TPTP Problem File: ITP221^3.p

%------------------------------------------------------------------------------
% File     : ITP221^3 : TPTP v8.2.0. Released v8.1.0.
% Domain   : Interactive Theorem Proving
% Problem  : Sledgehammer problem VEBT_Definitions 00493_022399
% 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    : 0063_VEBT_Definitions_00493_022399 [Des22]

% Status   : Theorem
% Rating   : 0.70 v8.2.0, 0.85 v8.1.0
% Syntax   : Number of formulae    : 11148 (6201 unt; 899 typ;   0 def)
%            Number of atoms       : 25899 (11696 equ;   0 cnn)
%            Maximal formula atoms :   71 (   2 avg)
%            Number of connectives : 104223 (2335   ~; 517   |;1426   &;91503   @)
%                                         (   0 <=>;8442  =>;   0  <=;   0 <~>)
%            Maximal formula depth :   39 (   5 avg)
%            Number of types       :   85 (  84 usr)
%            Number of type conns  : 2723 (2723   >;   0   *;   0   +;   0  <<)
%            Number of symbols     :  818 ( 815 usr;  57 con; 0-8 aty)
%            Number of variables   : 23152 (1825   ^;20748   !; 579   ?;23152   :)
% 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 17:39:15.009
%------------------------------------------------------------------------------
% Could-be-implicit typings (84)
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thf(ty_n_t__Option__Ooption_It__Num__Onum_J,type,
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thf(ty_n_t__Filter__Ofilter_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__String__Ochar_J,type,
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thf(ty_n_t__List__Olist_It__Real__Oreal_J,type,
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thf(ty_n_t__List__Olist_It__Num__Onum_J,type,
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thf(ty_n_t__List__Olist_It__Int__Oint_J,type,
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thf(ty_n_t__VEBT____Definitions__OVEBT,type,
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thf(ty_n_t__Set__Oset_It__Rat__Orat_J,type,
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thf(ty_n_t__Set__Oset_It__Num__Onum_J,type,
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thf(ty_n_t__Set__Oset_It__Nat__Onat_J,type,
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thf(ty_n_t__Set__Oset_It__Int__Oint_J,type,
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thf(ty_n_t__Extended____Nat__Oenat,type,
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thf(ty_n_t__Complex__Ocomplex,type,
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thf(ty_n_t__Set__Oset_I_Eo_J,type,
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thf(ty_n_t__String__Ochar,type,
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thf(ty_n_t__Real__Oreal,type,
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thf(ty_n_t__Rat__Orat,type,
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thf(ty_n_t__Num__Onum,type,
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thf(ty_n_t__Nat__Onat,type,
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thf(ty_n_t__Int__Oint,type,
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% Explicit typings (815)
thf(sy_c_Archimedean__Field_Oceiling_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oceiling_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Ofloor__ceiling__class_Ofloor_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Ofrac_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Ofrac_001t__Real__Oreal,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Rat__Orat,type,
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thf(sy_c_Archimedean__Field_Oround_001t__Real__Oreal,type,
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thf(sy_c_Binomial_Obinomial,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Complex__Ocomplex,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Int__Oint,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Nat__Onat,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Rat__Orat,type,
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thf(sy_c_Binomial_Ogbinomial_001t__Real__Oreal,type,
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thf(sy_c_Bit__Operations_Oand__int__rel,type,
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thf(sy_c_Bit__Operations_Oand__not__num,type,
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thf(sy_c_Bit__Operations_Oand__not__num__rel,type,
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thf(sy_c_Bit__Operations_Oconcat__bit,type,
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thf(sy_c_Bit__Operations_Oor__not__num__neg,type,
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thf(sy_c_Bit__Operations_Oor__not__num__neg__rel,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Odrop__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oflip__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Omask_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Int__Oint,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oor_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Otake__bit_001t__Nat__Onat,type,
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thf(sy_c_Bit__Operations_Osemiring__bit__operations__class_Oxor_001t__Nat__Onat,type,
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thf(sy_c_Complex_Ocomplex_ORe,type,
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thf(sy_c_Filter_Oat__top_001t__Real__Oreal,type,
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thf(sy_c_If_001t__Nat__Onat,type,
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thf(sy_c_If_001t__Rat__Orat,type,
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thf(sy_c_If_001t__Real__Oreal,type,
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thf(sy_c_If_001t__Set__Oset_It__Int__Oint_J,type,
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thf(sy_c_Int_OAbs__Integ,type,
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thf(sy_c_Int_ORep__Integ,type,
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thf(sy_c_Int_Oint__ge__less__than2,type,
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thf(sy_c_Int_Onat,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Complex__Ocomplex,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Rat__Orat,type,
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thf(sy_c_Int_Oring__1__class_OInts_001t__Real__Oreal,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Complex__Ocomplex,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Int__Oint,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Rat__Orat,type,
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thf(sy_c_Int_Oring__1__class_Oof__int_001t__Real__Oreal,type,
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thf(sy_c_Lattices_Oinf__class_Oinf_001t__Extended____Nat__Oenat,type,
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thf(sy_c_Lattices_Osemilattice__neutr__order_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Nat__Onat,type,
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thf(sy_c_Lattices_Osup__class_Osup_001t__Set__Oset_It__Nat__Onat_J,type,
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thf(sy_c_Lattices__Big_Olinorder__class_OMax_001t__Nat__Onat,type,
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thf(sy_c_Limits_OBfun_001t__Nat__Onat_001t__Real__Oreal,type,
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thf(sy_c_Limits_Oat__infinity_001t__Real__Oreal,type,
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thf(sy_c_List_Oappend_001t__Int__Oint,type,
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thf(sy_c_List_Oappend_001t__Nat__Onat,type,
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thf(sy_c_List_Oconcat_001_Eo,type,
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thf(sy_c_List_Oconcat_001t__Int__Oint,type,
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thf(sy_c_List_Oconcat_001t__Nat__Onat,type,
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thf(sy_c_List_Oconcat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Odistinct_001t__Int__Oint,type,
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thf(sy_c_List_Odistinct_001t__Nat__Onat,type,
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thf(sy_c_List_Odrop_001t__Nat__Onat,type,
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thf(sy_c_List_Olinorder__class_Osort__key_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_List_Olinorder__class_Osorted__list__of__set_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_OCons_001t__Int__Oint,type,
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thf(sy_c_List_Olist_OCons_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_ONil_001t__Int__Oint,type,
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thf(sy_c_List_Olist_ONil_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Ohd_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Omap_001t__Nat__Onat_001t__Nat__Onat,type,
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thf(sy_c_List_Olist_Oset_001_Eo,type,
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thf(sy_c_List_Olist_Oset_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Olist_Oset_001t__Int__Oint,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_I_Eo_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Int__Oint_J,type,
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thf(sy_c_List_Olist_Oset_001t__List__Olist_It__Nat__Onat_J,type,
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thf(sy_c_List_Olist_Oset_001t__Real__Oreal,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_Onth_001_Eo,type,
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thf(sy_c_List_Onth_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_List_Onth_001t__Complex__Ocomplex,type,
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thf(sy_c_List_Onth_001t__Int__Oint,type,
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thf(sy_c_List_Onth_001t__List__Olist_It__Nat__Onat_J,type,
    nth_list_nat: list_list_nat > nat > list_nat ).

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

thf(sy_c_List_Onth_001t__Num__Onum,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__Num__Onum_Mt__Num__Onum_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J,type,
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thf(sy_c_List_Onth_001t__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__VEBT____Definitions__OVEBT_J,type,
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thf(sy_c_List_Onth_001t__Real__Oreal,type,
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thf(sy_c_List_Onth_001t__VEBT____Definitions__OVEBT,type,
    nth_VEBT_VEBT: list_VEBT_VEBT > nat > vEBT_VEBT ).

