nat2 @ ( abs_abs_int @ ( times_times_int @ W2 @ Z2 ) ) )
      = ( times_times_nat @ ( nat2 @ ( abs_abs_int @ W2 ) ) @ ( nat2 @ ( abs_abs_int @ Z2 ) ) ) ) ).

% nat_abs_mult_distrib
thf(fact_9067_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_9068_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I2: int] : ( if_int @ ( I2 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I2 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_9069_powr__real__of__int,axiom,
    ! [X3: real,N2: int] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N2 ) )
            = ( power_power_real @ X3 @ ( nat2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N2 ) )
            = ( inverse_inverse_real @ ( power_power_real @ X3 @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_9070_nat__less__eq__zless,axiom,
    ! [W2: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
        = ( ord_less_int @ W2 @ Z2 ) ) ) ).

% nat_less_eq_zless
thf(fact_9071_nat__le__eq__zle,axiom,
    ! [W2: int,Z2: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W2 )
        | ( ord_less_eq_int @ zero_zero_int @ Z2 ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W2 ) @ ( nat2 @ Z2 ) )
        = ( ord_less_eq_int @ W2 @ Z2 ) ) ) ).

% nat_le_eq_zle
thf(fact_9072_nat__eq__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ( nat2 @ W2 )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_9073_nat__eq__iff2,axiom,
    ! [M: nat,W2: int] :
      ( ( M
        = ( nat2 @ W2 ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( W2
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_9074_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_9075_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_9076_nat__add__distrib,axiom,
    ! [Z2: int,Z8: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
       => ( ( nat2 @ ( plus_plus_int @ Z2 @ Z8 ) )
          = ( plus_plus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_9077_nat__mult__distrib,axiom,
    ! [Z2: int,Z8: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( nat2 @ ( times_times_int @ Z2 @ Z8 ) )
        = ( times_times_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ).

% nat_mult_distrib
thf(fact_9078_Suc__as__int,axiom,
    ( suc
    = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_9079_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D4: real,E: real] :
          ( ( ord_less_real @ D4 @ E )
         => ( ( P @ D4 )
           => ( 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_9080_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D4: real,E: real] :
          ( ( ord_less_real @ D4 @ E )
         => ( ( P @ D4 )
           => ( 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_9081_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_9082_nat__diff__distrib,axiom,
    ! [Z8: int,Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z8 )
     => ( ( ord_less_eq_int @ Z8 @ Z2 )
       => ( ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) )
          = ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_9083_nat__diff__distrib_H,axiom,
    ! [X3: int,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
       => ( ( nat2 @ ( minus_minus_int @ X3 @ Y3 ) )
          = ( minus_minus_nat @ ( nat2 @ X3 ) @ ( nat2 @ Y3 ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_9084_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_9085_nat__floor__neg,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_9086_nat__power__eq,axiom,
    ! [Z2: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( nat2 @ ( power_power_int @ Z2 @ N2 ) )
        = ( power_power_nat @ ( nat2 @ Z2 ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_9087_ln__inverse,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X3 ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X3 ) ) ) ) ).

% ln_inverse
thf(fact_9088_floor__eq3,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
          = N2 ) ) ) ).

% floor_eq3
thf(fact_9089_le__nat__floor,axiom,
    ! [X3: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X3 ) @ A )
     => ( ord_less_eq_nat @ X3 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_9090_divide__int__def,axiom,
    ( divide_divide_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ zero_zero_int
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) )
            @ ( uminus_uminus_int
              @ ( semiri1314217659103216013at_int
                @ ( plus_plus_nat @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) )
                  @ ( zero_n2687167440665602831ol_nat
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) ) ) ) ) ) ) ) ).

% divide_int_def
thf(fact_9091_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_9092_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z2: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
     => ( ( suc @ ( nat2 @ Z2 ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z2 ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_9093_nat__less__iff,axiom,
    ! [W2: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W2 )
     => ( ( ord_less_nat @ ( nat2 @ W2 ) @ M )
        = ( ord_less_int @ W2 @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_9094_nat__mult__distrib__neg,axiom,
    ! [Z2: int,Z8: int] :
      ( ( ord_less_eq_int @ Z2 @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z2 @ Z8 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z2 ) ) @ ( nat2 @ ( uminus_uminus_int @ Z8 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_9095_nat__abs__int__diff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ B @ A ) ) )
      & ( ~ ( ord_less_eq_nat @ A @ B )
       => ( ( nat2 @ ( abs_abs_int @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) )
          = ( minus_minus_nat @ A @ B ) ) ) ) ).

% nat_abs_int_diff
thf(fact_9096_log__inverse,axiom,
    ! [A: real,X3: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( log @ A @ ( inverse_inverse_real @ X3 ) )
            = ( uminus_uminus_real @ ( log @ A @ X3 ) ) ) ) ) ) ).

% log_inverse
thf(fact_9097_floor__eq4,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X3 )
     => ( ( ord_less_real @ X3 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X3 ) )
          = N2 ) ) ) ).

% floor_eq4
thf(fact_9098_diff__nat__eq__if,axiom,
    ! [Z8: int,Z2: int] :
      ( ( ( ord_less_int @ Z8 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
          = ( nat2 @ Z2 ) ) )
      & ( ~ ( ord_less_int @ Z8 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z2 ) @ ( nat2 @ Z8 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z2 @ Z8 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z2 @ Z8 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_9099_nat__dvd__iff,axiom,
    ! [Z2: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z2 ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z2 )
         => ( dvd_dvd_int @ Z2 @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z2 )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_9100_eucl__rel__int__remainderI,axiom,
    ! [R2: int,L: int,K: int,Q4: int] :
      ( ( ( sgn_sgn_int @ R2 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q4 @ L ) @ R2 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q4 @ R2 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_9101_plus__inverse__ge__2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_9102_real__inv__sqrt__pow2,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X3 ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_9103_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A32: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A32 )
     => ( ( ( A23 = zero_zero_int )
         => ( A32
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q2: int] :
              ( ( A32
                = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q2 @ A23 ) ) ) )
         => ~ ! [R4: int,Q2: int] :
                ( ( A32
                  = ( product_Pair_int_int @ Q2 @ R4 ) )
               => ( ( ( sgn_sgn_int @ R4 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R4 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q2 @ A23 ) @ R4 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_9104_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A33: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A1 = K3 )
              & ( A22 = zero_zero_int )
              & ( A33
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q5: int] :
              ( ( A1 = K3 )
              & ( A22 = L2 )
              & ( A33
                = ( product_Pair_int_int @ Q5 @ zero_zero_int ) )
              & ( L2 != zero_zero_int )
              & ( K3
                = ( times_times_int @ Q5 @ L2 ) ) )
          | ? [R5: int,L2: int,K3: int,Q5: int] :
              ( ( A1 = K3 )
              & ( A22 = L2 )
              & ( A33
                = ( product_Pair_int_int @ Q5 @ R5 ) )
              & ( ( sgn_sgn_int @ R5 )
                = ( sgn_sgn_int @ L2 ) )
              & ( ord_less_int @ ( abs_abs_int @ R5 ) @ ( abs_abs_int @ L2 ) )
              & ( K3
                = ( plus_plus_int @ ( times_times_int @ Q5 @ L2 ) @ R5 ) ) ) ) ) ) ).

% eucl_rel_int.simps
thf(fact_9105_divide__int__unfold,axiom,
    ! [L: int,K: int,N2: nat,M: 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 @ M ) ) @ ( 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 @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) )
          & ( ( ( sgn_sgn_int @ K )
             != ( sgn_sgn_int @ L ) )
           => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( semiri1314217659103216013at_int @ M ) ) @ ( times_times_int @ ( sgn_sgn_int @ L ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
              = ( uminus_uminus_int
                @ ( semiri1314217659103216013at_int
                  @ ( plus_plus_nat @ ( divide_divide_nat @ M @ N2 )
                    @ ( zero_n2687167440665602831ol_nat
                      @ ~ ( dvd_dvd_nat @ N2 @ M ) ) ) ) ) ) ) ) ) ) ).

% divide_int_unfold
thf(fact_9106_sinh__ln__real,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( sinh_real @ ( ln_ln_real @ X3 ) )
        = ( divide_divide_real @ ( minus_minus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_9107_arcsin__lt__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y3 ) )
          & ( ord_less_real @ ( arcsin @ Y3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_9108_sinh__real__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ ( sinh_real @ X3 ) @ ( sinh_real @ Y3 ) )
      = ( ord_less_real @ X3 @ Y3 ) ) ).

% sinh_real_less_iff
thf(fact_9109_sinh__real__neg__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( sinh_real @ X3 ) @ zero_zero_real )
      = ( ord_less_real @ X3 @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_9110_sinh__real__pos__iff,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X3 ) )
      = ( ord_less_real @ zero_zero_real @ X3 ) ) ).

% sinh_real_pos_iff
thf(fact_9111_sinh__less__cosh__real,axiom,
    ! [X3: real] : ( ord_less_real @ ( sinh_real @ X3 ) @ ( cosh_real @ X3 ) ) ).

% sinh_less_cosh_real
thf(fact_9112_cosh__real__pos,axiom,
    ! [X3: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X3 ) ) ).

% cosh_real_pos
thf(fact_9113_cosh__real__strict__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_9114_cosh__real__nonneg__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_9115_cosh__real__nonpos__less__iff,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ X3 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y3 @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X3 ) @ ( cosh_real @ Y3 ) )
          = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_9116_sgn__real__def,axiom,
    ( sgn_sgn_real
    = ( ^ [A3: real] : ( if_real @ ( A3 = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ A3 ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_real_def
thf(fact_9117_arcsin__less__arcsin,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_9118_arcsin__less__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X3 ) @ ( arcsin @ Y3 ) )
          = ( ord_less_real @ X3 @ Y3 ) ) ) ) ).

% arcsin_less_mono
thf(fact_9119_sgn__power__injE,axiom,
    ! [A: real,N2: nat,X3: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
        = X3 )
     => ( ( X3
          = ( times_times_real @ ( sgn_sgn_real @ B ) @ ( power_power_real @ ( abs_abs_real @ B ) @ N2 ) ) )
       => ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ( A = B ) ) ) ) ).

% sgn_power_injE
thf(fact_9120_cos__arcsin__nonzero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X3 ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_9121_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X: real] :
          ( the_int
          @ ^ [Z6: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z6 ) @ X )
              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9122_cosh__ln__real,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( ( cosh_real @ ( ln_ln_real @ X3 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X3 @ ( inverse_inverse_real @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_9123_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X: rat] :
          ( the_int
          @ ^ [Z6: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z6 ) @ X )
              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z6 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9124_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_9125_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_9126_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_9127_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9128_sgn__rat__def,axiom,
    ( sgn_sgn_rat
    = ( ^ [A3: rat] : ( if_rat @ ( A3 = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ A3 ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_rat_def
thf(fact_9129_abs__rat__def,axiom,
    ( abs_abs_rat
    = ( ^ [A3: rat] : ( if_rat @ ( ord_less_rat @ A3 @ zero_zero_rat ) @ ( uminus_uminus_rat @ A3 ) @ A3 ) ) ) ).

% abs_rat_def
thf(fact_9130_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y: rat] :
          ( ( ord_less_rat @ X @ Y )
          | ( X = Y ) ) ) ) ).

% less_eq_rat_def
thf(fact_9131_obtain__pos__sum,axiom,
    ! [R2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R2 )
     => ~ ! [S2: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S2 )
           => ! [T6: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T6 )
               => ( R2
                 != ( plus_plus_rat @ S2 @ T6 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9132_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M2: nat] :
            ( ( ord_less_nat @ M2 @ N2 )
            & ( P @ M2 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9133_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M2: nat] :
            ( ( ord_less_nat @ M2 @ N2 )
           => ( P @ M2 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_9134_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9135_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9136_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9137_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_9138_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N7 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9139_atLeastLessThanSuc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = ( insert_nat @ N2 @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ N2 )
       => ( ( set_or4665077453230672383an_nat @ M @ ( suc @ N2 ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThanSuc
thf(fact_9140_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_9141_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_9142_atLeastLessThan__nat__numeral,axiom,
    ! [M: nat,K: num] :
      ( ( ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = ( insert_nat @ ( pred_numeral @ K ) @ ( set_or4665077453230672383an_nat @ M @ ( pred_numeral @ K ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ M @ ( pred_numeral @ K ) )
       => ( ( set_or4665077453230672383an_nat @ M @ ( numeral_numeral_nat @ K ) )
          = bot_bot_set_nat ) ) ) ).

% atLeastLessThan_nat_numeral
thf(fact_9143_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_9144_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A: nat > nat,B: nat > nat] :
      ( ! [I3: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I3 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_nat @ ( A @ I3 ) @ ( A @ J2 ) ) ) )
     => ( ! [I3: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I3 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I3 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N2
            @ ( groups3542108847815614940at_nat
              @ ^ [I2: nat] : ( times_times_nat @ ( A @ I2 ) @ ( B @ I2 ) )
              @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) )
          @ ( times_times_nat @ ( groups3542108847815614940at_nat @ A @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ B @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ) ) ) ).

% Chebyshev_sum_upper_nat
thf(fact_9145_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_9146_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_9147_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_9148_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_9149_rat__inverse__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( inverse_inverse_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,B3: int] : ( if_Pro3027730157355071871nt_int @ ( A3 = zero_zero_int ) @ ( product_Pair_int_int @ zero_zero_int @ one_one_int ) @ ( product_Pair_int_int @ ( times_times_int @ ( sgn_sgn_int @ A3 ) @ B3 ) @ ( abs_abs_int @ A3 ) ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_inverse_code
thf(fact_9150_normalize__negative,axiom,
    ! [Q4: int,P4: int] :
      ( ( ord_less_int @ Q4 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P4 ) @ ( uminus_uminus_int @ Q4 ) ) ) ) ) ).

% normalize_negative
thf(fact_9151_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X5: nat > real] :
        ! [J3: nat] :
        ? [M8: nat] :
        ! [M2: nat] :
          ( ( ord_less_eq_nat @ M8 @ M2 )
         => ! [N: nat] :
              ( ( ord_less_eq_nat @ M8 @ N )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X5 @ M2 ) @ ( X5 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_9152_vebt__mint_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_mint @ X3 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ X3 )
       => ( ! [A4: $o,B4: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( ( A4
                   => ( Y3
                      = ( some_nat @ zero_zero_nat ) ) )
                  & ( ~ A4
                   => ( ( B4
                       => ( Y3
                          = ( some_nat @ one_one_nat ) ) )
                      & ( ~ B4
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y3
                      = ( some_nat @ Mi2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_mint_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_mint.pelims
thf(fact_9153_quotient__of__number_I3_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( numeral_numeral_rat @ K ) )
      = ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) ) ).

% quotient_of_number(3)
thf(fact_9154_normalize__denom__zero,axiom,
    ! [P4: int] :
      ( ( normalize @ ( product_Pair_int_int @ P4 @ zero_zero_int ) )
      = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% normalize_denom_zero
thf(fact_9155_rat__one__code,axiom,
    ( ( quotient_of @ one_one_rat )
    = ( product_Pair_int_int @ one_one_int @ one_one_int ) ) ).

% rat_one_code
thf(fact_9156_rat__zero__code,axiom,
    ( ( quotient_of @ zero_zero_rat )
    = ( product_Pair_int_int @ zero_zero_int @ one_one_int ) ) ).

% rat_zero_code
thf(fact_9157_quotient__of__number_I5_J,axiom,
    ! [K: num] :
      ( ( quotient_of @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( product_Pair_int_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) ).

% quotient_of_number(5)
thf(fact_9158_quotient__of__number_I4_J,axiom,
    ( ( quotient_of @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( product_Pair_int_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int ) ) ).

% quotient_of_number(4)
thf(fact_9159_rat__divide__code,axiom,
    ! [P4: rat,Q4: rat] :
      ( ( quotient_of @ ( divide_divide_rat @ P4 @ Q4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ C3 @ B3 ) ) )
            @ ( quotient_of @ Q4 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_divide_code
thf(fact_9160_rat__times__code,axiom,
    ! [P4: rat,Q4: rat] :
      ( ( quotient_of @ ( times_times_rat @ P4 @ Q4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( times_times_int @ A3 @ B3 ) @ ( times_times_int @ C3 @ D5 ) ) )
            @ ( quotient_of @ Q4 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_times_code
thf(fact_9161_quotient__of__div,axiom,
    ! [R2: rat,N2: int,D: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ N2 @ D ) )
     => ( R2
        = ( divide_divide_rat @ ( ring_1_of_int_rat @ N2 ) @ ( ring_1_of_int_rat @ D ) ) ) ) ).

% quotient_of_div
thf(fact_9162_rat__minus__code,axiom,
    ! [P4: rat,Q4: rat] :
      ( ( quotient_of @ ( minus_minus_rat @ P4 @ Q4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( minus_minus_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D5 ) ) )
            @ ( quotient_of @ Q4 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_minus_code
thf(fact_9163_quotient__of__denom__pos,axiom,
    ! [R2: rat,P4: int,Q4: int] :
      ( ( ( quotient_of @ R2 )
        = ( product_Pair_int_int @ P4 @ Q4 ) )
     => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).