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

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

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

thf(sy_c_List_Oproduct_001_Eo_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oproduct_001t__Code____Numeral__Ointeger_001_Eo,type,
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thf(sy_c_List_Oproduct_001t__Nat__Onat_001_Eo,type,
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thf(sy_c_List_Oproduct_001t__Nat__Onat_001t__VEBT____Definitions__OVEBT,type,
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thf(sy_c_List_Oproduct_001t__Num__Onum_001t__Num__Onum,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001_Eo,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Int__Oint,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__Nat__Onat,type,
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thf(sy_c_List_Oproduct_001t__VEBT____Definitions__OVEBT_001t__VEBT____Definitions__OVEBT,type,
    produc4743750530478302277T_VEBT: list_VEBT_VEBT > list_VEBT_VEBT > list_P7413028617227757229T_VEBT ).

thf(sy_c_List_Oremdups_001t__Nat__Onat,type,
    remdups_nat: list_nat > list_nat ).

thf(sy_c_List_Oreplicate_001_Eo,type,
    replicate_o: nat > $o > list_o ).

thf(sy_c_List_Oreplicate_001t__Complex__Ocomplex,type,
    replicate_complex: nat > complex > list_complex ).

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

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

thf(sy_c_List_Oreplicate_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    replic4235873036481779905at_nat: nat > product_prod_nat_nat > list_P6011104703257516679at_nat ).

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

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

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

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

thf(sy_c_List_Osubseqs_001_Eo,type,
    subseqs_o: list_o > list_list_o ).

thf(sy_c_List_Osubseqs_001t__Int__Oint,type,
    subseqs_int: list_int > list_list_int ).

thf(sy_c_List_Osubseqs_001t__Nat__Onat,type,
    subseqs_nat: list_nat > list_list_nat ).

thf(sy_c_List_Osubseqs_001t__VEBT____Definitions__OVEBT,type,
    subseqs_VEBT_VEBT: list_VEBT_VEBT > list_list_VEBT_VEBT ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Nat_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__Code____Numeral__Ointeger_J,type,
    size_s3445333598471063425nteger: list_Code_integer > 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_I_Eo_J_J,type,
    size_s2710708370519433104list_o: list_list_o > nat ).

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__List__Olist_It__Int__Oint_J_J,type,
    size_s533118279054570080st_int: list_list_int > nat ).

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

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

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Int__Oint_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_I_Eo_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__Nat__Onat_Mt__VEBT____Definitions__OVEBT_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_M_Eo_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Int__Oint_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Product____Type__Oprod_It__VEBT____Definitions__OVEBT_Mt__Nat__Onat_J_J,type,
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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,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__Real__Oreal_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__List__Olist_It__VEBT____Definitions__OVEBT_J,type,
    size_s6755466524823107622T_VEBT: list_VEBT_VEBT > nat ).

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

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

thf(sy_c_Nat_Osize__class_Osize_001t__Option__Ooption_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_J,type,
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thf(sy_c_Nat_Osize__class_Osize_001t__VEBT____Definitions__OVEBT,type,
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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_001t__Code____Numeral__Ointeger,type,
    neg_nu8804712462038260780nteger: code_integer > code_integer ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Num_Onum_OOne,type,
    one: num ).

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Option_Ooption_ONone_001t__Num__Onum,type,
    none_num: option_num ).

thf(sy_c_Option_Ooption_ONone_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
    none_P5556105721700978146at_nat: option4927543243414619207at_nat ).

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

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

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

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

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

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

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

thf(sy_c_Option_Ooption_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_Orderings_Obot__class_Obot_001t__Extended____Nat__Oenat,type,
    bot_bo4199563552545308370d_enat: extended_enat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Nat__Onat,type,
    bot_bot_nat: nat ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Int__Oint_J,type,
    bot_bot_set_int: set_int ).

thf(sy_c_Orderings_Obot__class_Obot_001t__Set__Oset_It__Nat__Onat_J,type,
    bot_bot_set_nat: set_nat ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless_001_062_It__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_M_Eo_J,type,
    ord_le549003669493604880_nat_o: ( product_prod_nat_nat > $o ) > ( product_prod_nat_nat > $o ) > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Complex__Ocomplex_M_Eo_J,type,
    ord_le4573692005234683329plex_o: ( complex > $o ) > ( complex > $o ) > $o ).

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

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

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

thf(sy_c_Orderings_Oord__class_Oless__eq_001_062_It__Real__Oreal_M_Eo_J,type,
    ord_less_eq_real_o: ( real > $o ) > ( real > $o ) > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_I_Eo_J,type,
    top_top_set_o: set_o ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Nat__Onat_J,type,
    top_top_set_nat: set_nat ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Product____Type__Ounit_J,type,
    top_to1996260823553986621t_unit: set_Product_unit ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__Real__Oreal_J,type,
    top_top_set_real: set_real ).

thf(sy_c_Orderings_Otop__class_Otop_001t__Set__Oset_It__String__Ochar_J,type,
    top_top_set_char: set_char ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J,type,
    produc6916734918728496179nteger: ( code_integer > code_integer > produc8923325533196201883nteger ) > produc8923325533196201883nteger > produc8923325533196201883nteger ).

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

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

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

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

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

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

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

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

thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Nat__Onat_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Product__Type_Oprod_Ocase__prod_001t__Nat__Onat_001t__Num__Onum_001t__Option__Ooption_It__Num__Onum_J,type,
    produc478579273971653890on_num: ( nat > num > option_num ) > product_prod_nat_num > option_num ).

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

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

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

thf(sy_c_Product__Type_Oprod_Osnd_001t__Code____Numeral__Ointeger_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Int__Oint_001t__Int__Oint,type,
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thf(sy_c_Product__Type_Oprod_Osnd_001t__Nat__Onat_001t__Nat__Onat,type,
    product_snd_nat_nat: product_prod_nat_nat > nat ).

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

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

thf(sy_c_Rat_Ofield__char__0__class_ORats_001t__Real__Oreal,type,
    field_5140801741446780682s_real: set_real ).

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

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

thf(sy_c_Real__Vector__Spaces_OReals_001t__Complex__Ocomplex,type,
    real_V2521375963428798218omplex: set_complex ).

thf(sy_c_Real__Vector__Spaces_OReals_001t__Real__Oreal,type,
    real_V470468836141973256s_real: set_real ).

thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Odist__class_Odist_001t__Real__Oreal,type,
    real_V975177566351809787t_real: real > real > real ).

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

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

thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_Oof__real_001t__Real__Oreal,type,
    real_V1803761363581548252l_real: real > real ).

thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Complex__Ocomplex,type,
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thf(sy_c_Real__Vector__Spaces_OscaleR__class_OscaleR_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Odivide__class_Odivide_001t__Complex__Ocomplex,type,
    divide1717551699836669952omplex: complex > complex > complex ).

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

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

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

thf(sy_c_Rings_Odivide__class_Odivide_001t__Real__Oreal,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Complex__Ocomplex,type,
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thf(sy_c_Rings_Odvd__class_Odvd_001t__Int__Oint,type,
    dvd_dvd_int: int > int > $o ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Series_Osummable_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osummable_001t__Int__Oint,type,
    summable_int: ( nat > int ) > $o ).

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

thf(sy_c_Series_Osummable_001t__Real__Oreal,type,
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thf(sy_c_Series_Osums_001t__Complex__Ocomplex,type,
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thf(sy_c_Series_Osums_001t__Int__Oint,type,
    sums_int: ( nat > int ) > int > $o ).

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

thf(sy_c_Series_Osums_001t__Real__Oreal,type,
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thf(sy_c_Set_OCollect_001_Eo,type,
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thf(sy_c_Set_OCollect_001t__Code____Numeral__Ointeger,type,
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thf(sy_c_Set_OCollect_001t__Complex__Ocomplex,type,
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thf(sy_c_Set_OCollect_001t__Int__Oint,type,
    collect_int: ( int > $o ) > set_int ).

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

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

thf(sy_c_Set_OCollect_001t__Num__Onum,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J,type,
    collec8663557070575231912omplex: ( produc4411394909380815293omplex > $o ) > set_Pr5085853215250843933omplex ).

thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J,type,
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thf(sy_c_Set_OCollect_001t__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J,type,
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thf(sy_c_Set_OCollect_001t__Rat__Orat,type,
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thf(sy_c_Set_OCollect_001t__Real__Oreal,type,
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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__VEBT____Definitions__OVEBT,type,
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thf(sy_c_Set_Oimage_001t__Int__Oint_001t__Int__Oint,type,
    image_int_int: ( int > int ) > set_int > set_int ).