% quotient_of_denom_pos
thf(fact_9164_rat__plus__code,axiom,
    ! [P4: rat,Q4: rat] :
      ( ( quotient_of @ ( plus_plus_rat @ P4 @ Q4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int,C3: int] :
            ( produc4245557441103728435nt_int
            @ ^ [B3: int,D5: int] : ( normalize @ ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ B3 @ C3 ) ) @ ( times_times_int @ C3 @ D5 ) ) )
            @ ( quotient_of @ Q4 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_plus_code
thf(fact_9165_normalize__denom__pos,axiom,
    ! [R2: product_prod_int_int,P4: int,Q4: int] :
      ( ( ( normalize @ R2 )
        = ( product_Pair_int_int @ P4 @ Q4 ) )
     => ( ord_less_int @ zero_zero_int @ Q4 ) ) ).

% normalize_denom_pos
thf(fact_9166_normalize__crossproduct,axiom,
    ! [Q4: int,S: int,P4: int,R2: int] :
      ( ( Q4 != zero_zero_int )
     => ( ( S != zero_zero_int )
       => ( ( ( normalize @ ( product_Pair_int_int @ P4 @ Q4 ) )
            = ( normalize @ ( product_Pair_int_int @ R2 @ S ) ) )
         => ( ( times_times_int @ P4 @ S )
            = ( times_times_int @ R2 @ Q4 ) ) ) ) ) ).

% normalize_crossproduct
thf(fact_9167_rat__uminus__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( uminus_uminus_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( uminus_uminus_int @ A3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_uminus_code
thf(fact_9168_rat__less__code,axiom,
    ( ord_less_rat
    = ( ^ [P6: rat,Q5: rat] :
          ( produc4947309494688390418_int_o
          @ ^ [A3: int,C3: int] :
              ( produc4947309494688390418_int_o
              @ ^ [B3: int,D5: int] : ( ord_less_int @ ( times_times_int @ A3 @ D5 ) @ ( times_times_int @ C3 @ B3 ) )
              @ ( quotient_of @ Q5 ) )
          @ ( quotient_of @ P6 ) ) ) ) ).

% rat_less_code
thf(fact_9169_rat__abs__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( abs_abs_rat @ P4 ) )
      = ( produc4245557441103728435nt_int
        @ ^ [A3: int] : ( product_Pair_int_int @ ( abs_abs_int @ A3 ) )
        @ ( quotient_of @ P4 ) ) ) ).

% rat_abs_code
thf(fact_9170_vebt__maxt_Opelims,axiom,
    ! [X3: vEBT_VEBT,Y3: option_nat] :
      ( ( ( vEBT_vebt_maxt @ X3 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ X3 )
       => ( ! [A4: $o,B4: $o] :
              ( ( X3
                = ( vEBT_Leaf @ A4 @ B4 ) )
             => ( ( ( B4
                   => ( Y3
                      = ( some_nat @ one_one_nat ) ) )
                  & ( ~ B4
                   => ( ( A4
                       => ( Y3
                          = ( some_nat @ zero_zero_nat ) ) )
                      & ( ~ A4
                       => ( Y3 = none_nat ) ) ) ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Leaf @ A4 @ B4 ) ) ) )
         => ( ! [Uu2: nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) )
               => ( ( Y3 = none_nat )
                 => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uu2 @ Uv2 @ Uw2 ) ) ) )
           => ~ ! [Mi2: nat,Ma2: nat,Ux2: nat,Uy2: list_VEBT_VEBT,Uz2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) )
                 => ( ( Y3
                      = ( some_nat @ Ma2 ) )
                   => ~ ( accp_VEBT_VEBT @ vEBT_vebt_maxt_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ Ux2 @ Uy2 @ Uz2 ) ) ) ) ) ) ) ) ).

% vebt_maxt.pelims
thf(fact_9171_quotient__of__int,axiom,
    ! [A: int] :
      ( ( quotient_of @ ( of_int @ A ) )
      = ( product_Pair_int_int @ A @ one_one_int ) ) ).

% quotient_of_int
thf(fact_9172_VEBT__internal_OminNull_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Y3: $o] :
      ( ( ( vEBT_VEBT_minNull @ X3 )
        = Y3 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
       => ( ( ( X3
              = ( vEBT_Leaf @ $false @ $false ) )
           => ( Y3
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) ) )
         => ( ! [Uv2: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ $true @ Uv2 ) )
               => ( ~ Y3
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) ) )
           => ( ! [Uu2: $o] :
                  ( ( X3
                    = ( vEBT_Leaf @ Uu2 @ $true ) )
                 => ( ~ Y3
                   => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) ) )
             => ( ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                    ( ( X3
                      = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
                   => ( Y3
                     => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) )
               => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                      ( ( X3
                        = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                     => ( ~ Y3
                       => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(1)
thf(fact_9173_VEBT__internal_OminNull_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ( vEBT_VEBT_minNull @ X3 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
       => ( ( ( X3
              = ( vEBT_Leaf @ $false @ $false ) )
           => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Uw2: nat,Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X3
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Uw2 @ Ux2 @ Uy2 ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(2)
thf(fact_9174_VEBT__internal_OminNull_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT] :
      ( ~ ( vEBT_VEBT_minNull @ X3 )
     => ( ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ X3 )
       => ( ! [Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ $true @ Uv2 ) )
             => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ $true @ Uv2 ) ) )
         => ( ! [Uu2: $o] :
                ( ( X3
                  = ( vEBT_Leaf @ Uu2 @ $true ) )
               => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Leaf @ Uu2 @ $true ) ) )
           => ~ ! [Uz2: product_prod_nat_nat,Va3: nat,Vb2: list_VEBT_VEBT,Vc2: vEBT_VEBT] :
                  ( ( X3
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) )
                 => ~ ( accp_VEBT_VEBT @ vEBT_V6963167321098673237ll_rel @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ Uz2 ) @ Va3 @ Vb2 @ Vc2 ) ) ) ) ) ) ) ).

% VEBT_internal.minNull.pelims(3)
thf(fact_9175_Frct__code__post_I5_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ one_one_int @ ( numeral_numeral_int @ K ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ K ) ) ) ).

% Frct_code_post(5)
thf(fact_9176_divmod__step__integer__def,axiom,
    ( unique4921790084139445826nteger
    = ( ^ [L2: num] :
          ( produc6916734918728496179nteger
          @ ^ [Q5: code_integer,R5: code_integer] : ( if_Pro6119634080678213985nteger @ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L2 ) @ R5 ) @ ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R5 @ ( numera6620942414471956472nteger @ L2 ) ) ) @ ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q5 ) @ R5 ) ) ) ) ) ).

% divmod_step_integer_def
thf(fact_9177_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M2: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M2 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9178_sgn__integer__code,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( K3 = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer ) ) ) ) ).

% sgn_integer_code
thf(fact_9179_exhaustive__integer_H_Ocases,axiom,
    ! [X3: produc8763457246119570046nteger] :
      ~ ! [F2: code_integer > option6357759511663192854e_term,D4: code_integer,I3: code_integer] :
          ( X3
         != ( produc6137756002093451184nteger @ F2 @ ( produc1086072967326762835nteger @ D4 @ I3 ) ) ) ).

% exhaustive_integer'.cases
thf(fact_9180_full__exhaustive__integer_H_Ocases,axiom,
    ! [X3: produc1908205239877642774nteger] :
      ~ ! [F2: produc6241069584506657477e_term > option6357759511663192854e_term,D4: code_integer,I3: code_integer] :
          ( X3
         != ( produc8603105652947943368nteger @ F2 @ ( produc1086072967326762835nteger @ D4 @ I3 ) ) ) ).

% full_exhaustive_integer'.cases
thf(fact_9181_zero__natural_Orsp,axiom,
    zero_zero_nat = zero_zero_nat ).

% zero_natural.rsp
thf(fact_9182_Frct__code__post_I1_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ zero_zero_int @ A ) )
      = zero_zero_rat ) ).

% Frct_code_post(1)
thf(fact_9183_Frct__code__post_I2_J,axiom,
    ! [A: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ zero_zero_int ) )
      = zero_zero_rat ) ).

% Frct_code_post(2)
thf(fact_9184_Frct__code__post_I8_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ A @ ( uminus_uminus_int @ B ) ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(8)
thf(fact_9185_Frct__code__post_I7_J,axiom,
    ! [A: int,B: int] :
      ( ( frct @ ( product_Pair_int_int @ ( uminus_uminus_int @ A ) @ B ) )
      = ( uminus_uminus_rat @ ( frct @ ( product_Pair_int_int @ A @ B ) ) ) ) ).

% Frct_code_post(7)
thf(fact_9186_Frct__code__post_I3_J,axiom,
    ( ( frct @ ( product_Pair_int_int @ one_one_int @ one_one_int ) )
    = one_one_rat ) ).

% Frct_code_post(3)
thf(fact_9187_Frct__code__post_I4_J,axiom,
    ! [K: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ one_one_int ) )
      = ( numeral_numeral_rat @ K ) ) ).

% Frct_code_post(4)
thf(fact_9188_Frct__code__post_I6_J,axiom,
    ! [K: num,L: num] :
      ( ( frct @ ( product_Pair_int_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_int @ L ) ) )
      = ( divide_divide_rat @ ( numeral_numeral_rat @ K ) @ ( numeral_numeral_rat @ L ) ) ) ).

% Frct_code_post(6)
thf(fact_9189_integer__of__int__code,axiom,
    ( code_integer_of_int
    = ( ^ [K3: int] :
          ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( code_integer_of_int @ ( uminus_uminus_int @ K3 ) ) )
          @ ( if_Code_integer @ ( K3 = zero_zero_int ) @ zero_z3403309356797280102nteger
            @ ( if_Code_integer
              @ ( ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = zero_zero_int )
              @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) )
              @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( code_integer_of_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ one_one_Code_integer ) ) ) ) ) ) ).

% integer_of_int_code
thf(fact_9190_pair__leqI2,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ S @ T )
       => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ) ).

% pair_leqI2
thf(fact_9191_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I2: nat] : ( ord_less_nat @ I2 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_9192_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9193_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I2: nat] : ( ord_less_eq_nat @ I2 @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat
thf(fact_9194_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_9195_abs__integer__code,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( if_Code_integer @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus1351360451143612070nteger @ K3 ) @ K3 ) ) ) ).

% abs_integer_code
thf(fact_9196_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9197_less__integer_Oabs__eq,axiom,
    ! [Xa2: int,X3: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
      = ( ord_less_int @ Xa2 @ X3 ) ) ).

% less_integer.abs_eq
thf(fact_9198_card__less,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9199_card__less__Suc,axiom,
    ! [M7: set_nat,I: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9200_card__less__Suc2,axiom,
    ! [M7: set_nat,I: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9201_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_9202_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N7: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N7 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N7 ) @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9203_card__sum__le__nat__sum,axiom,
    ! [S3: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S3 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ S3 ) ) ).

% card_sum_le_nat_sum
thf(fact_9204_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_9205_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_9206_pair__leqI1,axiom,
    ! [A: nat,B: nat,S: nat,T: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ ( product_Pair_nat_nat @ A @ S ) @ ( product_Pair_nat_nat @ B @ T ) ) @ fun_pair_leq ) ) ).

% pair_leqI1
thf(fact_9207_int__of__integer__code,axiom,
    ( code_int_of_integer
    = ( ^ [K3: code_integer] :
          ( if_int @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( uminus_uminus_int @ ( code_int_of_integer @ ( uminus1351360451143612070nteger @ K3 ) ) )
          @ ( if_int @ ( K3 = zero_z3403309356797280102nteger ) @ zero_zero_int
            @ ( produc1553301316500091796er_int
              @ ^ [L2: code_integer,J3: code_integer] : ( if_int @ ( J3 = zero_z3403309356797280102nteger ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( code_int_of_integer @ L2 ) ) @ one_one_int ) )
              @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% int_of_integer_code
thf(fact_9208_less__integer_Orep__eq,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_integer.rep_eq
thf(fact_9209_integer__less__iff,axiom,
    ( ord_le6747313008572928689nteger
    = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_iff
thf(fact_9210_divmod__integer__def,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ K3 @ L2 ) @ ( modulo364778990260209775nteger @ K3 @ L2 ) ) ) ) ).

% divmod_integer_def
thf(fact_9211_nat__of__integer__code,axiom,
    ( code_nat_of_integer
    = ( ^ [K3: code_integer] :
          ( if_nat @ ( ord_le3102999989581377725nteger @ K3 @ zero_z3403309356797280102nteger ) @ zero_zero_nat
          @ ( produc1555791787009142072er_nat
            @ ^ [L2: code_integer,J3: code_integer] : ( if_nat @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ ( plus_plus_nat @ ( plus_plus_nat @ ( code_nat_of_integer @ L2 ) @ ( code_nat_of_integer @ L2 ) ) @ one_one_nat ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% nat_of_integer_code
thf(fact_9212_bit__cut__integer__def,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( produc6677183202524767010eger_o @ ( divide6298287555418463151nteger @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
          @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ K3 ) ) ) ) ).

% bit_cut_integer_def
thf(fact_9213_divmod__abs__def,axiom,
    ( code_divmod_abs
    = ( ^ [K3: code_integer,L2: code_integer] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ).

% divmod_abs_def
thf(fact_9214_nat__of__integer__non__positive,axiom,
    ! [K: code_integer] :
      ( ( ord_le3102999989581377725nteger @ K @ zero_z3403309356797280102nteger )
     => ( ( code_nat_of_integer @ K )
        = zero_zero_nat ) ) ).

% nat_of_integer_non_positive
thf(fact_9215_nat__of__integer__code__post_I1_J,axiom,
    ( ( code_nat_of_integer @ zero_z3403309356797280102nteger )
    = zero_zero_nat ) ).

% nat_of_integer_code_post(1)
thf(fact_9216_divmod__abs__code_I6_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ zero_z3403309356797280102nteger @ J )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ) ).

% divmod_abs_code(6)
thf(fact_9217_divmod__abs__code_I5_J,axiom,
    ! [J: code_integer] :
      ( ( code_divmod_abs @ J @ zero_z3403309356797280102nteger )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ J ) ) ) ).

% divmod_abs_code(5)
thf(fact_9218_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S8: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S8 ) ) @ ( S8 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9219_divmod__integer__code,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ L2 )
            @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ ( code_divmod_abs @ K3 @ L2 )
              @ ( produc6916734918728496179nteger
                @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S8 ) ) )
                @ ( code_divmod_abs @ K3 @ L2 ) ) )
            @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
              @ ( produc6499014454317279255nteger @ uminus1351360451143612070nteger
                @ ( if_Pro6119634080678213985nteger @ ( ord_le6747313008572928689nteger @ K3 @ zero_z3403309356797280102nteger ) @ ( code_divmod_abs @ K3 @ L2 )
                  @ ( produc6916734918728496179nteger
                    @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S8 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9220_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_9221_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_9222_less__eq__nat_Osimps_I2_J,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
      = ( case_nat_o @ $false @ ( ord_less_eq_nat @ M ) @ N2 ) ) ).

% less_eq_nat.simps(2)
thf(fact_9223_diff__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [K3: nat] : K3
        @ ( minus_minus_nat @ M @ N2 ) ) ) ).

% diff_Suc
thf(fact_9224_binomial__def,axiom,
    ( binomial
    = ( ^ [N: nat,K3: 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 )
                  = K3 ) ) ) ) ) ) ).

% binomial_def
thf(fact_9225_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_9226_wmin__insertI,axiom,
    ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_weak ) ) ) ) ).

% wmin_insertI
thf(fact_9227_wmax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y3 @ YS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_leq )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_max_weak )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ XS ) @ YS ) @ fun_max_weak ) ) ) ) ).

% wmax_insertI
thf(fact_9228_bezw__0,axiom,
    ! [X3: nat] :
      ( ( bezw @ X3 @ zero_zero_nat )
      = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) ).

% bezw_0
thf(fact_9229_push__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se545348938243370406it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% push_bit_negative_int_iff
thf(fact_9230_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_9231_wmin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] : ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_weak ) ).

% wmin_emptyI
thf(fact_9232_wmax__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ X7 )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ X7 ) @ fun_max_weak ) ) ).

% wmax_emptyI
thf(fact_9233_bit__push__bit__iff__int,axiom,
    ! [M: nat,K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se545348938243370406it_int @ M @ K ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1146084159140164899it_int @ K @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_int
thf(fact_9234_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q4: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q4 ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1148574629649215175it_nat @ Q4 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_9235_smax__insertI,axiom,
    ! [Y3: product_prod_nat_nat,Y7: set_Pr1261947904930325089at_nat,X3: product_prod_nat_nat,X7: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ Y3 @ Y7 )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ Y7 ) @ fun_max_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ ( insert8211810215607154385at_nat @ X3 @ X7 ) @ Y7 ) @ fun_max_strict ) ) ) ) ).

% smax_insertI
thf(fact_9236_smin__insertI,axiom,
    ! [X3: product_prod_nat_nat,XS: set_Pr1261947904930325089at_nat,Y3: product_prod_nat_nat,YS: set_Pr1261947904930325089at_nat] :
      ( ( member8440522571783428010at_nat @ X3 @ XS )
     => ( ( member8206827879206165904at_nat @ ( produc6161850002892822231at_nat @ X3 @ Y3 ) @ fun_pair_less )
       => ( ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ YS ) @ fun_min_strict )
         => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ XS @ ( insert8211810215607154385at_nat @ Y3 @ YS ) ) @ fun_min_strict ) ) ) ) ).