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

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

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

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

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

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Complex__Ocomplex_Mt__Complex__Ocomplex_J_J,type,
    image_5971271580939081552omplex: ( real > filter6041513312241820739omplex ) > set_real > set_fi4554929511873752355omplex ).

thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Filter__Ofilter_It__Product____Type__Oprod_It__Real__Oreal_Mt__Real__Oreal_J_J,type,
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thf(sy_c_Set_Oimage_001t__Real__Oreal_001t__Real__Oreal,type,
    image_real_real: ( real > real ) > set_real > set_real ).

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

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

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

thf(sy_c_Set__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_OatLeastLessThan_001t__Int__Oint,type,
    set_or4662586982721622107an_int: int > int > set_int ).

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

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

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

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

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

thf(sy_c_Set__Interval_Oord__class_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_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_String_Oascii__of,type,
    ascii_of: char > char ).

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

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

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

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

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

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

thf(sy_c_Topological__Spaces_Omonoseq_001t__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__Complex__Ocomplex,type,
    topolo6517432010174082258omplex: ( nat > complex ) > $o ).

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

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Complex__Ocomplex,type,
    topolo896644834953643431omplex: filter6041513312241820739omplex ).

thf(sy_c_Topological__Spaces_Ouniformity__class_Ouniformity_001t__Real__Oreal,type,
    topolo1511823702728130853y_real: filter2146258269922977983l_real ).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_Transcendental_Oexp_001t__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__Complex__Ocomplex,type,
    sinh_complex: complex > complex ).

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

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

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

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

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

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

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

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

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

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

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

thf(sy_c_VEBT__Definitions_OVEBT__internal_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_Wellfounded_Oaccp_001t__List__Olist_It__Nat__Onat_J,type,
    accp_list_nat: ( list_nat > list_nat > $o ) > list_nat > $o ).

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

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

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

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

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

thf(sy_c_Wellfounded_Opred__nat,type,
    pred_nat: set_Pr1261947904930325089at_nat ).

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

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

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

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

thf(sy_c_member_001t__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_It__Nat__Onat_Mt__Nat__Onat_J,type,
    member8440522571783428010at_nat: product_prod_nat_nat > set_Pr1261947904930325089at_nat > $o ).

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

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

thf(sy_c_member_001t__Set__Oset_It__Int__Oint_J,type,
    member_set_int: set_int > set_set_int > $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_info,type,
    info: option4927543243414619207at_nat ).

thf(sy_v_m____,type,
    m: nat ).

thf(sy_v_ma____,type,
    ma: nat ).

thf(sy_v_mi____,type,
    mi: nat ).

thf(sy_v_n,type,
    n: nat ).

thf(sy_v_na____,type,
    na: nat ).

thf(sy_v_summary,type,
    summary: vEBT_VEBT ).

thf(sy_v_treeList,type,
    treeList: list_VEBT_VEBT ).

thf(sy_v_x,type,
    x: nat ).

% Relevant facts (10210)
thf(fact_0_both__member__options__def,axiom,
    ( vEBT_V8194947554948674370ptions
    = ( ^ [T: vEBT_VEBT,X: nat] :
          ( ( vEBT_V5719532721284313246member @ T @ X )
          | ( vEBT_VEBT_membermima @ T @ X ) ) ) ) ).

% both_member_options_def
thf(fact_1__092_060open_0621_A_060_Adeg_092_060close_062,axiom,
    ord_less_nat @ one_one_nat @ deg ).

% \<open>1 < deg\<close>
thf(fact_2_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_3__C5_Ohyps_C_I3_J,axiom,
    ( ( summary
      = ( vEBT_Node @ info @ deg @ treeList @ summary ) )
   => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ info @ deg @ treeList @ summary ) @ x ) ) ).

% "5.hyps"(3)
thf(fact_4__C5_Ohyps_C_I2_J,axiom,
    vEBT_invar_vebt @ summary @ m ).

% "5.hyps"(2)
thf(fact_5_assms_I1_J,axiom,
    vEBT_invar_vebt @ ( vEBT_Node @ info @ deg @ treeList @ summary ) @ n ).

% assms(1)
thf(fact_6_assms_I2_J,axiom,
    ord_less_nat @ x @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% assms(2)
thf(fact_7__092_060open_062naive__member_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_A_092_060Longrightarrow_062_Anaive__member_A_INode_Ainfo_Adeg_AtreeList_Asummary_J_Ax_092_060close_062,axiom,
    ( ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
   => ( vEBT_V5719532721284313246member @ ( vEBT_Node @ info @ deg @ treeList @ summary ) @ x ) ) ).

% \<open>naive_member (treeList ! high x (deg div 2)) (low x (deg div 2)) \<Longrightarrow> naive_member (Node info deg treeList summary) x\<close>
thf(fact_8__092_060open_062membermima_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_A_092_060Longrightarrow_062_Amembermima_A_INode_Ainfo_Adeg_AtreeList_Asummary_J_Ax_092_060close_062,axiom,
    ( ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
   => ( vEBT_VEBT_membermima @ ( vEBT_Node @ info @ deg @ treeList @ summary ) @ x ) ) ).

% \<open>membermima (treeList ! high x (deg div 2)) (low x (deg div 2)) \<Longrightarrow> membermima (Node info deg treeList summary) x\<close>
thf(fact_9__092_060open_062high_Ax_A_Ideg_Adiv_A2_J_A_060_A2_A_094_Am_092_060close_062,axiom,
    ord_less_nat @ ( vEBT_VEBT_high @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ).

% \<open>high x (deg div 2) < 2 ^ m\<close>
thf(fact_10__092_060open_062membermima_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_A_092_060or_062_Anaive__member_A_ItreeList_A_B_Ahigh_Ax_A_Ideg_Adiv_A2_J_J_A_Ilow_Ax_A_Ideg_Adiv_A2_J_J_092_060close_062,axiom,
    ( ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
    | ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% \<open>membermima (treeList ! high x (deg div 2)) (low x (deg div 2)) \<or> naive_member (treeList ! high x (deg div 2)) (low x (deg div 2))\<close>
thf(fact_11_assms_I3_J,axiom,
    vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ ( vEBT_VEBT_high @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ x @ ( divide_divide_nat @ deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% assms(3)
thf(fact_12__C5_Ohyps_C_I1_J,axiom,
    ! [X2: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
     => ( ( vEBT_invar_vebt @ X2 @ na )
        & ( ( X2
            = ( vEBT_Node @ info @ deg @ treeList @ summary ) )
         => ( vEBT_V8194947554948674370ptions @ ( vEBT_Node @ info @ deg @ treeList @ summary ) @ x ) ) ) ) ).

% "5.hyps"(1)
thf(fact_13__C5_Ohyps_C_I7_J,axiom,
    ! [I: nat] :
      ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
     => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ X3 ) )
        = ( vEBT_V8194947554948674370ptions @ summary @ I ) ) ) ).

% "5.hyps"(7)
thf(fact_14__C5_Ohyps_C_I6_J,axiom,
    ( deg
    = ( plus_plus_nat @ na @ m ) ) ).

% "5.hyps"(6)
thf(fact_15__C5_Ohyps_C_I10_J,axiom,
    ord_less_nat @ ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ deg ) ).

% "5.hyps"(10)
thf(fact_16_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_17_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_18__C5_Ohyps_C_I8_J,axiom,
    ( ( mi = ma )
   => ! [X2: vEBT_VEBT] :
        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ treeList ) )
       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ).

% "5.hyps"(8)
thf(fact_19__C5_Ohyps_C_I5_J,axiom,
    ( m
    = ( suc @ na ) ) ).

% "5.hyps"(5)
thf(fact_20__C5_Ohyps_C_I4_J,axiom,
    ( ( size_s6755466524823107622T_VEBT @ treeList )
    = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) ) ).

% "5.hyps"(4)
thf(fact_21_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_22_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_23__C5_Ohyps_C_I11_J,axiom,
    ( ( mi != ma )
   => ! [I: nat] :
        ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ m ) )
       => ( ( ( ( vEBT_VEBT_high @ ma @ na )
              = I )
           => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ ma @ na ) ) )
          & ! [X2: nat] :
              ( ( ( ( vEBT_VEBT_high @ X2 @ na )
                  = I )
                & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ treeList @ I ) @ ( vEBT_VEBT_low @ X2 @ na ) ) )
             => ( ( ord_less_nat @ mi @ X2 )
                & ( ord_less_eq_nat @ X2 @ ma ) ) ) ) ) ) ).

% "5.hyps"(11)
thf(fact_24_one__add__one,axiom,
    ( ( plus_plus_rat @ one_one_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_25_one__add__one,axiom,
    ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ one_on7984719198319812577d_enat )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_26_one__add__one,axiom,
    ( ( plus_plus_complex @ one_one_complex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_27_one__add__one,axiom,
    ( ( plus_plus_real @ one_one_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_28_one__add__one,axiom,
    ( ( plus_plus_nat @ one_one_nat @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_29_one__add__one,axiom,
    ( ( plus_plus_int @ one_one_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% one_add_one
thf(fact_30_power__strict__increasing__iff,axiom,
    ! [B: real,X4: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ ( power_power_real @ B @ X4 ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_nat @ X4 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_31_power__strict__increasing__iff,axiom,
    ! [B: rat,X4: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_rat @ ( power_power_rat @ B @ X4 ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_nat @ X4 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_32_power__strict__increasing__iff,axiom,
    ! [B: nat,X4: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_nat @ X4 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_33_power__strict__increasing__iff,axiom,
    ! [B: int,X4: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_nat @ X4 @ Y ) ) ) ).

% power_strict_increasing_iff
thf(fact_34_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_35_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_36_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_37_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_38_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_39_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_40_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_41_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_42_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_43_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_44_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_45_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_46_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_47_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_48_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_49_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_50_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_51_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_52_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_53_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_54_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_55_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_rat @ N2 )
        = one_one_rat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_56_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numera1916890842035813515d_enat @ N2 )
        = one_on7984719198319812577d_enat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_57_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numera6690914467698888265omplex @ N2 )
        = one_one_complex )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_58_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_real @ N2 )
        = one_one_real )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_59_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_nat @ N2 )
        = one_one_nat )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_60_numeral__eq__one__iff,axiom,
    ! [N2: num] :
      ( ( ( numeral_numeral_int @ N2 )
        = one_one_int )
      = ( N2 = one ) ) ).

% numeral_eq_one_iff
thf(fact_61_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_rat
        = ( numeral_numeral_rat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_62_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_on7984719198319812577d_enat
        = ( numera1916890842035813515d_enat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_63_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_complex
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_64_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_real
        = ( numeral_numeral_real @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_65_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_nat
        = ( numeral_numeral_nat @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_66_one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( one_one_int
        = ( numeral_numeral_int @ N2 ) )
      = ( one = N2 ) ) ).

% one_eq_numeral_iff
thf(fact_67_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_68_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_69_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_70_even__odd__cases,axiom,
    ! [X4: nat] :
      ( ! [N3: nat] :
          ( X4
         != ( plus_plus_nat @ N3 @ N3 ) )
     => ~ ! [N3: nat] :
            ( X4
           != ( plus_plus_nat @ N3 @ ( suc @ N3 ) ) ) ) ).

% even_odd_cases
thf(fact_71__C5_Ohyps_C_I9_J,axiom,
    ord_less_eq_nat @ mi @ ma ).

% "5.hyps"(9)
thf(fact_72_inthall,axiom,
    ! [Xs: list_real,P: real > $o,N2: nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_real2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_real @ Xs ) )
       => ( P @ ( nth_real @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_73_inthall,axiom,
    ! [Xs: list_complex,P: complex > $o,N2: nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ ( set_complex2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
       => ( P @ ( nth_complex @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_74_inthall,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,N2: nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s5460976970255530739at_nat @ Xs ) )
       => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_75_inthall,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,N2: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_76_inthall,axiom,
    ! [Xs: list_o,P: $o > $o,N2: nat] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( set_o2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
       => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_77_inthall,axiom,
    ! [Xs: list_nat,P: nat > $o,N2: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
       => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_78_inthall,axiom,
    ! [Xs: list_int,P: int > $o,N2: nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
         => ( P @ X5 ) )
     => ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
       => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).

% inthall
thf(fact_79_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numera1916890842035813515d_enat @ M )
        = ( numera1916890842035813515d_enat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_80_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numera6690914467698888265omplex @ M )
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_81_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_real @ M )
        = ( numeral_numeral_real @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_82_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_nat @ M )
        = ( numeral_numeral_nat @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_83_numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( numeral_numeral_int @ M )
        = ( numeral_numeral_int @ N2 ) )
      = ( M = N2 ) ) ).

% numeral_eq_iff
thf(fact_84_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_85_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_86_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_87_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_88_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_89_bits__div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% bits_div_by_1
thf(fact_90_bits__div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% bits_div_by_1
thf(fact_91_bits__div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% bits_div_by_1
thf(fact_92_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ one_one_rat @ N2 )
      = one_one_rat ) ).

% power_one
thf(fact_93_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ one_one_nat @ N2 )
      = one_one_nat ) ).

% power_one
thf(fact_94_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ one_one_real @ N2 )
      = one_one_real ) ).

% power_one
thf(fact_95_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ one_one_int @ N2 )
      = one_one_int ) ).

% power_one
thf(fact_96_power__one,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ one_one_complex @ N2 )
      = one_one_complex ) ).

% power_one
thf(fact_97_power__one__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_98_power__one__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_99_power__one__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_100_power__one__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ one_one_nat )
      = A ) ).

% power_one_right
thf(fact_101_set__n__deg__not__0,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,M: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
       => ( ord_less_eq_nat @ one_one_nat @ N2 ) ) ) ).

% set_n_deg_not_0
thf(fact_102_add__numeral__left,axiom,
    ! [V: num,W: num,Z: rat] :
      ( ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ ( plus_plus_rat @ ( numeral_numeral_rat @ W ) @ Z ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_103_add__numeral__left,axiom,
    ! [V: num,W: num,Z: extended_enat] :
      ( ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_104_add__numeral__left,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ V ) @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_105_add__numeral__left,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( plus_plus_real @ ( numeral_numeral_real @ V ) @ ( plus_plus_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_106_add__numeral__left,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( plus_plus_nat @ ( numeral_numeral_nat @ V ) @ ( plus_plus_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_107_add__numeral__left,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( plus_plus_int @ ( numeral_numeral_int @ V ) @ ( plus_plus_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) @ Z ) ) ).

% add_numeral_left
thf(fact_108_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_109_mem__Collect__eq,axiom,
    ! [A: complex,P: complex > $o] :
      ( ( member_complex @ A @ ( collect_complex @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_110_mem__Collect__eq,axiom,
    ! [A: real,P: real > $o] :
      ( ( member_real @ A @ ( collect_real @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_111_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_112_mem__Collect__eq,axiom,
    ! [A: nat,P: nat > $o] :
      ( ( member_nat @ A @ ( collect_nat @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_113_mem__Collect__eq,axiom,
    ! [A: int,P: int > $o] :
      ( ( member_int @ A @ ( collect_int @ P ) )
      = ( P @ A ) ) ).

% mem_Collect_eq
thf(fact_114_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_115_Collect__mem__eq,axiom,
    ! [A2: set_complex] :
      ( ( collect_complex
        @ ^ [X: complex] : ( member_complex @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_116_Collect__mem__eq,axiom,
    ! [A2: set_real] :
      ( ( collect_real
        @ ^ [X: real] : ( member_real @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_117_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_118_Collect__mem__eq,axiom,
    ! [A2: set_nat] :
      ( ( collect_nat
        @ ^ [X: nat] : ( member_nat @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_119_Collect__mem__eq,axiom,
    ! [A2: set_int] :
      ( ( collect_int
        @ ^ [X: int] : ( member_int @ X @ A2 ) )
      = A2 ) ).

% Collect_mem_eq
thf(fact_120_Collect__cong,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X5: complex] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_complex @ P )
        = ( collect_complex @ Q ) ) ) ).

% Collect_cong
thf(fact_121_Collect__cong,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_real @ P )
        = ( collect_real @ Q ) ) ) ).

% Collect_cong
thf(fact_122_Collect__cong,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_list_nat @ P )
        = ( collect_list_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_123_Collect__cong,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_nat @ P )
        = ( collect_nat @ Q ) ) ) ).

% Collect_cong
thf(fact_124_Collect__cong,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P @ X5 )
          = ( Q @ X5 ) )
     => ( ( collect_int @ P )
        = ( collect_int @ Q ) ) ) ).

% Collect_cong
thf(fact_125_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_126_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_127_numeral__plus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).