% smin_insertI
thf(fact_9237_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M2: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M2 @ K3 ) @ ( product_Pair_nat_nat @ M2 @ ( minus_minus_nat @ K3 @ M2 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M2 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9238_prod__decode__aux_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y3
            = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
         => ( Y3
            = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_9239_smax__emptyI,axiom,
    ! [Y7: set_Pr1261947904930325089at_nat] :
      ( ( finite6177210948735845034at_nat @ Y7 )
     => ( ( Y7 != bot_bo2099793752762293965at_nat )
       => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ Y7 ) @ fun_max_strict ) ) ) ).

% smax_emptyI
thf(fact_9240_smin__emptyI,axiom,
    ! [X7: set_Pr1261947904930325089at_nat] :
      ( ( X7 != bot_bo2099793752762293965at_nat )
     => ( member8757157785044589968at_nat @ ( produc2922128104949294807at_nat @ X7 @ bot_bo2099793752762293965at_nat ) @ fun_min_strict ) ) ).

% smin_emptyI
thf(fact_9241_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_9242_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_9243_finite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ? [R4: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R4 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S3 ) ) )
          & ! [N6: nat] :
              ( ( ord_less_nat @ N6 @ ( finite_card_nat @ S3 ) )
             => ( member_nat @ ( R4 @ N6 ) @ S3 ) ) ) ) ).

% finite_enumerate
thf(fact_9244_Sup__nat__empty,axiom,
    ( ( complete_Sup_Sup_nat @ bot_bot_set_nat )
    = zero_zero_nat ) ).

% Sup_nat_empty
thf(fact_9245_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_9246_quotient__of__denom__pos_H,axiom,
    ! [R2: rat] : ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ ( quotient_of @ R2 ) ) ) ).

% quotient_of_denom_pos'
thf(fact_9247_bezw__non__0,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y3 )
     => ( ( bezw @ X3 @ Y3 )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Y3 ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9248_bezw_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y3
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9249_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X: nat,Y: nat] : ( if_Pro3027730157355071871nt_int @ ( Y = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9250_rat__sgn__code,axiom,
    ! [P4: rat] :
      ( ( quotient_of @ ( sgn_sgn_rat @ P4 ) )
      = ( product_Pair_int_int @ ( sgn_sgn_int @ ( product_fst_int_int @ ( quotient_of @ P4 ) ) ) @ one_one_int ) ) ).

% rat_sgn_code
thf(fact_9251_bezw_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_int_int] :
      ( ( ( bezw @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y3
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X3 @ Xa2 ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).

% bezw.pelims
thf(fact_9252_prod__decode__aux_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa2 @ X3 )
               => ( Y3
                  = ( product_Pair_nat_nat @ Xa2 @ ( minus_minus_nat @ X3 @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa2 @ X3 )
               => ( Y3
                  = ( nat_prod_decode_aux @ ( suc @ X3 ) @ ( minus_minus_nat @ Xa2 @ ( suc @ X3 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_9253_normalize__def,axiom,
    ( normalize
    = ( ^ [P6: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P6 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P6 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P6 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P6 ) @ ( product_snd_int_int @ P6 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9254_gcd__1__int,axiom,
    ! [M: int] :
      ( ( gcd_gcd_int @ M @ one_one_int )
      = one_one_int ) ).

% gcd_1_int
thf(fact_9255_gcd__neg2__int,axiom,
    ! [X3: int,Y3: int] :
      ( ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y3 ) )
      = ( gcd_gcd_int @ X3 @ Y3 ) ) ).

% gcd_neg2_int
thf(fact_9256_gcd__neg1__int,axiom,
    ! [X3: int,Y3: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y3 )
      = ( gcd_gcd_int @ X3 @ Y3 ) ) ).

% gcd_neg1_int
thf(fact_9257_abs__gcd__int,axiom,
    ! [X3: int,Y3: int] :
      ( ( abs_abs_int @ ( gcd_gcd_int @ X3 @ Y3 ) )
      = ( gcd_gcd_int @ X3 @ Y3 ) ) ).

% abs_gcd_int
thf(fact_9258_gcd__abs1__int,axiom,
    ! [X3: int,Y3: int] :
      ( ( gcd_gcd_int @ ( abs_abs_int @ X3 ) @ Y3 )
      = ( gcd_gcd_int @ X3 @ Y3 ) ) ).

% gcd_abs1_int
thf(fact_9259_gcd__abs2__int,axiom,
    ! [X3: int,Y3: int] :
      ( ( gcd_gcd_int @ X3 @ ( abs_abs_int @ Y3 ) )
      = ( gcd_gcd_int @ X3 @ Y3 ) ) ).

% gcd_abs2_int
thf(fact_9260_gcd__pos__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_int @ zero_zero_int @ ( gcd_gcd_int @ M @ N2 ) )
      = ( ( M != zero_zero_int )
        | ( N2 != zero_zero_int ) ) ) ).

% gcd_pos_int
thf(fact_9261_gcd__neg__numeral__1__int,axiom,
    ! [N2: num,X3: int] :
      ( ( gcd_gcd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ X3 )
      = ( gcd_gcd_int @ ( numeral_numeral_int @ N2 ) @ X3 ) ) ).

% gcd_neg_numeral_1_int
thf(fact_9262_gcd__neg__numeral__2__int,axiom,
    ! [X3: int,N2: num] :
      ( ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( gcd_gcd_int @ X3 @ ( numeral_numeral_int @ N2 ) ) ) ).

% gcd_neg_numeral_2_int
thf(fact_9263_gcd__0__int,axiom,
    ! [X3: int] :
      ( ( gcd_gcd_int @ X3 @ zero_zero_int )
      = ( abs_abs_int @ X3 ) ) ).

% gcd_0_int
thf(fact_9264_gcd__0__left__int,axiom,
    ! [X3: int] :
      ( ( gcd_gcd_int @ zero_zero_int @ X3 )
      = ( abs_abs_int @ X3 ) ) ).

% gcd_0_left_int
thf(fact_9265_gcd__proj1__if__dvd__int,axiom,
    ! [X3: int,Y3: int] :
      ( ( dvd_dvd_int @ X3 @ Y3 )
     => ( ( gcd_gcd_int @ X3 @ Y3 )
        = ( abs_abs_int @ X3 ) ) ) ).

% gcd_proj1_if_dvd_int
thf(fact_9266_gcd__proj2__if__dvd__int,axiom,
    ! [Y3: int,X3: int] :
      ( ( dvd_dvd_int @ Y3 @ X3 )
     => ( ( gcd_gcd_int @ X3 @ Y3 )
        = ( abs_abs_int @ Y3 ) ) ) ).

% gcd_proj2_if_dvd_int
thf(fact_9267_gcd__idem__int,axiom,
    ! [X3: int] :
      ( ( gcd_gcd_int @ X3 @ X3 )
      = ( abs_abs_int @ X3 ) ) ).

% gcd_idem_int
thf(fact_9268_gcd__red__int,axiom,
    ( gcd_gcd_int
    = ( ^ [X: int,Y: int] : ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X @ Y ) ) ) ) ).

% gcd_red_int
thf(fact_9269_gcd__ge__0__int,axiom,
    ! [X3: int,Y3: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X3 @ Y3 ) ) ).

% gcd_ge_0_int
thf(fact_9270_bezout__int,axiom,
    ! [X3: int,Y3: int] :
    ? [U3: int,V2: int] :
      ( ( plus_plus_int @ ( times_times_int @ U3 @ X3 ) @ ( times_times_int @ V2 @ Y3 ) )
      = ( gcd_gcd_int @ X3 @ Y3 ) ) ).

% bezout_int
thf(fact_9271_gcd__mult__distrib__int,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( times_times_int @ ( abs_abs_int @ K ) @ ( gcd_gcd_int @ M @ N2 ) )
      = ( gcd_gcd_int @ ( times_times_int @ K @ M ) @ ( times_times_int @ K @ N2 ) ) ) ).

% gcd_mult_distrib_int
thf(fact_9272_gcd__le2__int,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ B ) ) ).

% gcd_le2_int
thf(fact_9273_gcd__le1__int,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ord_less_eq_int @ ( gcd_gcd_int @ A @ B ) @ A ) ) ).

% gcd_le1_int
thf(fact_9274_gcd__cases__int,axiom,
    ! [X3: int,Y3: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
         => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X3 )
         => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X3 @ ( uminus_uminus_int @ Y3 ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y3 )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ Y3 ) ) ) )
         => ( ( ( ord_less_eq_int @ X3 @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y3 @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X3 ) @ ( uminus_uminus_int @ Y3 ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X3 @ Y3 ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9275_gcd__unique__int,axiom,
    ! [D: int,A: int,B: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ D )
        & ( dvd_dvd_int @ D @ A )
        & ( dvd_dvd_int @ D @ B )
        & ! [E3: int] :
            ( ( ( dvd_dvd_int @ E3 @ A )
              & ( dvd_dvd_int @ E3 @ B ) )
           => ( dvd_dvd_int @ E3 @ D ) ) )
      = ( D
        = ( gcd_gcd_int @ A @ B ) ) ) ).

% gcd_unique_int
thf(fact_9276_gcd__non__0__int,axiom,
    ! [Y3: int,X3: int] :
      ( ( ord_less_int @ zero_zero_int @ Y3 )
     => ( ( gcd_gcd_int @ X3 @ Y3 )
        = ( gcd_gcd_int @ Y3 @ ( modulo_modulo_int @ X3 @ Y3 ) ) ) ) ).

% gcd_non_0_int
thf(fact_9277_gcd__code__int,axiom,
    ( gcd_gcd_int
    = ( ^ [K3: int,L2: int] : ( abs_abs_int @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( gcd_gcd_int @ L2 @ ( modulo_modulo_int @ ( abs_abs_int @ K3 ) @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ).

% gcd_code_int
thf(fact_9278_gcd__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( gcd_gcd_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.eq_neutr_iff
thf(fact_9279_gcd__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ A )
      = A ) ).

% gcd_nat.left_neutral
thf(fact_9280_gcd__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( gcd_gcd_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% gcd_nat.neutr_eq_iff
thf(fact_9281_gcd__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ zero_zero_nat )
      = A ) ).

% gcd_nat.right_neutral
thf(fact_9282_gcd__0__nat,axiom,
    ! [X3: nat] :
      ( ( gcd_gcd_nat @ X3 @ zero_zero_nat )
      = X3 ) ).

% gcd_0_nat
thf(fact_9283_gcd__0__left__nat,axiom,
    ! [X3: nat] :
      ( ( gcd_gcd_nat @ zero_zero_nat @ X3 )
      = X3 ) ).

% gcd_0_left_nat
thf(fact_9284_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9285_gcd__proj2__if__dvd__nat,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( dvd_dvd_nat @ Y3 @ X3 )
     => ( ( gcd_gcd_nat @ X3 @ Y3 )
        = Y3 ) ) ).

% gcd_proj2_if_dvd_nat
thf(fact_9286_gcd__proj1__if__dvd__nat,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ X3 @ Y3 )
     => ( ( gcd_gcd_nat @ X3 @ Y3 )
        = X3 ) ) ).

% gcd_proj1_if_dvd_nat
thf(fact_9287_gcd__nat_Obounded__iff,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ( dvd_dvd_nat @ A @ C ) ) ) ).

% gcd_nat.bounded_iff
thf(fact_9288_gcd__nat_Oabsorb2,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( gcd_gcd_nat @ A @ B )
        = B ) ) ).

% gcd_nat.absorb2
thf(fact_9289_gcd__nat_Oabsorb1,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( gcd_gcd_nat @ A @ B )
        = A ) ) ).

% gcd_nat.absorb1
thf(fact_9290_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9291_gcd__pos__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( ( M != zero_zero_nat )
        | ( N2 != zero_zero_nat ) ) ) ).

% gcd_pos_nat
thf(fact_9292_or__negative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ zero_zero_int )
      = ( ( ord_less_int @ K @ zero_zero_int )
        | ( ord_less_int @ L @ zero_zero_int ) ) ) ).

% or_negative_int_iff
thf(fact_9293_gcd__int__int__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_int_int_eq
thf(fact_9294_or__nat__numerals_I4_J,axiom,
    ! [X3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).

% or_nat_numerals(4)
thf(fact_9295_or__nat__numerals_I2_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_nat_numerals(2)
thf(fact_9296_gcd__nat__abs__right__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( gcd_gcd_nat @ N2 @ ( nat2 @ ( abs_abs_int @ K ) ) )
      = ( nat2 @ ( gcd_gcd_int @ ( semiri1314217659103216013at_int @ N2 ) @ K ) ) ) ).

% gcd_nat_abs_right_eq
thf(fact_9297_gcd__nat__abs__left__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ N2 )
      = ( nat2 @ ( gcd_gcd_int @ K @ ( semiri1314217659103216013at_int @ N2 ) ) ) ) ).

% gcd_nat_abs_left_eq
thf(fact_9298_or__nat__numerals_I3_J,axiom,
    ! [X3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X3 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X3 ) ) ) ).

% or_nat_numerals(3)
thf(fact_9299_or__nat__numerals_I1_J,axiom,
    ! [Y3: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y3 ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y3 ) ) ) ).

% or_nat_numerals(1)
thf(fact_9300_gcd__mult__distrib__nat,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( times_times_nat @ K @ ( gcd_gcd_nat @ M @ N2 ) )
      = ( gcd_gcd_nat @ ( times_times_nat @ K @ M ) @ ( times_times_nat @ K @ N2 ) ) ) ).

% gcd_mult_distrib_nat
thf(fact_9301_gcd__red__nat,axiom,
    ( gcd_gcd_nat
    = ( ^ [X: nat,Y: nat] : ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ).

% gcd_red_nat
thf(fact_9302_gcd__unique__nat,axiom,
    ! [D: nat,A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ D @ A )
        & ( dvd_dvd_nat @ D @ B )
        & ! [E3: nat] :
            ( ( ( dvd_dvd_nat @ E3 @ A )
              & ( dvd_dvd_nat @ E3 @ B ) )
           => ( dvd_dvd_nat @ E3 @ D ) ) )
      = ( D
        = ( gcd_gcd_nat @ A @ B ) ) ) ).

% gcd_unique_nat
thf(fact_9303_gcd__nat_Ostrict__coboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( ( dvd_dvd_nat @ B @ C )
        & ( B != C ) )
     => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
        & ( ( gcd_gcd_nat @ A @ B )
         != C ) ) ) ).

% gcd_nat.strict_coboundedI2
thf(fact_9304_gcd__nat_Ostrict__coboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ C )
        & ( A != C ) )
     => ( ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C )
        & ( ( gcd_gcd_nat @ A @ B )
         != C ) ) ) ).

% gcd_nat.strict_coboundedI1
thf(fact_9305_gcd__nat_Ostrict__order__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
      = ( ( A
          = ( gcd_gcd_nat @ A @ B ) )
        & ( A != B ) ) ) ).

% gcd_nat.strict_order_iff
thf(fact_9306_gcd__nat_Ostrict__boundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
        & ( A
         != ( gcd_gcd_nat @ B @ C ) ) )
     => ~ ( ( ( dvd_dvd_nat @ A @ B )
            & ( A != B ) )
         => ~ ( ( dvd_dvd_nat @ A @ C )
              & ( A != C ) ) ) ) ).

% gcd_nat.strict_boundedE
thf(fact_9307_gcd__nat_OcoboundedI2,axiom,
    ! [B: nat,C: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ C )
     => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).

% gcd_nat.coboundedI2
thf(fact_9308_gcd__nat_OcoboundedI1,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ C ) ) ).

% gcd_nat.coboundedI1
thf(fact_9309_gcd__nat_Oabsorb__iff2,axiom,
    ( dvd_dvd_nat
    = ( ^ [B3: nat,A3: nat] :
          ( ( gcd_gcd_nat @ A3 @ B3 )
          = B3 ) ) ) ).

% gcd_nat.absorb_iff2
thf(fact_9310_gcd__nat_Oabsorb__iff1,axiom,
    ( dvd_dvd_nat
    = ( ^ [A3: nat,B3: nat] :
          ( ( gcd_gcd_nat @ A3 @ B3 )
          = A3 ) ) ) ).

% gcd_nat.absorb_iff1
thf(fact_9311_gcd__nat_Ocobounded2,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ).

% gcd_nat.cobounded2
thf(fact_9312_gcd__nat_Ocobounded1,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ).

% gcd_nat.cobounded1
thf(fact_9313_gcd__nat_Oorder__iff,axiom,
    ( dvd_dvd_nat
    = ( ^ [A3: nat,B3: nat] :
          ( A3
          = ( gcd_gcd_nat @ A3 @ B3 ) ) ) ) ).

% gcd_nat.order_iff
thf(fact_9314_gcd__nat_OboundedI,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( ( dvd_dvd_nat @ A @ C )
       => ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) ) ) ) ).

% gcd_nat.boundedI
thf(fact_9315_gcd__nat_OboundedE,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ ( gcd_gcd_nat @ B @ C ) )
     => ~ ( ( dvd_dvd_nat @ A @ B )
         => ~ ( dvd_dvd_nat @ A @ C ) ) ) ).

% gcd_nat.boundedE
thf(fact_9316_gcd__nat_Oabsorb4,axiom,
    ! [B: nat,A: nat] :
      ( ( ( dvd_dvd_nat @ B @ A )
        & ( B != A ) )
     => ( ( gcd_gcd_nat @ A @ B )
        = B ) ) ).

% gcd_nat.absorb4
thf(fact_9317_gcd__nat_Oabsorb3,axiom,
    ! [A: nat,B: nat] :
      ( ( ( dvd_dvd_nat @ A @ B )
        & ( A != B ) )
     => ( ( gcd_gcd_nat @ A @ B )
        = A ) ) ).

% gcd_nat.absorb3
thf(fact_9318_gcd__nat_OorderI,axiom,
    ! [A: nat,B: nat] :
      ( ( A
        = ( gcd_gcd_nat @ A @ B ) )
     => ( dvd_dvd_nat @ A @ B ) ) ).

% gcd_nat.orderI
thf(fact_9319_gcd__nat_OorderE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ B )
     => ( A
        = ( gcd_gcd_nat @ A @ B ) ) ) ).

% gcd_nat.orderE
thf(fact_9320_gcd__nat_Omono,axiom,
    ! [A: nat,C: nat,B: nat,D: nat] :
      ( ( dvd_dvd_nat @ A @ C )
     => ( ( dvd_dvd_nat @ B @ D )
       => ( dvd_dvd_nat @ ( gcd_gcd_nat @ A @ B ) @ ( gcd_gcd_nat @ C @ D ) ) ) ) ).

% gcd_nat.mono
thf(fact_9321_gcd__le1__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ A ) ) ).

% gcd_le1_nat
thf(fact_9322_gcd__le2__nat,axiom,
    ! [B: nat,A: nat] :
      ( ( B != zero_zero_nat )
     => ( ord_less_eq_nat @ ( gcd_gcd_nat @ A @ B ) @ B ) ) ).

% gcd_le2_nat
thf(fact_9323_gcd__diff1__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ M @ N2 ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff1_nat
thf(fact_9324_gcd__diff2__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( gcd_gcd_nat @ ( minus_minus_nat @ N2 @ M ) @ N2 )
        = ( gcd_gcd_nat @ M @ N2 ) ) ) ).

% gcd_diff2_nat
thf(fact_9325_gcd__nat_Oelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: nat] :
      ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( Xa2 = zero_zero_nat )
         => ( Y3 = X3 ) )
        & ( ( Xa2 != zero_zero_nat )
         => ( Y3
            = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9326_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X: nat,Y: nat] : ( if_nat @ ( Y = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X @ Y ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9327_gcd__non__0__nat,axiom,
    ! [Y3: nat,X3: nat] :
      ( ( Y3 != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X3 @ Y3 )
        = ( gcd_gcd_nat @ Y3 @ ( modulo_modulo_nat @ X3 @ Y3 ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9328_gcd__integer_Orep__eq,axiom,
    ! [X3: code_integer,Xa2: code_integer] :
      ( ( code_int_of_integer @ ( gcd_gcd_Code_integer @ X3 @ Xa2 ) )
      = ( gcd_gcd_int @ ( code_int_of_integer @ X3 ) @ ( code_int_of_integer @ Xa2 ) ) ) ).

% gcd_integer.rep_eq
thf(fact_9329_gcd__integer_Oabs__eq,axiom,
    ! [Xa2: int,X3: int] :
      ( ( gcd_gcd_Code_integer @ ( code_integer_of_int @ Xa2 ) @ ( code_integer_of_int @ X3 ) )
      = ( code_integer_of_int @ ( gcd_gcd_int @ Xa2 @ X3 ) ) ) ).

% gcd_integer.abs_eq
thf(fact_9330_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X4: nat,Y5: nat] :
          ( ( times_times_nat @ A @ X4 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y5 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9331_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X4: nat,Y5: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y5 ) @ ( times_times_nat @ A @ X4 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X4 ) @ ( times_times_nat @ B @ Y5 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y5 ) @ ( times_times_nat @ B @ X4 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X4 ) @ ( times_times_nat @ A @ Y5 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9332_gcd__code__integer,axiom,
    ( gcd_gcd_Code_integer
    = ( ^ [K3: code_integer,L2: code_integer] : ( abs_abs_Code_integer @ ( if_Code_integer @ ( L2 = zero_z3403309356797280102nteger ) @ K3 @ ( gcd_gcd_Code_integer @ L2 @ ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K3 ) @ ( abs_abs_Code_integer @ L2 ) ) ) ) ) ) ) ).

% gcd_code_integer
thf(fact_9333_gcd__int__def,axiom,
    ( gcd_gcd_int
    = ( ^ [X: int,Y: int] : ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ ( nat2 @ ( abs_abs_int @ X ) ) @ ( nat2 @ ( abs_abs_int @ Y ) ) ) ) ) ) ).

% gcd_int_def
thf(fact_9334_bezw__aux,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( semiri1314217659103216013at_int @ ( gcd_gcd_nat @ X3 @ Y3 ) )
      = ( plus_plus_int @ ( times_times_int @ ( product_fst_int_int @ ( bezw @ X3 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ X3 ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ X3 @ Y3 ) ) @ ( semiri1314217659103216013at_int @ Y3 ) ) ) ) ).

% bezw_aux
thf(fact_9335_OR__upper,axiom,
    ! [X3: int,N2: nat,Y3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X3 )
     => ( ( ord_less_int @ X3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X3 @ Y3 ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% OR_upper
thf(fact_9336_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_9337_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_9338_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M2: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M2 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9339_gcd__nat_Opelims,axiom,
    ! [X3: nat,Xa2: nat,Y3: nat] :
      ( ( ( gcd_gcd_nat @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) )
       => ~ ( ( ( ( Xa2 = zero_zero_nat )
               => ( Y3 = X3 ) )
              & ( ( Xa2 != zero_zero_nat )
               => ( Y3
                  = ( gcd_gcd_nat @ Xa2 @ ( modulo_modulo_nat @ X3 @ Xa2 ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X3 @ Xa2 ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9340_gcd__nat_Oidem,axiom,
    ! [A: nat] :
      ( ( gcd_gcd_nat @ A @ A )
      = A ) ).

% gcd_nat.idem
thf(fact_9341_gcd__nat_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( gcd_gcd_nat @ A @ ( gcd_gcd_nat @ A @ B ) )
      = ( gcd_gcd_nat @ A @ B ) ) ).

% gcd_nat.left_idem
thf(fact_9342_gcd__nat_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( gcd_gcd_nat @ ( gcd_gcd_nat @ A @ B ) @ B )
      = ( gcd_gcd_nat @ A @ B ) ) ).

% gcd_nat.right_idem
thf(fact_9343_gcd__idem__nat,axiom,
    ! [X3: nat] :
      ( ( gcd_gcd_nat @ X3 @ X3 )
      = X3 ) ).

% gcd_idem_nat
thf(fact_9344_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M2: nat,N: nat] : ( if_nat @ ( M2 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M2 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M2 @ ( 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 @ M2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9345_max__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_max_nat @ M @ N2 ) ) ) ).

% max_Suc_Suc
thf(fact_9346_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_9347_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_9348_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_9349_max__nat_Oneutr__eq__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( zero_zero_nat
        = ( ord_max_nat @ A @ B ) )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.neutr_eq_iff
thf(fact_9350_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_9351_max__nat_Oeq__neutr__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ( ord_max_nat @ A @ B )
        = zero_zero_nat )
      = ( ( A = zero_zero_nat )
        & ( B = zero_zero_nat ) ) ) ).

% max_nat.eq_neutr_iff
thf(fact_9352_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_9353_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_9354_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q4 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q4 ) ) ) ).

% nat_add_max_right
thf(fact_9355_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q4 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q4 ) @ ( plus_plus_nat @ N2 @ Q4 ) ) ) ).

% nat_add_max_left
thf(fact_9356_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q4 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).

% nat_mult_max_left
thf(fact_9357_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q4 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q4 ) ) ) ).

% nat_mult_max_right
thf(fact_9358_nat__minus__add__max,axiom,
    ! [N2: nat,M: nat] :
      ( ( plus_plus_nat @ ( minus_minus_nat @ N2 @ M ) @ M )
      = ( ord_max_nat @ N2 @ M ) ) ).

% nat_minus_add_max
thf(fact_9359_max__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ M4 @ N2 ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9360_max__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M4: nat] : ( suc @ ( ord_max_nat @ N2 @ M4 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9361_divmod__integer__eq__cases,axiom,
    ( code_divmod_integer
    = ( ^ [K3: code_integer,L2: code_integer] :
          ( if_Pro6119634080678213985nteger @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger )
          @ ( if_Pro6119634080678213985nteger @ ( L2 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ K3 )
            @ ( comp_C1593894019821074884nteger @ ( comp_C8797469213163452608nteger @ produc6499014454317279255nteger @ times_3573771949741848930nteger ) @ sgn_sgn_Code_integer @ L2
              @ ( if_Pro6119634080678213985nteger
                @ ( ( sgn_sgn_Code_integer @ K3 )
                  = ( sgn_sgn_Code_integer @ L2 ) )
                @ ( code_divmod_abs @ K3 @ L2 )
                @ ( produc6916734918728496179nteger
                  @ ^ [R5: code_integer,S8: code_integer] : ( if_Pro6119634080678213985nteger @ ( S8 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( abs_abs_Code_integer @ L2 ) @ S8 ) ) )
                  @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ).

% divmod_integer_eq_cases
thf(fact_9362_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_9363_root__powr__inverse,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( root @ N2 @ X3 )
          = ( powr_real @ X3 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9364_drop__bit__negative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) @ zero_zero_int )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% drop_bit_negative_int_iff
thf(fact_9365_real__root__Suc__0,axiom,
    ! [X3: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X3 )
      = X3 ) ).

% real_root_Suc_0
thf(fact_9366_root__0,axiom,
    ! [X3: real] :
      ( ( root @ zero_zero_nat @ X3 )
      = zero_zero_real ) ).

% root_0
thf(fact_9367_real__root__eq__iff,axiom,
    ! [N2: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X3 )
          = ( root @ N2 @ Y3 ) )
        = ( X3 = Y3 ) ) ) ).

% real_root_eq_iff
thf(fact_9368_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_9369_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_9370_real__root__eq__0__iff,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X3 )
          = zero_zero_real )
        = ( X3 = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9371_real__root__less__iff,axiom,
    ! [N2: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) )
        = ( ord_less_real @ X3 @ Y3 ) ) ) ).

% real_root_less_iff
thf(fact_9372_real__root__le__iff,axiom,
    ! [N2: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) )
        = ( ord_less_eq_real @ X3 @ Y3 ) ) ) ).

% real_root_le_iff
thf(fact_9373_real__root__eq__1__iff,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X3 )
          = one_one_real )
        = ( X3 = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9374_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_9375_real__root__lt__0__iff,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X3 ) @ zero_zero_real )
        = ( ord_less_real @ X3 @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9376_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y3 ) )
        = ( ord_less_real @ zero_zero_real @ Y3 ) ) ) ).

% real_root_gt_0_iff
thf(fact_9377_real__root__le__0__iff,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X3 @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9378_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y3 ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y3 ) ) ) ).

% real_root_ge_0_iff
thf(fact_9379_real__root__lt__1__iff,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X3 ) @ one_one_real )
        = ( ord_less_real @ X3 @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9380_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y3 ) )
        = ( ord_less_real @ one_one_real @ Y3 ) ) ) ).

% real_root_gt_1_iff
thf(fact_9381_real__root__le__1__iff,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ one_one_real )
        = ( ord_less_eq_real @ X3 @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9382_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y3 ) )
        = ( ord_less_eq_real @ one_one_real @ Y3 ) ) ) ).

% real_root_ge_1_iff
thf(fact_9383_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_9384_real__root__pow__pos2,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( power_power_real @ ( root @ N2 @ X3 ) @ N2 )
          = X3 ) ) ) ).

% real_root_pow_pos2
thf(fact_9385_real__root__less__mono,axiom,
    ! [N2: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ord_less_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) ) ) ) ).

% real_root_less_mono
thf(fact_9386_real__root__le__mono,axiom,
    ! [N2: nat,X3: real,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ X3 @ Y3 )
       => ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ ( root @ N2 @ Y3 ) ) ) ) ).

% real_root_le_mono
thf(fact_9387_real__root__power,axiom,
    ! [N2: nat,X3: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X3 @ K ) )
        = ( power_power_real @ ( root @ N2 @ X3 ) @ K ) ) ) ).

% real_root_power
thf(fact_9388_real__root__abs,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( abs_abs_real @ X3 ) )
        = ( abs_abs_real @ ( root @ N2 @ X3 ) ) ) ) ).

% real_root_abs
thf(fact_9389_sgn__root,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( sgn_sgn_real @ ( root @ N2 @ X3 ) )
        = ( sgn_sgn_real @ X3 ) ) ) ).

% sgn_root
thf(fact_9390_real__root__gt__zero,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X3 ) ) ) ) ).

% real_root_gt_zero
thf(fact_9391_real__root__strict__decreasing,axiom,
    ! [N2: nat,N7: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N7 )
       => ( ( ord_less_real @ one_one_real @ X3 )
         => ( ord_less_real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9392_root__abs__power,axiom,
    ! [N2: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y3 @ N2 ) ) )
        = ( abs_abs_real @ Y3 ) ) ) ).

% root_abs_power
thf(fact_9393_real__root__pos__pos,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X3 ) ) ) ) ).

% real_root_pos_pos
thf(fact_9394_real__root__strict__increasing,axiom,
    ! [N2: nat,N7: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N7 )
       => ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( ( ord_less_real @ X3 @ one_one_real )
           => ( ord_less_real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9395_real__root__decreasing,axiom,
    ! [N2: nat,N7: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N7 )
       => ( ( ord_less_eq_real @ one_one_real @ X3 )
         => ( ord_less_eq_real @ ( root @ N7 @ X3 ) @ ( root @ N2 @ X3 ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9396_real__root__pow__pos,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( ( power_power_real @ ( root @ N2 @ X3 ) @ N2 )
          = X3 ) ) ) ).

% real_root_pow_pos
thf(fact_9397_real__root__pos__unique,axiom,
    ! [N2: nat,Y3: real,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y3 )
       => ( ( ( power_power_real @ Y3 @ N2 )
            = X3 )
         => ( ( root @ N2 @ X3 )
            = Y3 ) ) ) ) ).

% real_root_pos_unique
thf(fact_9398_real__root__power__cancel,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
       => ( ( root @ N2 @ ( power_power_real @ X3 @ N2 ) )
          = X3 ) ) ) ).

% real_root_power_cancel
thf(fact_9399_real__root__increasing,axiom,
    ! [N2: nat,N7: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N7 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X3 )
         => ( ( ord_less_eq_real @ X3 @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N2 @ X3 ) @ ( root @ N7 @ X3 ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9400_root__sgn__power,axiom,
    ! [N2: nat,Y3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y3 ) @ ( power_power_real @ ( abs_abs_real @ Y3 ) @ N2 ) ) )
        = Y3 ) ) ).

% root_sgn_power
thf(fact_9401_sgn__power__root,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X3 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X3 ) ) @ N2 ) )
        = X3 ) ) ).

% sgn_power_root
thf(fact_9402_ln__root,axiom,
    ! [N2: nat,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ln_ln_real @ ( root @ N2 @ B ) )
          = ( divide_divide_real @ ( ln_ln_real @ B ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% ln_root
thf(fact_9403_log__root,axiom,
    ! [N2: nat,A: real,B: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( log @ B @ ( root @ N2 @ A ) )
          = ( divide_divide_real @ ( log @ B @ A ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ).

% log_root
thf(fact_9404_log__base__root,axiom,
    ! [N2: nat,B: real,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N2 @ B ) @ X3 )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X3 ) ) ) ) ) ).

% log_base_root
thf(fact_9405_split__root,axiom,
    ! [P: real > $o,N2: nat,X3: real] :
      ( ( P @ ( root @ N2 @ X3 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ! [Y: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
                = X3 )
             => ( P @ Y ) ) ) ) ) ).

% split_root
thf(fact_9406_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_9407_min__Suc__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ ( suc @ M ) @ ( suc @ N2 ) )
      = ( suc @ ( ord_min_nat @ M @ N2 ) ) ) ).

% min_Suc_Suc
thf(fact_9408_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_9409_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_9410_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_9411_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_9412_min__diff,axiom,
    ! [M: nat,I: nat,N2: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I ) @ ( minus_minus_nat @ N2 @ I ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I ) ) ).

% min_diff
thf(fact_9413_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q4 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q4 ) @ ( times_times_nat @ N2 @ Q4 ) ) ) ).

% nat_mult_min_left
thf(fact_9414_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q4: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q4 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q4 ) ) ) ).

% nat_mult_min_right
thf(fact_9415_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_9416_min__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ M4 @ N2 ) )
        @ M ) ) ).

% min_Suc2
thf(fact_9417_min__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M4: nat] : ( suc @ ( ord_min_nat @ N2 @ M4 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_9418_inf__int__def,axiom,
    inf_inf_int = ord_min_int ).