% numeral_plus_numeral
thf(fact_128_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_129_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_130_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_131_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_132_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_133_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_134_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_135_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_136_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_137_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_138_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_139_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_140_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_141_Suc__numeral,axiom,
    ! [N2: num] :
      ( ( suc @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( plus_plus_num @ N2 @ one ) ) ) ).

% Suc_numeral
thf(fact_142_power__increasing__iff,axiom,
    ! [B: real,X4: nat,Y: nat] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_eq_real @ ( power_power_real @ B @ X4 ) @ ( power_power_real @ B @ Y ) )
        = ( ord_less_eq_nat @ X4 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_143_power__increasing__iff,axiom,
    ! [B: rat,X4: nat,Y: nat] :
      ( ( ord_less_rat @ one_one_rat @ B )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ B @ X4 ) @ ( power_power_rat @ B @ Y ) )
        = ( ord_less_eq_nat @ X4 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_144_power__increasing__iff,axiom,
    ! [B: nat,X4: nat,Y: nat] :
      ( ( ord_less_nat @ one_one_nat @ B )
     => ( ( ord_less_eq_nat @ ( power_power_nat @ B @ X4 ) @ ( power_power_nat @ B @ Y ) )
        = ( ord_less_eq_nat @ X4 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_145_power__increasing__iff,axiom,
    ! [B: int,X4: nat,Y: nat] :
      ( ( ord_less_int @ one_one_int @ B )
     => ( ( ord_less_eq_int @ ( power_power_int @ B @ X4 ) @ ( power_power_int @ B @ Y ) )
        = ( ord_less_eq_nat @ X4 @ Y ) ) ) ).

% power_increasing_iff
thf(fact_146_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_147_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_148_Suc__1,axiom,
    ( ( suc @ one_one_nat )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% Suc_1
thf(fact_149_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_150_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_151_add__One__commute,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ N2 )
      = ( plus_plus_num @ N2 @ one ) ) ).

% add_One_commute
thf(fact_152_le__numeral__extra_I4_J,axiom,
    ord_less_eq_real @ one_one_real @ one_one_real ).

% le_numeral_extra(4)
thf(fact_153_le__numeral__extra_I4_J,axiom,
    ord_less_eq_rat @ one_one_rat @ one_one_rat ).

% le_numeral_extra(4)
thf(fact_154_le__numeral__extra_I4_J,axiom,
    ord_less_eq_nat @ one_one_nat @ one_one_nat ).

% le_numeral_extra(4)
thf(fact_155_le__numeral__extra_I4_J,axiom,
    ord_less_eq_int @ one_one_int @ one_one_int ).

% le_numeral_extra(4)
thf(fact_156_div__le__dividend,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( divide_divide_nat @ M @ N2 ) @ M ) ).

% div_le_dividend
thf(fact_157_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_158_power__increasing,axiom,
    ! [N2: nat,N4: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N4 )
     => ( ( ord_less_eq_real @ one_one_real @ A )
       => ( ord_less_eq_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_159_power__increasing,axiom,
    ! [N2: nat,N4: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N4 )
     => ( ( ord_less_eq_rat @ one_one_rat @ A )
       => ( ord_less_eq_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_160_power__increasing,axiom,
    ! [N2: nat,N4: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N4 )
     => ( ( ord_less_eq_nat @ one_one_nat @ A )
       => ( ord_less_eq_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_161_power__increasing,axiom,
    ! [N2: nat,N4: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N4 )
     => ( ( ord_less_eq_int @ one_one_int @ A )
       => ( ord_less_eq_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N4 ) ) ) ) ).

% power_increasing
thf(fact_162_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_163_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_164_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_165_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_166_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_167_one__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% one_le_numeral
thf(fact_168_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ one_one_real @ ( numeral_numeral_real @ N2 ) ) ).

% one_le_numeral
thf(fact_169_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ one_one_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% one_le_numeral
thf(fact_170_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% one_le_numeral
thf(fact_171_one__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ one_one_int @ ( numeral_numeral_int @ N2 ) ) ).

% one_le_numeral
thf(fact_172_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_173_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_174_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_175_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_176_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_177_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_178_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_179_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_180_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_181_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_182_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_183_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_184_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_185_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_186_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_187_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_188_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_real @ one_one_real @ one_one_real ) ).

% less_numeral_extra(4)
thf(fact_189_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_rat @ one_one_rat @ one_one_rat ) ).

% less_numeral_extra(4)
thf(fact_190_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_nat @ one_one_nat @ one_one_nat ) ).

% less_numeral_extra(4)
thf(fact_191_less__numeral__extra_I4_J,axiom,
    ~ ( ord_less_int @ one_one_int @ one_one_int ) ).

% less_numeral_extra(4)
thf(fact_192_power__divide,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( divide_divide_real @ A @ B ) @ N2 )
      = ( divide_divide_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).

% power_divide
thf(fact_193_power__divide,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ A @ B ) @ N2 )
      = ( divide1717551699836669952omplex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).

% power_divide
thf(fact_194_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ one_one_rat ) ).

% not_numeral_less_one
thf(fact_195_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ one_on7984719198319812577d_enat ) ).

% not_numeral_less_one
thf(fact_196_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ one_one_real ) ).

% not_numeral_less_one
thf(fact_197_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ one_one_nat ) ).

% not_numeral_less_one
thf(fact_198_not__numeral__less__one,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ).

% not_numeral_less_one
thf(fact_199_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_rat @ one_one_rat @ ( numeral_numeral_rat @ X4 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X4 ) @ one_one_rat ) ) ).

% one_plus_numeral_commute
thf(fact_200_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_p3455044024723400733d_enat @ one_on7984719198319812577d_enat @ ( numera1916890842035813515d_enat @ X4 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X4 ) @ one_on7984719198319812577d_enat ) ) ).

% one_plus_numeral_commute
thf(fact_201_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_complex @ one_one_complex @ ( numera6690914467698888265omplex @ X4 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X4 ) @ one_one_complex ) ) ).

% one_plus_numeral_commute
thf(fact_202_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_real @ one_one_real @ ( numeral_numeral_real @ X4 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).

% one_plus_numeral_commute
thf(fact_203_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_nat @ one_one_nat @ ( numeral_numeral_nat @ X4 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).

% one_plus_numeral_commute
thf(fact_204_one__plus__numeral__commute,axiom,
    ! [X4: num] :
      ( ( plus_plus_int @ one_one_int @ ( numeral_numeral_int @ X4 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).

% one_plus_numeral_commute
thf(fact_205_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_206_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera1916890842035813515d_enat @ ( bit0 @ N2 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ ( numera1916890842035813515d_enat @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_207_numeral__Bit0,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_Bit0
thf(fact_208_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_209_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_210_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_211_numeral__One,axiom,
    ( ( numeral_numeral_rat @ one )
    = one_one_rat ) ).

% numeral_One
thf(fact_212_numeral__One,axiom,
    ( ( numera1916890842035813515d_enat @ one )
    = one_on7984719198319812577d_enat ) ).

% numeral_One
thf(fact_213_numeral__One,axiom,
    ( ( numera6690914467698888265omplex @ one )
    = one_one_complex ) ).

% numeral_One
thf(fact_214_numeral__One,axiom,
    ( ( numeral_numeral_real @ one )
    = one_one_real ) ).

% numeral_One
thf(fact_215_numeral__One,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numeral_One
thf(fact_216_numeral__One,axiom,
    ( ( numeral_numeral_int @ one )
    = one_one_int ) ).

% numeral_One
thf(fact_217_divide__numeral__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_218_divide__numeral__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% divide_numeral_1
thf(fact_219_power__one__over,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( divide_divide_rat @ one_one_rat @ A ) @ N2 )
      = ( divide_divide_rat @ one_one_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_220_power__one__over,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( divide_divide_real @ one_one_real @ A ) @ N2 )
      = ( divide_divide_real @ one_one_real @ ( power_power_real @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_221_power__one__over,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ ( divide1717551699836669952omplex @ one_one_complex @ A ) @ N2 )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_one_over
thf(fact_222_numerals_I1_J,axiom,
    ( ( numeral_numeral_nat @ one )
    = one_one_nat ) ).