% inf_int_def
thf(fact_9419_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X: nat,Y: nat] : ( ord_less_eq_nat @ Y @ X )
    @ ^ [X: nat,Y: nat] : ( ord_less_nat @ Y @ X ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9420_gcd__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ gcd_gcd_nat @ zero_zero_nat @ dvd_dvd_nat
    @ ^ [M2: nat,N: nat] :
        ( ( dvd_dvd_nat @ M2 @ N )
        & ( M2 != N ) ) ) ).

% gcd_nat.semilattice_neutr_order_axioms
thf(fact_9421_times__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) )
          @ Xa2
          @ X3 ) ) ) ).

% times_int.abs_eq
thf(fact_9422_int_Oabs__induct,axiom,
    ! [P: int > $o,X3: int] :
      ( ! [Y5: product_prod_nat_nat] : ( P @ ( abs_Integ @ Y5 ) )
     => ( P @ X3 ) ) ).

% int.abs_induct
thf(fact_9423_eq__Abs__Integ,axiom,
    ! [Z2: int] :
      ~ ! [X4: nat,Y5: nat] :
          ( Z2
         != ( abs_Integ @ ( product_Pair_nat_nat @ X4 @ Y5 ) ) ) ).

% eq_Abs_Integ
thf(fact_9424_nat_Oabs__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( nat2 @ ( abs_Integ @ X3 ) )
      = ( produc6842872674320459806at_nat @ minus_minus_nat @ X3 ) ) ).

% nat.abs_eq
thf(fact_9425_zero__int__def,axiom,
    ( zero_zero_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ zero_zero_nat @ zero_zero_nat ) ) ) ).

% zero_int_def
thf(fact_9426_int__def,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] : ( abs_Integ @ ( product_Pair_nat_nat @ N @ zero_zero_nat ) ) ) ) ).

% int_def
thf(fact_9427_uminus__int_Oabs__eq,axiom,
    ! [X3: product_prod_nat_nat] :
      ( ( uminus_uminus_int @ ( abs_Integ @ X3 ) )
      = ( abs_Integ
        @ ( produc2626176000494625587at_nat
          @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X )
          @ X3 ) ) ) ).

% uminus_int.abs_eq
thf(fact_9428_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9429_less__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
        @ Xa2
        @ X3 ) ) ).

% less_int.abs_eq
thf(fact_9430_less__eq__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) )
        @ Xa2
        @ X3 ) ) ).

% less_eq_int.abs_eq
thf(fact_9431_plus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) )
          @ Xa2
          @ X3 ) ) ) ).

% plus_int.abs_eq
thf(fact_9432_minus__int_Oabs__eq,axiom,
    ! [Xa2: product_prod_nat_nat,X3: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa2 ) @ ( abs_Integ @ X3 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) )
          @ Xa2
          @ X3 ) ) ) ).

% minus_int.abs_eq
thf(fact_9433_max__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_max_strict @ fun_max_weak ) ).

% max_rpair_set
thf(fact_9434_min__rpair__set,axiom,
    fun_re2478310338295953701at_nat @ ( produc9060074326276436823at_nat @ fun_min_strict @ fun_min_weak ) ).

% min_rpair_set
thf(fact_9435_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9436_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y: nat,Z6: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y @ V4 ) @ ( plus_plus_nat @ U2 @ Z6 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9437_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M2: nat,N: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M2 @ N ) ) @ M2 ) ) ) ).

% prod_encode_def
thf(fact_9438_le__prod__encode__2,axiom,
    ! [B: nat,A: nat] : ( ord_less_eq_nat @ B @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_2
thf(fact_9439_le__prod__encode__1,axiom,
    ! [A: nat,B: nat] : ( ord_less_eq_nat @ A @ ( nat_prod_encode @ ( product_Pair_nat_nat @ A @ B ) ) ) ).

% le_prod_encode_1
thf(fact_9440_nat_Orep__eq,axiom,
    ( nat2
    = ( ^ [X: int] : ( produc6842872674320459806at_nat @ minus_minus_nat @ ( rep_Integ @ X ) ) ) ) ).

% nat.rep_eq
thf(fact_9441_prod__encode__prod__decode__aux,axiom,
    ! [K: nat,M: nat] :
      ( ( nat_prod_encode @ ( nat_prod_decode_aux @ K @ M ) )
      = ( plus_plus_nat @ ( nat_triangle @ K ) @ M ) ) ).

% prod_encode_prod_decode_aux
thf(fact_9442_uminus__int__def,axiom,
    ( uminus_uminus_int
    = ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ
      @ ( produc2626176000494625587at_nat
        @ ^ [X: nat,Y: nat] : ( product_Pair_nat_nat @ Y @ X ) ) ) ) ).

% uminus_int_def
thf(fact_9443_times__int__def,axiom,
    ( times_times_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y @ U2 ) ) ) ) ) ) ) ).

% times_int_def
thf(fact_9444_minus__int__def,axiom,
    ( minus_minus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y @ U2 ) ) ) ) ) ) ).

% minus_int_def
thf(fact_9445_plus__int__def,axiom,
    ( plus_plus_int
    = ( map_fu4960017516451851995nt_int @ rep_Integ @ ( map_fu3667384564859982768at_int @ rep_Integ @ abs_Integ )
      @ ( produc27273713700761075at_nat
        @ ^ [X: nat,Y: nat] :
            ( produc2626176000494625587at_nat
            @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y @ V4 ) ) ) ) ) ) ).

% plus_int_def
thf(fact_9446_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_9447_Gcd__dvd__nat,axiom,
    ! [A: nat,A2: set_nat] :
      ( ( member_nat @ A @ A2 )
     => ( dvd_dvd_nat @ ( gcd_Gcd_nat @ A2 ) @ A ) ) ).

% Gcd_dvd_nat
thf(fact_9448_Gcd__greatest__nat,axiom,
    ! [A2: set_nat,A: nat] :
      ( ! [B4: nat] :
          ( ( member_nat @ B4 @ A2 )
         => ( dvd_dvd_nat @ A @ B4 ) )
     => ( dvd_dvd_nat @ A @ ( gcd_Gcd_nat @ A2 ) ) ) ).

% Gcd_greatest_nat
thf(fact_9449_Gcd__nat__eq__one,axiom,
    ! [N7: set_nat] :
      ( ( member_nat @ one_one_nat @ N7 )
     => ( ( gcd_Gcd_nat @ N7 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_9450_Gcd__in,axiom,
    ! [A2: set_nat] :
      ( ! [A4: nat,B4: nat] :
          ( ( member_nat @ A4 @ A2 )
         => ( ( member_nat @ B4 @ A2 )
           => ( member_nat @ ( gcd_gcd_nat @ A4 @ B4 ) @ A2 ) ) )
     => ( ( A2 != bot_bot_set_nat )
       => ( member_nat @ ( gcd_Gcd_nat @ A2 ) @ A2 ) ) ) ).

% Gcd_in
thf(fact_9451_sub__BitM__One__eq,axiom,
    ! [N2: num] :
      ( ( neg_numeral_sub_int @ ( bitM @ N2 ) @ one )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ).

% sub_BitM_One_eq
thf(fact_9452_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_9453_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M2: nat,N: nat] :
            ( N
            = ( suc @ M2 ) ) ) ) ) ).

% pred_nat_def
thf(fact_9454_abs__Gcd__eq,axiom,
    ! [K5: set_int] :
      ( ( abs_abs_int @ ( gcd_Gcd_int @ K5 ) )
      = ( gcd_Gcd_int @ K5 ) ) ).

% abs_Gcd_eq
thf(fact_9455_Gcd__dvd__int,axiom,
    ! [A: int,A2: set_int] :
      ( ( member_int @ A @ A2 )
     => ( dvd_dvd_int @ ( gcd_Gcd_int @ A2 ) @ A ) ) ).

% Gcd_dvd_int
thf(fact_9456_Gcd__greatest__int,axiom,
    ! [A2: set_int,A: int] :
      ( ! [B4: int] :
          ( ( member_int @ B4 @ A2 )
         => ( dvd_dvd_int @ A @ B4 ) )
     => ( dvd_dvd_int @ A @ ( gcd_Gcd_int @ A2 ) ) ) ).

% Gcd_greatest_int
thf(fact_9457_Gcd__int__greater__eq__0,axiom,
    ! [K5: set_int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_int_greater_eq_0
thf(fact_9458_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9459_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_9460_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_9461_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_9462_num__of__nat__plus__distrib,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( num_of_nat @ ( plus_plus_nat @ M @ N2 ) )
          = ( plus_plus_num @ ( num_of_nat @ M ) @ ( num_of_nat @ N2 ) ) ) ) ) ).

% num_of_nat_plus_distrib
thf(fact_9463_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( ( 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
                  @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                    & ! [X: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi3: nat,Ma3: nat] :
                        ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                        & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I2: nat] :
                            ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                        & ( ( Mi3 = Ma3 )
                         => ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        & ( ( Mi3 != Ma3 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                            & ! [X: nat] :
                                ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                 => ( ( ord_less_nat @ Mi3 @ X )
                                    & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9464_VEBT__internal_Ovalid_H_Osimps_I2_J,axiom,
    ! [Mima2: option4927543243414619207at_nat,Deg: nat,TreeList2: list_VEBT_VEBT,Summary: vEBT_VEBT,Deg4: nat] :
      ( ( vEBT_VEBT_valid @ ( vEBT_Node @ Mima2 @ Deg @ TreeList2 @ Summary ) @ Deg4 )
      = ( ( Deg = Deg4 )
        & ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
           => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( vEBT_VEBT_valid @ Summary @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        & ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( case_o184042715313410164at_nat
          @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X5 )
            & ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi3: nat,Ma3: nat] :
                ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I2: nat] :
                    ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I2 ) @ X5 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I2 ) ) )
                & ( ( Mi3 = Ma3 )
                 => ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                & ( ( Mi3 != Ma3 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma3 )
                    & ! [X: nat] :
                        ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                       => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ X )
                         => ( ( ord_less_nat @ Mi3 @ X )
                            & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9465_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
        = Y3 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y3
            = ( Xa2 != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
             => ( Y3
                = ( ~ ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9466_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X3
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa2 != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary2: vEBT_VEBT] :
              ( ( X3
                = ( 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
                    @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi3: nat,Ma3: nat] :
                          ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                          & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I2: nat] :
                              ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                          & ( ( Mi3 = Ma3 )
                           => ! [X: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                          & ( ( Mi3 != Ma3 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                              & ! [X: nat] :
                                  ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                   => ( ( ord_less_nat @ Mi3 @ X )
                                      & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9467_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat,Y3: $o] :
      ( ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y3
                  = ( 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] :
                ( ( X3
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( Y3
                    = ( ( Deg2 = Xa2 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary2 @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( case_o184042715313410164at_nat
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9468_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( 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] :
                ( ( X3
                  = ( 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
                        @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi3: nat,Ma3: nat] :
                              ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                              & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I2: nat] :
                                  ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                              & ( ( Mi3 = Ma3 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                              & ( ( Mi3 != Ma3 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                  & ! [X: nat] :
                                      ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                       => ( ( ord_less_nat @ Mi3 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9469_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X3: vEBT_VEBT,Xa2: nat] :
      ( ~ ( vEBT_VEBT_valid @ X3 @ Xa2 )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X3 @ Xa2 ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X3
                = ( 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] :
                ( ( X3
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary2 ) @ Xa2 ) )
                 => ( ( 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
                      @ ( ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X5 )
                        & ! [X: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi3: nat,Ma3: nat] :
                            ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                            & ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I2: nat] :
                                ( ( ord_less_nat @ I2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I2 ) @ X5 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary2 @ I2 ) ) )
                            & ( ( Mi3 = Ma3 )
                             => ! [X: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X5: nat] : ( vEBT_V8194947554948674370ptions @ X @ X5 ) ) )
                            & ( ( Mi3 != Ma3 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma3 )
                                & ! [X: nat] :
                                    ( ( ord_less_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                   => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ X )
                                     => ( ( ord_less_nat @ Mi3 @ X )
                                        & ( ord_less_eq_nat @ X @ Ma3 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9470_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9471_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9472_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_12: nat] : ( P @ X_12 )
     => ( ! [Y5: nat] :
            ( ( P @ Y5 )
           => ( ord_less_eq_nat @ Y5 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9473_take__bit__num__simps_I1_J,axiom,
    ! [M: num] :
      ( ( bit_take_bit_num @ zero_zero_nat @ M )
      = none_num ) ).

% take_bit_num_simps(1)
thf(fact_9474_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_9475_take__bit__num__simps_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ N2 @ M ) ) ) ).

% take_bit_num_simps(3)
thf(fact_9476_take__bit__num__simps_I4_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ ( suc @ N2 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N2 @ M ) ) ) ) ).

% take_bit_num_simps(4)
thf(fact_9477_take__bit__num__simps_I6_J,axiom,
    ! [R2: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R2 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R2 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9478_Code__Abstract__Nat_Otake__bit__num__code_I2_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ N2 @ ( bit0 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N: nat] :
            ( case_o6005452278849405969um_num @ none_num
            @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
            @ ( bit_take_bit_num @ N @ M ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(2)
thf(fact_9479_Code__Abstract__Nat_Otake__bit__num__code_I1_J,axiom,
    ! [N2: nat] :
      ( ( bit_take_bit_num @ N2 @ one )
      = ( case_nat_option_num @ none_num
        @ ^ [N: nat] : ( some_num @ one )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(1)
thf(fact_9480_Code__Abstract__Nat_Otake__bit__num__code_I3_J,axiom,
    ! [N2: nat,M: num] :
      ( ( bit_take_bit_num @ N2 @ ( bit1 @ M ) )
      = ( case_nat_option_num @ none_num
        @ ^ [N: nat] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ N @ M ) ) )
        @ N2 ) ) ).

% Code_Abstract_Nat.take_bit_num_code(3)
thf(fact_9481_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M2: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M2 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M2 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9482_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M2: num] :
          ( produc478579273971653890on_num
          @ ^ [A3: nat,X: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P6: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
                      @ ( bit_take_bit_num @ O @ P6 ) )
                  @ ^ [P6: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P6 ) ) )
                  @ X )
              @ A3 )
          @ ( product_Pair_nat_num @ N @ M2 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9483_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I2: int,N: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I2 ) @ ( semiri5074537144036343181t_real @ N ) ) )
          & ( N != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_9484_Rats__dense__in__real,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_real @ X3 @ Y3 )
     => ? [X4: real] :
          ( ( member_real @ X4 @ field_5140801741446780682s_real )
          & ( ord_less_real @ X3 @ X4 )
          & ( ord_less_real @ X4 @ Y3 ) ) ) ).

% Rats_dense_in_real
thf(fact_9485_Rats__no__bot__less,axiom,
    ! [X3: real] :
    ? [X4: real] :
      ( ( member_real @ X4 @ field_5140801741446780682s_real )
      & ( ord_less_real @ X4 @ X3 ) ) ).

% Rats_no_bot_less
thf(fact_9486_rat__floor__lemma,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( divide_divide_int @ A @ B ) ) @ ( fract @ A @ B ) )
      & ( ord_less_rat @ ( fract @ A @ B ) @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( divide_divide_int @ A @ B ) @ one_one_int ) ) ) ) ).

% rat_floor_lemma
thf(fact_9487_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y3: nat,X3: nat] :
      ( ( ( ord_less_nat @ C @ Y3 )
       => ( ( image_nat_nat
            @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
            @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X3 @ C ) @ ( minus_minus_nat @ Y3 @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y3 )
       => ( ( ( ord_less_nat @ X3 @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
                @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X3 @ Y3 )
           => ( ( image_nat_nat
                @ ^ [I2: nat] : ( minus_minus_nat @ I2 @ C )
                @ ( set_or4665077453230672383an_nat @ X3 @ Y3 ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9488_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_9489_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_9490_less__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% less_rat
thf(fact_9491_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_9492_Rat__induct__pos,axiom,
    ! [P: rat > $o,Q4: rat] :
      ( ! [A4: int,B4: int] :
          ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( P @ ( fract @ A4 @ B4 ) ) )
     => ( P @ Q4 ) ) ).

% Rat_induct_pos
thf(fact_9493_quotient__of__eq,axiom,
    ! [A: int,B: int,P4: int,Q4: int] :
      ( ( ( quotient_of @ ( fract @ A @ B ) )
        = ( product_Pair_int_int @ P4 @ Q4 ) )
     => ( ( fract @ P4 @ Q4 )
        = ( fract @ A @ B ) ) ) ).

% quotient_of_eq
thf(fact_9494_normalize__eq,axiom,
    ! [A: int,B: int,P4: int,Q4: int] :
      ( ( ( normalize @ ( product_Pair_int_int @ A @ B ) )
        = ( product_Pair_int_int @ P4 @ Q4 ) )
     => ( ( fract @ P4 @ Q4 )
        = ( fract @ A @ B ) ) ) ).

% normalize_eq
thf(fact_9495_quotient__of__Fract,axiom,
    ! [A: int,B: int] :
      ( ( quotient_of @ ( fract @ A @ B ) )
      = ( normalize @ ( product_Pair_int_int @ A @ B ) ) ) ).

% quotient_of_Fract
thf(fact_9496_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_9497_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_9498_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_9499_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_9500_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_9501_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_9502_Fract__less__zero__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ zero_zero_rat )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Fract_less_zero_iff
thf(fact_9503_zero__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ zero_zero_int @ A ) ) ) ).

% zero_less_Fract_iff
thf(fact_9504_one__less__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_int @ B @ A ) ) ) ).

% one_less_Fract_iff
thf(fact_9505_Fract__less__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_int @ A @ B ) ) ) ).

% Fract_less_one_iff
thf(fact_9506_zero__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ).

% zero_le_Fract_iff
thf(fact_9507_Fract__le__zero__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ zero_zero_rat )
        = ( ord_less_eq_int @ A @ zero_zero_int ) ) ) ).

% Fract_le_zero_iff
thf(fact_9508_Fract__le__one__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ one_one_rat )
        = ( ord_less_eq_int @ A @ B ) ) ) ).

% Fract_le_one_iff
thf(fact_9509_one__le__Fract__iff,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ zero_zero_int @ B )
     => ( ( ord_less_eq_rat @ one_one_rat @ ( fract @ A @ B ) )
        = ( ord_less_eq_int @ B @ A ) ) ) ).

% one_le_Fract_iff
thf(fact_9510_Gcd__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Gcd_int @ ( image_int_int @ abs_abs_int @ K5 ) )
      = ( gcd_Gcd_int @ K5 ) ) ).

% Gcd_abs_eq
thf(fact_9511_Gcd__int__eq,axiom,
    ! [N7: set_nat] :
      ( ( gcd_Gcd_int @ ( image_nat_int @ semiri1314217659103216013at_int @ N7 ) )
      = ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ N7 ) ) ) ).

% Gcd_int_eq
thf(fact_9512_Gcd__nat__abs__eq,axiom,
    ! [K5: set_int] :
      ( ( gcd_Gcd_nat
        @ ( image_int_nat
          @ ^ [K3: int] : ( nat2 @ ( abs_abs_int @ K3 ) )
          @ K5 ) )
      = ( nat2 @ ( gcd_Gcd_int @ K5 ) ) ) ).

% Gcd_nat_abs_eq
thf(fact_9513_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S7: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_9514_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S7: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S7 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_9515_infinite__int__iff__infinite__nat__abs,axiom,
    ! [S3: set_int] :
      ( ( ~ ( finite_finite_int @ S3 ) )
      = ( ~ ( finite_finite_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ S3 ) ) ) ) ).

% infinite_int_iff_infinite_nat_abs
thf(fact_9516_Gcd__int__def,axiom,
    ( gcd_Gcd_int
    = ( ^ [K7: set_int] : ( semiri1314217659103216013at_int @ ( gcd_Gcd_nat @ ( image_int_nat @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) @ K7 ) ) ) ) ) ).

% Gcd_int_def
thf(fact_9517_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_9518_range__abs__Nats,axiom,
    ( ( image_int_int @ abs_abs_int @ top_top_set_int )
    = semiring_1_Nats_int ) ).

% range_abs_Nats
thf(fact_9519_infinite__UNIV__int,axiom,
    ~ ( finite_finite_int @ top_top_set_int ) ).

% infinite_UNIV_int
thf(fact_9520_nat__not__finite,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% nat_not_finite
thf(fact_9521_infinite__UNIV__nat,axiom,
    ~ ( finite_finite_nat @ top_top_set_nat ) ).

% infinite_UNIV_nat
thf(fact_9522_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_9523_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_9524_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M2: nat] : ( modulo_modulo_nat @ M2 @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_9525_root__def,axiom,
    ( root
    = ( ^ [N: nat,X: real] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N ) )
            @ X ) ) ) ) ).

% root_def
thf(fact_9526_DERIV__even__real__root,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( ord_less_real @ X3 @ 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 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9527_DERIV__real__root__generic,axiom,
    ! [N2: nat,X3: real,D6: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( X3 != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
           => ( ( ord_less_real @ zero_zero_real @ X3 )
             => ( D6
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
             => ( ( ord_less_real @ X3 @ zero_zero_real )
               => ( D6
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( 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 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D6 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9528_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D4 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9529_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D4 )
                 => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9530_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D4 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9531_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X3: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D4 )
                 => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9532_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ? [Y6: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9533_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ? [Y6: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9534_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X3: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D4 )
                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9535_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X3: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X3 @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D4 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9536_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X3: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D4 )
                   => ( ord_less_real @ ( F @ X3 ) @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9537_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X3: real,S3: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ S3 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D4: real] :
            ( ( ord_less_real @ zero_zero_real @ D4 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X3 @ H4 ) @ S3 )
                 => ( ( ord_less_real @ H4 @ D4 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X3 @ H4 ) ) @ ( F @ X3 ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9538_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
       => ? [Z3: real] :
            ( ( ord_less_real @ A @ Z3 )
            & ( ord_less_real @ Z3 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F5 @ Z3 ) ) ) ) ) ) ).

% MVT2
thf(fact_9539_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X3: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y5: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D )
             => ( ( F @ X3 )
                = ( F @ Y5 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9540_DERIV__ln,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9541_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X3: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y5: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y5 ) @ ( F @ X3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9542_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X3: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y5: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X3 @ Y5 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X3 ) @ ( F @ Y5 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9543_DERIV__ln__divide,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X3 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9544_DERIV__pow,axiom,
    ! [N2: nat,X3: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X: real] : ( power_power_real @ X @ N2 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X3 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X3 @ S ) ) ).

% DERIV_pow
thf(fact_9545_has__real__derivative__powr,axiom,
    ! [Z2: real,R2: real] :
      ( ( ord_less_real @ zero_zero_real @ Z2 )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z6: real] : ( powr_real @ Z6 @ R2 )
        @ ( times_times_real @ R2 @ ( powr_real @ Z2 @ ( minus_minus_real @ R2 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9546_DERIV__log,axiom,
    ! [X3: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X3 ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9547_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X3: real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R2 )
          @ ( times_times_real @ ( times_times_real @ R2 @ ( powr_real @ ( G @ X3 ) @ ( minus_minus_real @ R2 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9548_DERIV__powr,axiom,
    ! [G: real > real,M: real,X3: real,F: real > real,R2: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X3 ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R2 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X3 ) @ ( F @ X3 ) ) @ ( plus_plus_real @ ( times_times_real @ R2 @ ( ln_ln_real @ ( G @ X3 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X3 ) ) @ ( G @ X3 ) ) ) )
            @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9549_arcosh__real__has__field__derivative,axiom,
    ! [X3: real,A2: set_real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A2 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9550_DERIV__real__sqrt,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ zero_zero_real @ X3 )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9551_DERIV__real__sqrt__generic,axiom,
    ! [X3: real,D6: real] :
      ( ( X3 != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X3 )
         => ( D6
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X3 @ zero_zero_real )
           => ( D6
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X3 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D6 @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9552_artanh__real__has__field__derivative,axiom,
    ! [X3: real,A2: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ A2 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9553_DERIV__real__root,axiom,
    ! [N2: nat,X3: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X3 )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9554_DERIV__arcsin,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9555_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X3: real,N2: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
     => ? [T6: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
          & ( ( F @ X3 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9556_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X3: real,N2: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
       => ? [T6: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
            & ( ( F @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9557_DERIV__odd__real__root,axiom,
    ! [N2: nat,X3: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( X3 != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X3 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9558_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
                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
                      @ ( 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_9559_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
                    @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
                    @ ( 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_9560_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
                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ H2 @ M2 ) )
                      @ ( 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_9561_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X3: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( X3 != zero_zero_real )
         => ( ! [M3: nat,X4: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M3 ) @ ( Diff @ ( suc @ M3 ) @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
           => ? [T6: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T6 ) )
                & ( ord_less_real @ ( abs_abs_real @ T6 ) @ ( abs_abs_real @ X3 ) )
                & ( ( F @ X3 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9562_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X3: 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 @ X3 ) ) )
           => ( 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 @ X3 ) )
            & ( ( F @ X3 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ X3 @ M2 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X3 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9563_Taylor,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X3: 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 @ B ) )
             => ( 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 @ B )
             => ( ( ord_less_eq_real @ A @ X3 )
               => ( ( ord_less_eq_real @ X3 @ B )
                 => ( ( X3 != C )
                   => ? [T6: real] :
                        ( ( ( ord_less_real @ X3 @ C )
                         => ( ( ord_less_real @ X3 @ T6 )
                            & ( ord_less_real @ T6 @ C ) ) )
                        & ( ~ ( ord_less_real @ X3 @ C )
                         => ( ( ord_less_real @ C @ T6 )
                            & ( ord_less_real @ T6 @ X3 ) ) )
                        & ( ( F @ X3 )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ M2 ) )
                              @ ( set_ord_lessThan_nat @ N2 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X3 @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9564_Taylor__up,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: 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 @ B ) )
             => ( 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 @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ C @ T6 )
                  & ( ord_less_real @ T6 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M2 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T6 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9565_Taylor__down,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: 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 @ B ) )
             => ( 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 @ B )
             => ? [T6: real] :
                  ( ( ord_less_real @ A @ T6 )
                  & ( ord_less_real @ T6 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M2: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M2 @ C ) @ ( semiri2265585572941072030t_real @ M2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M2 ) )
                        @ ( 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_9566_Maclaurin__lemma2,axiom,
    ! [N2: nat,H2: real,Diff: nat > real > real,K: nat,B2: 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 ) )
       => ! [M6: nat,T7: real] :
            ( ( ( ord_less_nat @ M6 @ N2 )
              & ( ord_less_eq_real @ zero_zero_real @ T7 )
              & ( ord_less_eq_real @ T7 @ H2 ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M6 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M6 @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ U2 @ P6 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M6 ) ) )
                    @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M6 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M6 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M6 ) @ T7 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M6 ) @ P6 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P6 ) ) @ ( power_power_real @ T7 @ P6 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M6 ) ) ) )
                  @ ( times_times_real @ B2 @ ( divide_divide_real @ ( power_power_real @ T7 @ ( minus_minus_nat @ N2 @ ( suc @ M6 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M6 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T7 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9567_DERIV__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X8: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X8 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X3 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9568_DERIV__arccos,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9569_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X4: real] :
          ( ( member_real @ X4 @ ( 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 @ X4 @ N ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] :
                ( suminf_real
                @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X @ ( 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_9570_arccos__less__arccos,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ Y3 )
       => ( ( ord_less_eq_real @ Y3 @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y3 ) @ ( arccos @ X3 ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_9571_arccos__less__mono,axiom,
    ! [X3: real,Y3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y3 ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X3 ) @ ( arccos @ Y3 ) )
          = ( ord_less_real @ Y3 @ X3 ) ) ) ) ).

% arccos_less_mono
thf(fact_9572_DERIV__isconst3,axiom,
    ! [A: real,B: real,X3: real,Y3: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( member_real @ X3 @ ( set_or1633881224788618240n_real @ A @ B ) )
       => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ! [X4: real] :
                ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
           => ( ( F @ X3 )
              = ( F @ Y3 ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9573_arccos__lt__bounded,axiom,
    ! [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y3 )
     => ( ( ord_less_real @ Y3 @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y3 ) )
          & ( ord_less_real @ ( arccos @ Y3 ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_9574_sin__arccos__nonzero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X3 ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_9575_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_9576_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_9577_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_9578_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9579_LIM__fun__less__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R4 ) )
               => ( ord_less_real @ ( F @ X2 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_9580_LIM__fun__not__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( L != zero_zero_real )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R4 ) )
               => ( ( F @ X2 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9581_LIM__fun__gt__zero,axiom,
    ! [F: real > real,L: real,C: real] :
      ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ L ) @ ( topolo2177554685111907308n_real @ C @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [R4: real] :
            ( ( ord_less_real @ zero_zero_real @ R4 )
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R4 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9582_isCont__inverse__function2,axiom,
    ! [A: real,X3: real,B: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X3 )
     => ( ( ord_less_real @ X3 @ B )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ A @ Z3 )
             => ( ( ord_less_eq_real @ Z3 @ B )
               => ( ( G @ ( F @ Z3 ) )
                  = Z3 ) ) )
         => ( ! [Z3: real] :
                ( ( ord_less_eq_real @ A @ Z3 )
               => ( ( ord_less_eq_real @ Z3 @ B )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_9583_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_9584_isCont__arcosh,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_9585_DERIV__inverse__function,axiom,
    ! [F: real > real,D6: real,G: real > real,X3: real,A: real,B: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D6 @ ( topolo2177554685111907308n_real @ ( G @ X3 ) @ top_top_set_real ) )
     => ( ( D6 != zero_zero_real )
       => ( ( ord_less_real @ A @ X3 )
         => ( ( ord_less_real @ X3 @ B )
           => ( ! [Y5: real] :
                  ( ( ord_less_real @ A @ Y5 )
                 => ( ( ord_less_real @ Y5 @ B )
                   => ( ( F @ ( G @ Y5 ) )
                      = Y5 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D6 ) @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_9586_isCont__arccos,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_9587_isCont__arcsin,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_9588_LIM__less__bound,axiom,
    ! [B: real,X3: real,F: real > real] :
      ( ( ord_less_real @ B @ X3 )
     => ( ! [X4: real] :
            ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ B @ X3 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X3 ) ) ) ) ) ).

% LIM_less_bound
thf(fact_9589_isCont__artanh,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X3 @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_9590_isCont__inverse__function,axiom,
    ! [D: real,X3: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X3 ) ) @ D )
           => ( ( G @ ( F @ Z3 ) )
              = Z3 ) )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z3 @ X3 ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X3 ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_9591_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F5: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z3: real] :
            ( ( ord_less_eq_real @ A @ Z3 )
           => ( ( ord_less_eq_real @ Z3 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z3: real] :
              ( ( ord_less_eq_real @ A @ Z3 )
             => ( ( ord_less_eq_real @ Z3 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z3: real] :
                ( ( ord_less_real @ A @ Z3 )
               => ( ( ord_less_real @ Z3 @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
           => ( ! [Z3: real] :
                  ( ( ord_less_real @ A @ Z3 )
                 => ( ( ord_less_real @ Z3 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F5 @ Z3 ) @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) )
             => ? [C2: real] :
                  ( ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C2 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F5 @ C2 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_9592_summable__Leibniz_I2_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ zero_zero_real @ ( A @ zero_zero_nat ) )
         => ! [N6: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_9593_summable__Leibniz_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( ( ord_less_real @ ( A @ zero_zero_nat ) @ zero_zero_real )
         => ! [N6: nat] :
              ( member_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_9594_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
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_9595_trivial__limit__sequentially,axiom,
    at_top_nat != bot_bot_filter_nat ).

% trivial_limit_sequentially
thf(fact_9596_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_9597_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_9598_mult__nat__right__at__top,axiom,
    ! [C: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ C )
     => ( filterlim_nat_nat
        @ ^ [X: nat] : ( times_times_nat @ X @ C )
        @ at_top_nat
        @ at_top_nat ) ) ).

% mult_nat_right_at_top
thf(fact_9599_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] :
                ( ! [N6: nat] : ( ord_less_eq_real @ ( F @ N6 ) @ L4 )
                & ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat )
                & ! [N6: nat] : ( ord_less_eq_real @ L4 @ ( G @ N6 ) )
                & ( filterlim_nat_real @ G @ ( topolo2815343760600316023s_real @ L4 ) @ at_top_nat ) ) ) ) ) ) ).

% nested_sequence_unique
thf(fact_9600_LIMSEQ__inverse__zero,axiom,
    ! [X7: nat > real] :
      ( ! [R4: real] :
        ? [N8: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N8 @ N3 )
         => ( ord_less_real @ R4 @ ( X7 @ N3 ) ) )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( X7 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_9601_LIMSEQ__root__const,axiom,
    ! [C: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( root @ N @ C )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat ) ) ).

% LIMSEQ_root_const
thf(fact_9602_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_9603_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_9604_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 )
             => ? [N6: nat] : ( ord_less_eq_real @ L @ ( plus_plus_real @ ( F @ N6 ) @ E ) ) )
         => ( filterlim_nat_real @ F @ ( topolo2815343760600316023s_real @ L ) @ at_top_nat ) ) ) ) ).

% increasing_LIMSEQ
thf(fact_9605_LIMSEQ__realpow__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ord_less_real @ X3 @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X3 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_9606_LIMSEQ__divide__realpow__zero,axiom,
    ! [X3: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( divide_divide_real @ A @ ( power_power_real @ X3 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_9607_LIMSEQ__abs__realpow__zero2,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ C ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero2
thf(fact_9608_LIMSEQ__abs__realpow__zero,axiom,
    ! [C: real] :
      ( ( ord_less_real @ ( abs_abs_real @ C ) @ one_one_real )
     => ( filterlim_nat_real @ ( power_power_real @ ( abs_abs_real @ C ) ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ).

% LIMSEQ_abs_realpow_zero
thf(fact_9609_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X3: real] :
      ( ( ord_less_real @ one_one_real @ X3 )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ X3 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_9610_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_9611_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_9612_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_9613_zeroseq__arctan__series,axiom,
    ! [X3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X3 ) @ 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 @ X3 @ ( 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_9614_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
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_9615_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
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_9616_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] :
              ( ! [N6: nat] :
                  ( ord_less_eq_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N6: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_9617_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
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_9618_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
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I2: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I2 ) @ ( A @ I2 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_9619_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I2: nat] : ( P @ ( suc @ I2 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_9620_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_9621_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X4: nat] :
          ( ( ord_less_eq_nat @ C @ X4 )
         => ( P @ X4 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_9622_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_9623_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_9624_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I2: nat] : ( P @ ( plus_plus_nat @ I2 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_9625_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ B @ X4 )
         => ? [Y6: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
              & ( ord_less_real @ Y6 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_9626_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_9627_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9628_eventually__at__left__real,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ B @ A ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5984915006950818249n_real @ A ) ) ) ) ).