% numerals(1)
thf(fact_223_numeral__Bit0__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% numeral_Bit0_div_2
thf(fact_224_numeral__Bit0__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% numeral_Bit0_div_2
thf(fact_225_numeral__Bit0__div__2,axiom,
    ! [N2: num] :
      ( ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% numeral_Bit0_div_2
thf(fact_226_power__strict__increasing,axiom,
    ! [N2: nat,N4: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N4 )
     => ( ( ord_less_real @ one_one_real @ A )
       => ( ord_less_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_227_power__strict__increasing,axiom,
    ! [N2: nat,N4: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N4 )
     => ( ( ord_less_rat @ one_one_rat @ A )
       => ( ord_less_rat @ ( power_power_rat @ A @ N2 ) @ ( power_power_rat @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_228_power__strict__increasing,axiom,
    ! [N2: nat,N4: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N4 )
     => ( ( ord_less_nat @ one_one_nat @ A )
       => ( ord_less_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_229_power__strict__increasing,axiom,
    ! [N2: nat,N4: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N4 )
     => ( ( ord_less_int @ one_one_int @ A )
       => ( ord_less_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ A @ N4 ) ) ) ) ).

% power_strict_increasing
thf(fact_230_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_231_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_232_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_233_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_234_one__power2,axiom,
    ( ( power_power_rat @ one_one_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_rat ) ).

% one_power2
thf(fact_235_one__power2,axiom,
    ( ( power_power_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_power2
thf(fact_236_one__power2,axiom,
    ( ( power_power_real @ one_one_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_real ) ).

% one_power2
thf(fact_237_one__power2,axiom,
    ( ( power_power_int @ one_one_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_power2
thf(fact_238_one__power2,axiom,
    ( ( power_power_complex @ one_one_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_complex ) ).

% one_power2
thf(fact_239_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_240_less__exp,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% less_exp
thf(fact_241_semiring__norm_I76_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit0 @ N2 ) ) ).

% semiring_norm(76)
thf(fact_242_semiring__norm_I2_J,axiom,
    ( ( plus_plus_num @ one @ one )
    = ( bit0 @ one ) ) ).

% semiring_norm(2)
thf(fact_243_field__less__half__sum,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ord_less_rat @ X4 @ ( divide_divide_rat @ ( plus_plus_rat @ X4 @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_244_field__less__half__sum,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ord_less_real @ X4 @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% field_less_half_sum
thf(fact_245_semiring__norm_I75_J,axiom,
    ! [M: num] :
      ~ ( ord_less_num @ M @ one ) ).

% semiring_norm(75)
thf(fact_246_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_247_semiring__norm_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(6)
thf(fact_248_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_249_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_250_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_251_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_252_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_253_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_254_semiring__norm_I87_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit0 @ M )
        = ( bit0 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(87)
thf(fact_255_nat_Oinject,axiom,
    ! [X22: nat,Y2: nat] :
      ( ( ( suc @ X22 )
        = ( suc @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% nat.inject
thf(fact_256_old_Onat_Oinject,axiom,
    ! [Nat: nat,Nat2: nat] :
      ( ( ( suc @ Nat )
        = ( suc @ Nat2 ) )
      = ( Nat = Nat2 ) ) ).

% old.nat.inject
thf(fact_257_semiring__norm_I85_J,axiom,
    ! [M: num] :
      ( ( bit0 @ M )
     != one ) ).

% semiring_norm(85)
thf(fact_258_semiring__norm_I83_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit0 @ N2 ) ) ).

% semiring_norm(83)
thf(fact_259_lessI,axiom,
    ! [N2: nat] : ( ord_less_nat @ N2 @ ( suc @ N2 ) ) ).

% lessI
thf(fact_260_Suc__mono,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ ( suc @ M ) @ ( suc @ N2 ) ) ) ).

% Suc_mono
thf(fact_261_semiring__norm_I71_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(71)
thf(fact_262_semiring__norm_I68_J,axiom,
    ! [N2: num] : ( ord_less_eq_num @ one @ N2 ) ).

% semiring_norm(68)
thf(fact_263_semiring__norm_I69_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit0 @ M ) @ one ) ).

% semiring_norm(69)
thf(fact_264_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_265_enat__less__induct,axiom,
    ! [P: extended_enat > $o,N2: extended_enat] :
      ( ! [N3: extended_enat] :
          ( ! [M2: extended_enat] :
              ( ( ord_le72135733267957522d_enat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% enat_less_induct
thf(fact_266_le__num__One__iff,axiom,
    ! [X4: num] :
      ( ( ord_less_eq_num @ X4 @ one )
      = ( X4 = one ) ) ).

% le_num_One_iff
thf(fact_267_two__realpow__ge__one,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ).

% two_realpow_ge_one
thf(fact_268_Suc__inject,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( suc @ X4 )
        = ( suc @ Y ) )
     => ( X4 = Y ) ) ).

% Suc_inject
thf(fact_269_n__not__Suc__n,axiom,
    ! [N2: nat] :
      ( N2
     != ( suc @ N2 ) ) ).

% n_not_Suc_n
thf(fact_270_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_271_less__not__refl,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_not_refl
thf(fact_272_less__not__refl2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ N2 @ M )
     => ( M != N2 ) ) ).

% less_not_refl2
thf(fact_273_less__not__refl3,axiom,
    ! [S: nat,T2: nat] :
      ( ( ord_less_nat @ S @ T2 )
     => ( S != T2 ) ) ).

% less_not_refl3
thf(fact_274_less__irrefl__nat,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ N2 ) ).

% less_irrefl_nat
thf(fact_275_nat__less__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% nat_less_induct
thf(fact_276_infinite__descent,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ~ ( P @ N3 )
         => ? [M2: nat] :
              ( ( ord_less_nat @ M2 @ N3 )
              & ~ ( P @ M2 ) ) )
     => ( P @ N2 ) ) ).

% infinite_descent
thf(fact_277_linorder__neqE__nat,axiom,
    ! [X4: nat,Y: nat] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_nat @ X4 @ Y )
       => ( ord_less_nat @ Y @ X4 ) ) ) ).

% linorder_neqE_nat
thf(fact_278_le__refl,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ N2 ) ).

% le_refl
thf(fact_279_le__trans,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ J @ K )
       => ( ord_less_eq_nat @ I2 @ K ) ) ) ).

% le_trans
thf(fact_280_eq__imp__le,axiom,
    ! [M: nat,N2: nat] :
      ( ( M = N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% eq_imp_le
thf(fact_281_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_282_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_283_Nat_Oex__has__greatest__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ? [X5: nat] :
            ( ( P @ X5 )
            & ! [Y4: nat] :
                ( ( P @ Y4 )
               => ( ord_less_eq_nat @ Y4 @ X5 ) ) ) ) ) ).

% Nat.ex_has_greatest_nat
thf(fact_284_size__neq__size__imp__neq,axiom,
    ! [X4: list_VEBT_VEBT,Y: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ X4 )
       != ( size_s6755466524823107622T_VEBT @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_285_size__neq__size__imp__neq,axiom,
    ! [X4: list_o,Y: list_o] :
      ( ( ( size_size_list_o @ X4 )
       != ( size_size_list_o @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_286_size__neq__size__imp__neq,axiom,
    ! [X4: list_nat,Y: list_nat] :
      ( ( ( size_size_list_nat @ X4 )
       != ( size_size_list_nat @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_287_size__neq__size__imp__neq,axiom,
    ! [X4: list_int,Y: list_int] :
      ( ( ( size_size_list_int @ X4 )
       != ( size_size_list_int @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_288_size__neq__size__imp__neq,axiom,
    ! [X4: num,Y: num] :
      ( ( ( size_size_num @ X4 )
       != ( size_size_num @ Y ) )
     => ( X4 != Y ) ) ).

% size_neq_size_imp_neq
thf(fact_289_Nat_OlessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ K )
     => ( ( K
         != ( suc @ I2 ) )
       => ~ ! [J2: nat] :
              ( ( ord_less_nat @ I2 @ J2 )
             => ( K
               != ( suc @ J2 ) ) ) ) ) ).

% Nat.lessE
thf(fact_290_Suc__lessD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_nat @ M @ N2 ) ) ).

% Suc_lessD
thf(fact_291_Suc__lessE,axiom,
    ! [I2: nat,K: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ K )
     => ~ ! [J2: nat] :
            ( ( ord_less_nat @ I2 @ J2 )
           => ( K
             != ( suc @ J2 ) ) ) ) ).