% eventually_at_left_real
thf(fact_9629_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_9630_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
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
            @ at_top_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_9631_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_9632_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X4: real] :
          ( ( ord_less_eq_real @ X4 @ B )
         => ? [Y6: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y6 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_9633_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
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
            @ at_bot_real
            @ F3 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_9634_GMVT,axiom,
    ! [A: real,B: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ( ord_less_eq_real @ A @ X4 )
              & ( ord_less_eq_real @ X4 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F ) )
       => ( ! [X4: real] :
              ( ( ( ord_less_real @ A @ X4 )
                & ( ord_less_real @ X4 @ B ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
         => ( ! [X4: real] :
                ( ( ( ord_less_eq_real @ A @ X4 )
                  & ( ord_less_eq_real @ X4 @ B ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G ) )
           => ( ! [X4: real] :
                  ( ( ( ord_less_real @ A @ X4 )
                    & ( ord_less_real @ X4 @ B ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C2: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C2 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C2 )
                  & ( ord_less_real @ C2 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ G_c )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ F_c ) ) ) ) ) ) ) ) ).

% GMVT
thf(fact_9635_MVT,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X4: real] :
              ( ( ord_less_real @ A @ X4 )
             => ( ( ord_less_real @ X4 @ B )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z3: real] :
              ( ( ord_less_real @ A @ Z3 )
              & ( ord_less_real @ Z3 @ B )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_9636_Rolle__deriv,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X4: real] :
                ( ( ord_less_real @ A @ X4 )
               => ( ( ord_less_real @ X4 @ B )
                 => ( has_de1759254742604945161l_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
           => ? [Z3: real] :
                ( ( ord_less_real @ A @ Z3 )
                & ( ord_less_real @ Z3 @ B )
                & ( ( F5 @ Z3 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_9637_mvt,axiom,
    ! [A: real,B: real,F: real > real,F5: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X4: real] :
              ( ( ord_less_real @ A @ X4 )
             => ( ( ord_less_real @ X4 @ B )
               => ( has_de1759254742604945161l_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A @ Xi )
               => ( ( ord_less_real @ Xi @ B )
                 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                   != ( F5 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_9638_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_real @ A @ X4 )
           => ( ( ord_less_real @ X4 @ B )
             => ? [Y6: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y6 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_9639_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X4: real] :
            ( ( ord_less_real @ A @ X4 )
           => ( ( ord_less_real @ X4 @ B )
             => ? [Y6: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y6 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
                  & ( ord_less_real @ Y6 @ zero_zero_real ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ) ).

% DERIV_neg_imp_decreasing_open
thf(fact_9640_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X4: real] :
              ( ( ord_less_real @ A @ X4 )
             => ( ( ord_less_real @ X4 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
         => ( ( F @ B )
            = ( F @ A ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_9641_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: real > real,X3: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X4: real] :
              ( ( ord_less_real @ A @ X4 )
             => ( ( ord_less_real @ X4 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A @ X3 )
           => ( ( ord_less_eq_real @ X3 @ B )
             => ( ( F @ X3 )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_9642_Rolle,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X4: real] :
                ( ( ord_less_real @ A @ X4 )
               => ( ( ord_less_real @ X4 @ B )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) )
           => ? [Z3: real] :
                ( ( ord_less_real @ A @ Z3 )
                & ( ord_less_real @ Z3 @ B )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z3 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_9643_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_9644_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_9645_eventually__at__right__real,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( eventually_real
        @ ^ [X: real] : ( member_real @ X @ ( set_or1633881224788618240n_real @ A @ B ) )
        @ ( topolo2177554685111907308n_real @ A @ ( set_or5849166863359141190n_real @ A ) ) ) ) ).

% eventually_at_right_real
thf(fact_9646_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_9647_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_9648_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_9649_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_9650_Gcd__eq__Max,axiom,
    ! [M7: set_nat] :
      ( ( finite_finite_nat @ M7 )
     => ( ( M7 != bot_bot_set_nat )
       => ( ~ ( member_nat @ zero_zero_nat @ M7 )
         => ( ( gcd_Gcd_nat @ M7 )
            = ( lattic8265883725875713057ax_nat
              @ ( comple7806235888213564991et_nat
                @ ( image_nat_set_nat
                  @ ^ [M2: nat] :
                      ( collect_nat
                      @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ M2 ) )
                  @ M7 ) ) ) ) ) ) ) ).

% Gcd_eq_Max
thf(fact_9651_Max__divisors__self__nat,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] : ( dvd_dvd_nat @ D5 @ N2 ) ) )
        = N2 ) ) ).

% Max_divisors_self_nat
thf(fact_9652_bdd__above__nat,axiom,
    condit2214826472909112428ve_nat = finite_finite_nat ).

% bdd_above_nat
thf(fact_9653_card__le__Suc__Max,axiom,
    ! [S3: set_nat] :
      ( ( finite_finite_nat @ S3 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S3 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S3 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_9654_Sup__nat__def,axiom,
    ( complete_Sup_Sup_nat
    = ( ^ [X5: set_nat] : ( if_nat @ ( X5 = bot_bot_set_nat ) @ zero_zero_nat @ ( lattic8265883725875713057ax_nat @ X5 ) ) ) ) ).

% Sup_nat_def
thf(fact_9655_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M2: nat,N: nat] :
          ( if_nat @ ( N = zero_zero_nat ) @ zero_zero_nat
          @ ( lattic8265883725875713057ax_nat
            @ ( collect_nat
              @ ^ [K3: nat] : ( ord_less_eq_nat @ ( times_times_nat @ K3 @ N ) @ M2 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_9656_gcd__is__Max__divisors__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( gcd_gcd_nat @ M @ N2 )
        = ( lattic8265883725875713057ax_nat
          @ ( collect_nat
            @ ^ [D5: nat] :
                ( ( dvd_dvd_nat @ D5 @ M )
                & ( dvd_dvd_nat @ D5 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_nat
thf(fact_9657_Max__divisors__self__int,axiom,
    ! [N2: int] :
      ( ( N2 != zero_zero_int )
     => ( ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D5: int] : ( dvd_dvd_int @ D5 @ N2 ) ) )
        = ( abs_abs_int @ N2 ) ) ) ).

% Max_divisors_self_int
thf(fact_9658_gcd__is__Max__divisors__int,axiom,
    ! [N2: int,M: int] :
      ( ( N2 != zero_zero_int )
     => ( ( gcd_gcd_int @ M @ N2 )
        = ( lattic8263393255366662781ax_int
          @ ( collect_int
            @ ^ [D5: int] :
                ( ( dvd_dvd_int @ D5 @ M )
                & ( dvd_dvd_int @ D5 @ N2 ) ) ) ) ) ) ).

% gcd_is_Max_divisors_int
thf(fact_9659_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X: complex,Y: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_9660_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X: real,Y: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_9661_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] :
          ! [M2: nat] :
            ( ( ord_less_eq_nat @ N5 @ M2 )
           => ! [N: nat] :
                ( ( ord_less_eq_nat @ N5 @ N )
               => ( P @ ( product_Pair_nat_nat @ N @ M2 ) ) ) ) ) ) ).

% eventually_prod_sequentially
thf(fact_9662_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_9663_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_9664_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M2: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M2 ) @ M2 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_9665_less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi6264000038957366511cl_nat @ pred_nat ) )
      = ( ord_less_nat @ M @ N2 ) ) ).

% less_eq
thf(fact_9666_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( inj_on_real_real
        @ ^ [Y: real] : ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_9667_log__inj,axiom,
    ! [B: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( inj_on_real_real @ ( log @ B ) @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% log_inj
thf(fact_9668_inj__Suc,axiom,
    ! [N7: set_nat] : ( inj_on_nat_nat @ suc @ N7 ) ).

% inj_Suc
thf(fact_9669_inj__on__diff__nat,axiom,
    ! [N7: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ N7 )
         => ( ord_less_eq_nat @ K @ N3 ) )
     => ( inj_on_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
        @ N7 ) ) ).

% inj_on_diff_nat
thf(fact_9670_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_9671_pred__nat__trancl__eq__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( member8440522571783428010at_nat @ ( product_Pair_nat_nat @ M @ N2 ) @ ( transi2905341329935302413cl_nat @ pred_nat ) )
      = ( ord_less_eq_nat @ M @ N2 ) ) ).

% pred_nat_trancl_eq_le
thf(fact_9672_measure__function__int,axiom,
    fun_is_measure_int @ ( comp_int_nat_int @ nat2 @ abs_abs_int ) ).

% measure_function_int
thf(fact_9673_powr__real__of__int_H,axiom,
    ! [X3: real,N2: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ X3 )
     => ( ( ( X3 != zero_zero_real )
          | ( ord_less_int @ zero_zero_int @ N2 ) )
       => ( ( powr_real @ X3 @ ( ring_1_of_int_real @ N2 ) )
          = ( power_int_real @ X3 @ N2 ) ) ) ) ).

% powr_real_of_int'
thf(fact_9674_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_9675_min__weak__def,axiom,
    ( fun_min_weak
    = ( sup_su5525570899277871387at_nat @ ( min_ex6901939911449802026at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% min_weak_def
thf(fact_9676_sup__int__def,axiom,
    sup_sup_int = ord_max_int ).

% sup_int_def
thf(fact_9677_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_9678_max__weak__def,axiom,
    ( fun_max_weak
    = ( sup_su5525570899277871387at_nat @ ( max_ex8135407076693332796at_nat @ fun_pair_leq ) @ ( insert9069300056098147895at_nat @ ( produc2922128104949294807at_nat @ bot_bo2099793752762293965at_nat @ bot_bo2099793752762293965at_nat ) @ bot_bo228742789529271731at_nat ) ) ) ).

% max_weak_def
thf(fact_9679_pos__deriv__imp__strict__mono,axiom,
    ! [F: real > real,F5: real > real] :
      ( ! [X4: real] : ( has_fi5821293074295781190e_real @ F @ ( F5 @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( F5 @ X4 ) )
       => ( order_7092887310737990675l_real @ F ) ) ) ).

% pos_deriv_imp_strict_mono
thf(fact_9680_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_9681_infinite__enumerate,axiom,
    ! [S3: set_nat] :
      ( ~ ( finite_finite_nat @ S3 )
     => ? [R4: nat > nat] :
          ( ( order_5726023648592871131at_nat @ R4 )
          & ! [N6: nat] : ( member_nat @ ( R4 @ N6 ) @ S3 ) ) ) ).

% infinite_enumerate
thf(fact_9682_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_9683_less__int__def,axiom,
    ( ord_less_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).

% less_int_def
thf(fact_9684_nat__def,axiom,
    ( nat2
    = ( map_fu2345160673673942751at_nat @ rep_Integ @ id_nat @ ( produc6842872674320459806at_nat @ minus_minus_nat ) ) ) ).

% nat_def
thf(fact_9685_less__eq__int__def,axiom,
    ( ord_less_eq_int
    = ( map_fu434086159418415080_int_o @ rep_Integ @ ( map_fu4826362097070443709at_o_o @ rep_Integ @ id_o )
      @ ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y ) ) ) ) ) ) ).

% less_eq_int_def
thf(fact_9686_positive__rat,axiom,
    ! [A: int,B: int] :
      ( ( positive @ ( fract @ A @ B ) )
      = ( ord_less_int @ zero_zero_int @ ( times_times_int @ A @ B ) ) ) ).

% positive_rat
thf(fact_9687_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y: rat] : ( positive @ ( minus_minus_rat @ Y @ X ) ) ) ) ).

% less_rat_def
thf(fact_9688_Rat_Opositive_Orep__eq,axiom,
    ( positive
    = ( ^ [X: rat] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ ( rep_Rat @ X ) ) @ ( product_snd_int_int @ ( rep_Rat @ X ) ) ) ) ) ) ).

% Rat.positive.rep_eq
thf(fact_9689_Rat_Opositive__def,axiom,
    ( positive
    = ( map_fu898904425404107465nt_o_o @ rep_Rat @ id_o
      @ ^ [X: product_prod_int_int] : ( ord_less_int @ zero_zero_int @ ( times_times_int @ ( product_fst_int_int @ X ) @ ( product_snd_int_int @ X ) ) ) ) ) ).

% Rat.positive_def
thf(fact_9690_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_9691_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_9692_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_9693_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_9694_list__encode_Oelims,axiom,
    ! [X3: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X3 )
        = Y3 )
     => ( ( ( X3 = nil_nat )
         => ( Y3 != zero_zero_nat ) )
       => ~ ! [X4: nat,Xs3: list_nat] :
              ( ( X3
                = ( cons_nat @ X4 @ Xs3 ) )
             => ( Y3
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_9695_list__encode_Osimps_I1_J,axiom,
    ( ( nat_list_encode @ nil_nat )
    = zero_zero_nat ) ).

% list_encode.simps(1)
thf(fact_9696_list__encode_Osimps_I2_J,axiom,
    ! [X3: nat,Xs: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X3 @ Xs ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X3 @ ( nat_list_encode @ Xs ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_9697_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I2: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I2 ) @ Js @ ( upto_aux @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9698_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_9699_upto_Opelims,axiom,
    ! [X3: int,Xa2: int,Y3: list_int] :
      ( ( ( upto @ X3 @ Xa2 )
        = Y3 )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) )
       => ~ ( ( ( ( ord_less_eq_int @ X3 @ Xa2 )
               => ( Y3
                  = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
              & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
               => ( Y3 = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X3 @ Xa2 ) ) ) ) ) ).

% upto.pelims
thf(fact_9700_upto__empty,axiom,
    ! [J: int,I: int] :
      ( ( ord_less_int @ J @ I )
     => ( ( upto @ I @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9701_upto__Nil2,axiom,
    ! [I: int,J: int] :
      ( ( nil_int
        = ( upto @ I @ J ) )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil2
thf(fact_9702_upto__Nil,axiom,
    ! [I: int,J: int] :
      ( ( ( upto @ I @ J )
        = nil_int )
      = ( ord_less_int @ J @ I ) ) ).

% upto_Nil
thf(fact_9703_upto__single,axiom,
    ! [I: int] :
      ( ( upto @ I @ I )
      = ( cons_int @ I @ nil_int ) ) ).

% upto_single
thf(fact_9704_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_9705_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_9706_upto__rec__numeral_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(1)
thf(fact_9707_upto__rec__numeral_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( numeral_numeral_int @ M ) @ ( upto @ ( plus_plus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(2)
thf(fact_9708_upto__rec__numeral_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( numeral_numeral_int @ N2 ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(3)
thf(fact_9709_upto__rec__numeral_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = ( cons_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( upto @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
       => ( ( upto @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
          = nil_int ) ) ) ).

% upto_rec_numeral(4)
thf(fact_9710_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_9711_distinct__upto,axiom,
    ! [I: int,J: int] : ( distinct_int @ ( upto @ I @ J ) ) ).

% distinct_upto
thf(fact_9712_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I2: int,J3: int] : ( append_int @ ( upto @ I2 @ J3 ) ) ) ) ).

% upto_aux_def
thf(fact_9713_upto__code,axiom,
    ( upto
    = ( ^ [I2: int,J3: int] : ( upto_aux @ I2 @ J3 @ nil_int ) ) ) ).

% upto_code
thf(fact_9714_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_9715_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_9716_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ I2 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9717_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_9718_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_9719_upto_Osimps,axiom,
    ( upto
    = ( ^ [I2: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I2 @ J3 ) @ ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9720_upto_Oelims,axiom,
    ! [X3: int,Xa2: int,Y3: list_int] :
      ( ( ( upto @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( ord_less_eq_int @ X3 @ Xa2 )
         => ( Y3
            = ( cons_int @ X3 @ ( upto @ ( plus_plus_int @ X3 @ one_one_int ) @ Xa2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ X3 @ Xa2 )
         => ( Y3 = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9721_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_9722_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I2: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_9723_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_9724_list__encode_Opelims,axiom,
    ! [X3: list_nat,Y3: nat] :
      ( ( ( nat_list_encode @ X3 )
        = Y3 )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X3 )
       => ( ( ( X3 = nil_nat )
           => ( ( Y3 = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X4: nat,Xs3: list_nat] :
                ( ( X3
                  = ( cons_nat @ X4 @ Xs3 ) )
               => ( ( Y3
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs3 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X4 @ Xs3 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_9725_remdups__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( remdups_nat @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% remdups_upt
thf(fact_9726_hd__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( ord_less_nat @ I @ J )
     => ( ( hd_nat @ ( upt @ I @ J ) )
        = I ) ) ).

% hd_upt
thf(fact_9727_drop__upt,axiom,
    ! [M: nat,I: nat,J: nat] :
      ( ( drop_nat @ M @ ( upt @ I @ J ) )
      = ( upt @ ( plus_plus_nat @ I @ M ) @ J ) ) ).

% drop_upt
thf(fact_9728_length__upt,axiom,
    ! [I: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I @ J ) )
      = ( minus_minus_nat @ J @ I ) ) ).