% Suc_lessE
thf(fact_292_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_293_less__SucE,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( suc @ N2 ) )
     => ( ~ ( ord_less_nat @ M @ N2 )
       => ( M = N2 ) ) ) ).

% less_SucE
thf(fact_294_less__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_nat @ M @ ( suc @ N2 ) ) ) ).

% less_SucI
thf(fact_295_Ex__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ N2 )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
            & ( P @ I3 ) ) ) ) ).

% Ex_less_Suc
thf(fact_296_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_297_not__less__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( ~ ( ord_less_nat @ M @ N2 ) )
      = ( ord_less_nat @ N2 @ ( suc @ M ) ) ) ).

% not_less_eq
thf(fact_298_All__less__Suc,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ N2 )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
           => ( P @ I3 ) ) ) ) ).

% All_less_Suc
thf(fact_299_Suc__less__eq2,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_less_nat @ ( suc @ N2 ) @ M )
      = ( ? [M3: nat] :
            ( ( M
              = ( suc @ M3 ) )
            & ( ord_less_nat @ N2 @ M3 ) ) ) ) ).

% Suc_less_eq2
thf(fact_300_less__antisym,axiom,
    ! [N2: nat,M: nat] :
      ( ~ ( ord_less_nat @ N2 @ M )
     => ( ( ord_less_nat @ N2 @ ( suc @ M ) )
       => ( M = N2 ) ) ) ).

% less_antisym
thf(fact_301_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_302_less__trans__Suc,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ J @ K )
       => ( ord_less_nat @ ( suc @ I2 ) @ K ) ) ) ).

% less_trans_Suc
thf(fact_303_less__Suc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > nat > $o] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ! [I4: nat] : ( P @ I4 @ ( suc @ I4 ) )
       => ( ! [I4: nat,J2: nat,K2: nat] :
              ( ( ord_less_nat @ I4 @ J2 )
             => ( ( ord_less_nat @ J2 @ K2 )
               => ( ( P @ I4 @ J2 )
                 => ( ( P @ J2 @ K2 )
                   => ( P @ I4 @ K2 ) ) ) ) )
         => ( P @ I2 @ J ) ) ) ) ).

% less_Suc_induct
thf(fact_304_strict__inc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ! [I4: nat] :
            ( ( J
              = ( suc @ I4 ) )
           => ( P @ I4 ) )
       => ( ! [I4: nat] :
              ( ( ord_less_nat @ I4 @ J )
             => ( ( P @ ( suc @ I4 ) )
               => ( P @ I4 ) ) )
         => ( P @ I2 ) ) ) ) ).

% strict_inc_induct
thf(fact_305_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_306_Suc__leD,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ M ) @ N2 )
     => ( ord_less_eq_nat @ M @ N2 ) ) ).

% Suc_leD
thf(fact_307_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_308_le__SucI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ M @ ( suc @ N2 ) ) ) ).

% le_SucI
thf(fact_309_Suc__le__D,axiom,
    ! [N2: nat,M4: nat] :
      ( ( ord_less_eq_nat @ ( suc @ N2 ) @ M4 )
     => ? [M5: nat] :
          ( M4
          = ( suc @ M5 ) ) ) ).

% Suc_le_D
thf(fact_310_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_311_Suc__n__not__le__n,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_nat @ ( suc @ N2 ) @ N2 ) ).

% Suc_n_not_le_n
thf(fact_312_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_313_full__nat__induct,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ ( suc @ M2 ) @ N3 )
             => ( P @ M2 ) )
         => ( P @ N3 ) )
     => ( P @ N2 ) ) ).

% full_nat_induct
thf(fact_314_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_315_transitive__stepwise__le,axiom,
    ! [M: nat,N2: nat,R: nat > nat > $o] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ! [X5: nat] : ( R @ X5 @ X5 )
       => ( ! [X5: nat,Y3: nat,Z2: nat] :
              ( ( R @ X5 @ Y3 )
             => ( ( R @ Y3 @ Z2 )
               => ( R @ X5 @ Z2 ) ) )
         => ( ! [N3: nat] : ( R @ N3 @ ( suc @ N3 ) )
           => ( R @ M @ N2 ) ) ) ) ) ).

% transitive_stepwise_le
thf(fact_316_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_317_add__Suc,axiom,
    ! [M: nat,N2: nat] :
      ( ( plus_plus_nat @ ( suc @ M ) @ N2 )
      = ( suc @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% add_Suc
thf(fact_318_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_319_nat__less__le,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N: nat] :
          ( ( ord_less_eq_nat @ M6 @ N )
          & ( M6 != N ) ) ) ) ).

% nat_less_le
thf(fact_320_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_321_le__eq__less__or__eq,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N: nat] :
          ( ( ord_less_nat @ M6 @ N )
          | ( M6 = N ) ) ) ) ).

% le_eq_less_or_eq
thf(fact_322_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_323_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_324_less__mono__imp__le__mono,axiom,
    ! [F: nat > nat,I2: nat,J: nat] :
      ( ! [I4: nat,J2: nat] :
          ( ( ord_less_nat @ I4 @ J2 )
         => ( ord_less_nat @ ( F @ I4 ) @ ( F @ J2 ) ) )
     => ( ( ord_less_eq_nat @ I2 @ J )
       => ( ord_less_eq_nat @ ( F @ I2 ) @ ( F @ J ) ) ) ) ).

% less_mono_imp_le_mono
thf(fact_325_add__lessD1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
     => ( ord_less_nat @ I2 @ K ) ) ).

% add_lessD1
thf(fact_326_add__less__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ K @ L )
       => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_less_mono
thf(fact_327_not__add__less1,axiom,
    ! [I2: nat,J: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ I2 @ J ) @ I2 ) ).

% not_add_less1
thf(fact_328_not__add__less2,axiom,
    ! [J: nat,I2: nat] :
      ~ ( ord_less_nat @ ( plus_plus_nat @ J @ I2 ) @ I2 ) ).

% not_add_less2
thf(fact_329_add__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_less_mono1
thf(fact_330_trans__less__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_less_add1
thf(fact_331_trans__less__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ord_less_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_less_add2
thf(fact_332_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_333_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_334_le__add1,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ N2 @ M ) ) ).

% le_add1
thf(fact_335_le__add2,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ).

% le_add2
thf(fact_336_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_337_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_338_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_339_add__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ) ).

% add_le_mono
thf(fact_340_add__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ K ) ) ) ).

% add_le_mono1
thf(fact_341_trans__le__add1,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ J @ M ) ) ) ).

% trans_le_add1
thf(fact_342_trans__le__add2,axiom,
    ! [I2: nat,J: nat,M: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ I2 @ ( plus_plus_nat @ M @ J ) ) ) ).

% trans_le_add2
thf(fact_343_nat__le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [M6: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( plus_plus_nat @ M6 @ K3 ) ) ) ) ).

% nat_le_iff_add
thf(fact_344_lift__Suc__mono__less,axiom,
    ! [F: nat > real,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_real @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_real @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_345_lift__Suc__mono__less,axiom,
    ! [F: nat > rat,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_346_lift__Suc__mono__less,axiom,
    ! [F: nat > num,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_347_lift__Suc__mono__less,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_348_lift__Suc__mono__less,axiom,
    ! [F: nat > int,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_nat @ N2 @ N5 )
       => ( ord_less_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_less
thf(fact_349_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_350_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_351_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_352_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_353_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_354_lift__Suc__mono__le,axiom,
    ! [F: nat > set_int,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_set_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_355_lift__Suc__mono__le,axiom,
    ! [F: nat > rat,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_rat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_356_lift__Suc__mono__le,axiom,
    ! [F: nat > num,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_num @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_357_lift__Suc__mono__le,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_nat @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_358_lift__Suc__mono__le,axiom,
    ! [F: nat > int,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( F @ ( suc @ N3 ) ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_int @ ( F @ N2 ) @ ( F @ N5 ) ) ) ) ).