% length_upt
thf(fact_9729_take__upt,axiom,
    ! [I: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I @ M ) @ N2 )
     => ( ( take_nat @ M @ ( upt @ I @ N2 ) )
        = ( upt @ I @ ( plus_plus_nat @ I @ M ) ) ) ) ).

% take_upt
thf(fact_9730_upt__conv__Nil,axiom,
    ! [J: nat,I: nat] :
      ( ( ord_less_eq_nat @ J @ I )
     => ( ( upt @ I @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_9731_sorted__list__of__set__range,axiom,
    ! [M: nat,N2: nat] :
      ( ( linord2614967742042102400et_nat @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sorted_list_of_set_range
thf(fact_9732_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_9733_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_9734_upt__rec__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = ( cons_nat @ ( numeral_numeral_nat @ M ) @ ( upt @ ( suc @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) )
      & ( ~ ( ord_less_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
       => ( ( upt @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
          = nil_nat ) ) ) ).

% upt_rec_numeral
thf(fact_9735_upt__0,axiom,
    ! [I: nat] :
      ( ( upt @ I @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_9736_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N2: nat,Ns: list_nat,Q4: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
        = ( upt @ M @ Q4 ) )
      = ( ( cons_nat @ N2 @ Ns )
        = ( upt @ ( suc @ M ) @ Q4 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_9737_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I2: nat] : ( plus_plus_nat @ I2 @ N2 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_9738_map__Suc__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat @ suc @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% map_Suc_upt
thf(fact_9739_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).

% atMost_upto
thf(fact_9740_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ ( suc @ M2 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_9741_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).

% atLeast_upt
thf(fact_9742_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ N @ ( suc @ M2 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_9743_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I2: nat,J3: nat] : ( set_nat2 @ ( upt @ I2 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_9744_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N: nat,M2: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ M2 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_9745_distinct__upt,axiom,
    ! [I: nat,J: nat] : ( distinct_nat @ ( upt @ I @ J ) ) ).

% distinct_upt
thf(fact_9746_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_9747_map__decr__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( map_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) )
        @ ( upt @ ( suc @ M ) @ ( suc @ N2 ) ) )
      = ( upt @ M @ N2 ) ) ).

% map_decr_upt
thf(fact_9748_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_9749_upt__eq__Cons__conv,axiom,
    ! [I: nat,J: nat,X3: nat,Xs: list_nat] :
      ( ( ( upt @ I @ J )
        = ( cons_nat @ X3 @ Xs ) )
      = ( ( ord_less_nat @ I @ J )
        & ( I = X3 )
        & ( ( upt @ ( plus_plus_nat @ I @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_9750_upt__rec,axiom,
    ( upt
    = ( ^ [I2: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I2 @ J3 ) @ ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_9751_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_9752_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_9753_sum__list__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups4561878855575611511st_nat @ ( upt @ M @ N2 ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : X
          @ ( set_or4665077453230672383an_nat @ M @ N2 ) ) ) ) ).

% sum_list_upt
thf(fact_9754_card__length__sum__list__rec,axiom,
    ! [M: nat,N7: nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ M )
     => ( ( finite_card_list_nat
          @ ( collect_list_nat
            @ ^ [L2: list_nat] :
                ( ( ( size_size_list_nat @ L2 )
                  = M )
                & ( ( groups4561878855575611511st_nat @ L2 )
                  = N7 ) ) ) )
        = ( plus_plus_nat
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = ( minus_minus_nat @ M @ one_one_nat ) )
                  & ( ( groups4561878855575611511st_nat @ L2 )
                    = N7 ) ) ) )
          @ ( finite_card_list_nat
            @ ( collect_list_nat
              @ ^ [L2: list_nat] :
                  ( ( ( size_size_list_nat @ L2 )
                    = M )
                  & ( ( plus_plus_nat @ ( groups4561878855575611511st_nat @ L2 ) @ one_one_nat )
                    = N7 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_9755_card__length__sum__list,axiom,
    ! [M: nat,N7: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N7 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N7 @ M ) @ one_one_nat ) @ N7 ) ) ).

% card_length_sum_list
thf(fact_9756_tl__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( tl_nat @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ N2 ) ) ).

% tl_upt
thf(fact_9757_sorted__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).

% sorted_upt
thf(fact_9758_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).

% sorted_wrt_upt
thf(fact_9759_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_9760_sorted__upto,axiom,
    ! [M: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N2 ) ) ).

% sorted_upto
thf(fact_9761_sorted__wrt__upto,axiom,
    ! [I: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I @ J ) ) ).

% sorted_wrt_upto
thf(fact_9762_pairs__le__eq__Sigma,axiom,
    ! [M: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I2: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ J3 ) @ M ) ) )
      = ( produc457027306803732586at_nat @ ( set_ord_atMost_nat @ M )
        @ ^ [R5: nat] : ( set_ord_atMost_nat @ ( minus_minus_nat @ M @ R5 ) ) ) ) ).

% pairs_le_eq_Sigma
thf(fact_9763_natLess__def,axiom,
    ( bNF_Ca8459412986667044542atLess
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ) ).

% natLess_def
thf(fact_9764_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_9765_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_9766_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_9767_set__decode__div__2,axiom,
    ! [X3: nat] :
      ( ( nat_set_decode @ ( divide_divide_nat @ X3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( vimage_nat_nat @ suc @ ( nat_set_decode @ X3 ) ) ) ).

% set_decode_div_2
thf(fact_9768_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_9769_Restr__natLeq,axiom,
    ! [N2: nat] :
      ( ( inf_in2572325071724192079at_nat @ bNF_Ca8665028551170535155natLeq
        @ ( produc457027306803732586at_nat
          @ ( collect_nat
            @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) )
          @ ^ [Uu3: nat] :
              ( collect_nat
              @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) ) ) )
      = ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ N2 )
              & ( ord_less_nat @ Y @ N2 )
              & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% Restr_natLeq
thf(fact_9770_natLeq__def,axiom,
    ( bNF_Ca8665028551170535155natLeq
    = ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_eq_nat ) ) ) ).

% natLeq_def
thf(fact_9771_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
          @ ^ [X: nat,Y: nat] :
              ( ( ord_less_nat @ X @ N2 )
              & ( ord_less_nat @ Y @ N2 )
              & ( ord_less_eq_nat @ X @ Y ) ) ) ) ) ).

% Restr_natLeq2
thf(fact_9772_natLeq__underS__less,axiom,
    ! [N2: nat] :
      ( ( order_underS_nat @ bNF_Ca8665028551170535155natLeq @ N2 )
      = ( collect_nat
        @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) ) ) ).

% natLeq_underS_less
thf(fact_9773_Arg__bounded,axiom,
    ! [Z2: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
      & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ).

% Arg_bounded
thf(fact_9774_Arg__correct,axiom,
    ! [Z2: complex] :
      ( ( Z2 != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z2 )
          = ( cis @ ( arg @ Z2 ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z2 ) )
        & ( ord_less_eq_real @ ( arg @ Z2 ) @ pi ) ) ) ).

% Arg_correct
thf(fact_9775_Field__natLeq__on,axiom,
    ! [N2: nat] :
      ( ( field_nat
        @ ( collec3392354462482085612at_nat
          @ ( produc6081775807080527818_nat_o
            @ ^ [X: nat,Y: nat] :
                ( ( ord_less_nat @ X @ N2 )
                & ( ord_less_nat @ Y @ N2 )
                & ( ord_less_eq_nat @ X @ Y ) ) ) ) )
      = ( collect_nat
        @ ^ [X: nat] : ( ord_less_nat @ X @ N2 ) ) ) ).

% Field_natLeq_on
thf(fact_9776_cis__Arg__unique,axiom,
    ! [Z2: complex,X3: real] :
      ( ( ( sgn_sgn_complex @ Z2 )
        = ( cis @ X3 ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X3 )
       => ( ( ord_less_eq_real @ X3 @ pi )
         => ( ( arg @ Z2 )
            = X3 ) ) ) ) ).

% cis_Arg_unique
thf(fact_9777_bij__betw__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( bij_betw_nat_complex
        @ ^ [K3: nat] : ( cis @ ( divide_divide_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( semiri5074537144036343181t_real @ K3 ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
        @ ( set_ord_lessThan_nat @ N2 )
        @ ( collect_complex
          @ ^ [Z6: complex] :
              ( ( power_power_complex @ Z6 @ N2 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9778_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_9779_bij__betw__Suc,axiom,
    ! [M7: set_nat,N7: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N7 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N7 ) ) ).

% bij_betw_Suc
thf(fact_9780_Arg__def,axiom,
    ( arg
    = ( ^ [Z6: complex] :
          ( if_real @ ( Z6 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A3: real] :
                ( ( ( sgn_sgn_complex @ Z6 )
                  = ( cis @ A3 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
                & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_9781_not__negative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_int @ ( bit_ri7919022796975470100ot_int @ K ) @ zero_zero_int )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% not_negative_int_iff
thf(fact_9782_not__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ord_less_int @ K @ zero_zero_int ) ) ).

% not_nonnegative_int_iff
thf(fact_9783_wf__int__ge__less__than2,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than2 @ D ) ) ).

% wf_int_ge_less_than2
thf(fact_9784_wf__int__ge__less__than,axiom,
    ! [D: int] : ( wf_int @ ( int_ge_less_than @ D ) ) ).

% wf_int_ge_less_than
thf(fact_9785_wf__less,axiom,
    wf_nat @ ( collec3392354462482085612at_nat @ ( produc6081775807080527818_nat_o @ ord_less_nat ) ) ).

% wf_less
thf(fact_9786_and__not__num__eq__None__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = none_num )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
        = zero_zero_int ) ) ).

% and_not_num_eq_None_iff
thf(fact_9787_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9788_and__not__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_and_not_num @ one @ ( bit1 @ N2 ) )
      = none_num ) ).

% and_not_num.simps(3)
thf(fact_9789_and__not__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y3: option_num] :
      ( ( ( bit_and_not_num @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y3 != none_num ) ) )
       => ( ( ( X3 = one )
           => ( ? [N3: num] :
                  ( Xa2
                  = ( bit0 @ N3 ) )
             => ( Y3
               != ( some_num @ one ) ) ) )
         => ( ( ( X3 = one )
             => ( ? [N3: num] :
                    ( Xa2
                    = ( bit1 @ N3 ) )
               => ( Y3 != none_num ) ) )
           => ( ! [M3: num] :
                  ( ( X3
                    = ( bit0 @ M3 ) )
                 => ( ( Xa2 = one )
                   => ( Y3
                     != ( some_num @ ( bit0 @ M3 ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X3
                      = ( bit0 @ M3 ) )
                   => ! [N3: num] :
                        ( ( Xa2
                          = ( bit0 @ N3 ) )
                       => ( Y3
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X3
                        = ( bit0 @ M3 ) )
                     => ! [N3: num] :
                          ( ( Xa2
                            = ( bit1 @ N3 ) )
                         => ( Y3
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X3
                          = ( bit1 @ M3 ) )
                       => ( ( Xa2 = one )
                         => ( Y3
                           != ( some_num @ ( bit0 @ M3 ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X3
                            = ( bit1 @ M3 ) )
                         => ! [N3: num] :
                              ( ( Xa2
                                = ( bit0 @ N3 ) )
                             => ( Y3
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                  @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X3
                              = ( bit1 @ M3 ) )
                           => ! [N3: num] :
                                ( ( Xa2
                                  = ( bit1 @ N3 ) )
                               => ( Y3
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_9790_and__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y3: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y3
             != ( some_num @ one ) ) ) )
       => ( ( ( X3 = one )
           => ( ? [N3: num] :
                  ( Xa2
                  = ( bit0 @ N3 ) )
             => ( Y3 != none_num ) ) )
         => ( ( ( X3 = one )
             => ( ? [N3: num] :
                    ( Xa2
                    = ( bit1 @ N3 ) )
               => ( Y3
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M3: num] :
                    ( X3
                    = ( bit0 @ M3 ) )
               => ( ( Xa2 = one )
                 => ( Y3 != none_num ) ) )
             => ( ! [M3: num] :
                    ( ( X3
                      = ( bit0 @ M3 ) )
                   => ! [N3: num] :
                        ( ( Xa2
                          = ( bit0 @ N3 ) )
                       => ( Y3
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X3
                        = ( bit0 @ M3 ) )
                     => ! [N3: num] :
                          ( ( Xa2
                            = ( bit1 @ N3 ) )
                         => ( Y3
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
                 => ( ( ? [M3: num] :
                          ( X3
                          = ( bit1 @ M3 ) )
                     => ( ( Xa2 = one )
                       => ( Y3
                         != ( some_num @ one ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X3
                            = ( bit1 @ M3 ) )
                         => ! [N3: num] :
                              ( ( Xa2
                                = ( bit0 @ N3 ) )
                             => ( Y3
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X3
                              = ( bit1 @ M3 ) )
                           => ! [N3: num] :
                                ( ( Xa2
                                  = ( bit1 @ N3 ) )
                               => ( Y3
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_9791_and__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
      = none_num ) ).

% and_num.simps(4)
thf(fact_9792_and__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N2 ) )
      = none_num ) ).

% and_num.simps(2)
thf(fact_9793_xor__num_Oelims,axiom,
    ! [X3: num,Xa2: num,Y3: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X3 @ Xa2 )
        = Y3 )
     => ( ( ( X3 = one )
         => ( ( Xa2 = one )
           => ( Y3 != none_num ) ) )
       => ( ( ( X3 = one )
           => ! [N3: num] :
                ( ( Xa2
                  = ( bit0 @ N3 ) )
               => ( Y3
                 != ( some_num @ ( bit1 @ N3 ) ) ) ) )
         => ( ( ( X3 = one )
             => ! [N3: num] :
                  ( ( Xa2
                    = ( bit1 @ N3 ) )
                 => ( Y3
                   != ( some_num @ ( bit0 @ N3 ) ) ) ) )
           => ( ! [M3: num] :
                  ( ( X3
                    = ( bit0 @ M3 ) )
                 => ( ( Xa2 = one )
                   => ( Y3
                     != ( some_num @ ( bit1 @ M3 ) ) ) ) )
             => ( ! [M3: num] :
                    ( ( X3
                      = ( bit0 @ M3 ) )
                   => ! [N3: num] :
                        ( ( Xa2
                          = ( bit0 @ N3 ) )
                       => ( Y3
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) )
               => ( ! [M3: num] :
                      ( ( X3
                        = ( bit0 @ M3 ) )
                     => ! [N3: num] :
                          ( ( Xa2
                            = ( bit1 @ N3 ) )
                         => ( Y3
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) )
                 => ( ! [M3: num] :
                        ( ( X3
                          = ( bit1 @ M3 ) )
                       => ( ( Xa2 = one )
                         => ( Y3
                           != ( some_num @ ( bit0 @ M3 ) ) ) ) )
                   => ( ! [M3: num] :
                          ( ( X3
                            = ( bit1 @ M3 ) )
                         => ! [N3: num] :
                              ( ( Xa2
                                = ( bit0 @ N3 ) )
                             => ( Y3
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) )
                     => ~ ! [M3: num] :
                            ( ( X3
                              = ( bit1 @ M3 ) )
                           => ! [N3: num] :
                                ( ( Xa2
                                  = ( bit1 @ N3 ) )
                               => ( Y3
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M3 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_9794_xor__num_Osimps_I1_J,axiom,
    ( ( bit_un2480387367778600638or_num @ one @ one )
    = none_num ) ).

% xor_num.simps(1)

% Helper facts (34)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X3: int,Y3: int] :
      ( ( if_int @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X3: int,Y3: int] :
      ( ( if_int @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X3: nat,Y3: nat] :
      ( ( if_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( if_rat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X3: rat,Y3: rat] :
      ( ( if_rat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y3: real] :
      ( ( if_real @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X3: real,Y3: real] :
      ( ( if_real @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X5: real] : ( P @ X5 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( if_complex @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X3: complex,Y3: complex] :
      ( ( if_complex @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Extended____Nat__Oenat_T,axiom,
    ! [X3: extended_enat,Y3: extended_enat] :
      ( ( if_Extended_enat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X3: code_integer,Y3: code_integer] :
      ( ( if_Code_integer @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( if_set_int @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X3: set_int,Y3: set_int] :
      ( ( if_set_int @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X3: list_int,Y3: list_int] :
      ( ( if_list_int @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X3: list_int,Y3: list_int] :
      ( ( if_list_int @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X3: list_nat,Y3: list_nat] :
      ( ( if_list_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: int > int,Y3: int > int] :
      ( ( if_int_int @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: int > int,Y3: int > int] :
      ( ( if_int_int @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X3: option_num,Y3: option_num] :
      ( ( if_option_num @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X3: option_num,Y3: option_num] :
      ( ( if_option_num @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X3: product_prod_int_int,Y3: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X3: product_prod_nat_nat,Y3: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X3: produc6271795597528267376eger_o,Y3: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X3 @ Y3 )
      = X3 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X3 @ Y3 )
      = Y3 ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X3: produc8923325533196201883nteger,Y3: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X3 @ Y3 )
      = X3 ) ).

% Conjectures (2)
thf(conj_0,hypothesis,
    ! [X2: nat] :
      ( ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ maxs ) @ X2 )
     => thesis ) ).

thf(conj_1,conjecture,
    thesis ).

%------------------------------------------------------------------------------