% lift_Suc_mono_le
thf(fact_359_lift__Suc__antimono__le,axiom,
    ! [F: nat > set_int,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_set_int @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_360_lift__Suc__antimono__le,axiom,
    ! [F: nat > rat,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_rat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_361_lift__Suc__antimono__le,axiom,
    ! [F: nat > num,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_num @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_362_lift__Suc__antimono__le,axiom,
    ! [F: nat > nat,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_nat @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_363_lift__Suc__antimono__le,axiom,
    ! [F: nat > int,N2: nat,N5: nat] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ ( suc @ N3 ) ) @ ( F @ N3 ) )
     => ( ( ord_less_eq_nat @ N2 @ N5 )
       => ( ord_less_eq_int @ ( F @ N5 ) @ ( F @ N2 ) ) ) ) ).

% lift_Suc_antimono_le
thf(fact_364_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_365_less__eq__Suc__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat] : ( ord_less_eq_nat @ ( suc @ N ) ) ) ) ).

% less_eq_Suc_le
thf(fact_366_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_367_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_368_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_369_inc__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( P @ J )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I2 @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ ( suc @ N3 ) )
                 => ( P @ N3 ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% inc_induct
thf(fact_370_dec__induct,axiom,
    ! [I2: nat,J: nat,P: nat > $o] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( P @ I2 )
       => ( ! [N3: nat] :
              ( ( ord_less_eq_nat @ I2 @ N3 )
             => ( ( ord_less_nat @ N3 @ J )
               => ( ( P @ N3 )
                 => ( P @ ( suc @ N3 ) ) ) ) )
         => ( P @ J ) ) ) ) ).

% dec_induct
thf(fact_371_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_372_Suc__leI,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( suc @ M ) @ N2 ) ) ).

% Suc_leI
thf(fact_373_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_374_less__add__Suc1,axiom,
    ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ I2 @ M ) ) ) ).

% less_add_Suc1
thf(fact_375_less__add__Suc2,axiom,
    ! [I2: nat,M: nat] : ( ord_less_nat @ I2 @ ( suc @ ( plus_plus_nat @ M @ I2 ) ) ) ).

% less_add_Suc2
thf(fact_376_less__iff__Suc__add,axiom,
    ( ord_less_nat
    = ( ^ [M6: nat,N: nat] :
        ? [K3: nat] :
          ( N
          = ( suc @ ( plus_plus_nat @ M6 @ K3 ) ) ) ) ) ).

% less_iff_Suc_add
thf(fact_377_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_378_mono__nat__linear__lb,axiom,
    ! [F: nat > nat,M: nat,K: nat] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_nat @ M5 @ N3 )
         => ( ord_less_nat @ ( F @ M5 ) @ ( F @ N3 ) ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ ( F @ M ) @ K ) @ ( F @ ( plus_plus_nat @ M @ K ) ) ) ) ).

% mono_nat_linear_lb
thf(fact_379_Suc__eq__plus1,axiom,
    ( suc
    = ( ^ [N: nat] : ( plus_plus_nat @ N @ one_one_nat ) ) ) ).

% Suc_eq_plus1
thf(fact_380_plus__1__eq__Suc,axiom,
    ( ( plus_plus_nat @ one_one_nat )
    = suc ) ).

% plus_1_eq_Suc
thf(fact_381_Suc__eq__plus1__left,axiom,
    ( suc
    = ( plus_plus_nat @ one_one_nat ) ) ).

% Suc_eq_plus1_left
thf(fact_382_field__sum__of__halves,axiom,
    ! [X4: rat] :
      ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ ( divide_divide_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
      = X4 ) ).

% field_sum_of_halves
thf(fact_383_field__sum__of__halves,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( divide_divide_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = X4 ) ).

% field_sum_of_halves
thf(fact_384_div__by__1,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ one_one_rat )
      = A ) ).

% div_by_1
thf(fact_385_div__by__1,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ one_one_nat )
      = A ) ).

% div_by_1
thf(fact_386_div__by__1,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ one_one_int )
      = A ) ).

% div_by_1
thf(fact_387_div__by__1,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ one_one_real )
      = A ) ).

% div_by_1
thf(fact_388_div__by__1,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ one_one_complex )
      = A ) ).

% div_by_1
thf(fact_389_div__by__1,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ one_one_Code_integer )
      = A ) ).

% div_by_1
thf(fact_390_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_391_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_392_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_393_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_394_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_395_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_396_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_397_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_398_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_399_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_400_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_401_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_402_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_403_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_404_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_405_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_406_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_407_nth__mem,axiom,
    ! [N2: nat,Xs: list_complex] :
      ( ( ord_less_nat @ N2 @ ( size_s3451745648224563538omplex @ Xs ) )
     => ( member_complex @ ( nth_complex @ Xs @ N2 ) @ ( set_complex2 @ Xs ) ) ) ).

% nth_mem
thf(fact_408_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_409_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_410_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_411_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_412_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_413_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o] :
      ( ( ord_less_nat @ N2 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
     => ( ! [X5: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_VEBT_VEBT @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_414_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_o,P: $o > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Xs ) )
     => ( ! [X5: $o] :
            ( ( member_o @ X5 @ ( set_o2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_o @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_415_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_nat,P: nat > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_nat @ Xs ) )
     => ( ! [X5: nat] :
            ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_nat @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_416_list__ball__nth,axiom,
    ! [N2: nat,Xs: list_int,P: int > $o] :
      ( ( ord_less_nat @ N2 @ ( size_size_list_int @ Xs ) )
     => ( ! [X5: int] :
            ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
           => ( P @ X5 ) )
       => ( P @ ( nth_int @ Xs @ N2 ) ) ) ) ).

% list_ball_nth
thf(fact_417_in__set__conv__nth,axiom,
    ! [X4: real,Xs: list_real] :
      ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs ) )
            & ( ( nth_real @ Xs @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_418_in__set__conv__nth,axiom,
    ! [X4: complex,Xs: list_complex] :
      ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs ) )
            & ( ( nth_complex @ Xs @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_419_in__set__conv__nth,axiom,
    ! [X4: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s5460976970255530739at_nat @ Xs ) )
            & ( ( nth_Pr7617993195940197384at_nat @ Xs @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_420_in__set__conv__nth,axiom,
    ! [X4: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
            & ( ( nth_VEBT_VEBT @ Xs @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_421_in__set__conv__nth,axiom,
    ! [X4: $o,Xs: list_o] :
      ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
            & ( ( nth_o @ Xs @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_422_in__set__conv__nth,axiom,
    ! [X4: nat,Xs: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
            & ( ( nth_nat @ Xs @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_423_in__set__conv__nth,axiom,
    ! [X4: int,Xs: list_int] :
      ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
      = ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
            & ( ( nth_int @ Xs @ I3 )
              = X4 ) ) ) ) ).

% in_set_conv_nth
thf(fact_424_all__nth__imp__all__set,axiom,
    ! [Xs: list_real,P: real > $o,X4: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_real @ Xs ) )
         => ( P @ ( nth_real @ Xs @ I4 ) ) )
     => ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_425_all__nth__imp__all__set,axiom,
    ! [Xs: list_complex,P: complex > $o,X4: complex] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s3451745648224563538omplex @ Xs ) )
         => ( P @ ( nth_complex @ Xs @ I4 ) ) )
     => ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_426_all__nth__imp__all__set,axiom,
    ! [Xs: list_P6011104703257516679at_nat,P: product_prod_nat_nat > $o,X4: product_prod_nat_nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s5460976970255530739at_nat @ Xs ) )
         => ( P @ ( nth_Pr7617993195940197384at_nat @ Xs @ I4 ) ) )
     => ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_427_all__nth__imp__all__set,axiom,
    ! [Xs: list_VEBT_VEBT,P: vEBT_VEBT > $o,X4: vEBT_VEBT] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
         => ( P @ ( nth_VEBT_VEBT @ Xs @ I4 ) ) )
     => ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_428_all__nth__imp__all__set,axiom,
    ! [Xs: list_o,P: $o > $o,X4: $o] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
         => ( P @ ( nth_o @ Xs @ I4 ) ) )
     => ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_429_all__nth__imp__all__set,axiom,
    ! [Xs: list_nat,P: nat > $o,X4: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
         => ( P @ ( nth_nat @ Xs @ I4 ) ) )
     => ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_430_all__nth__imp__all__set,axiom,
    ! [Xs: list_int,P: int > $o,X4: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
         => ( P @ ( nth_int @ Xs @ I4 ) ) )
     => ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
       => ( P @ X4 ) ) ) ).

% all_nth_imp_all_set
thf(fact_431_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 ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( P @ ( nth_VEBT_VEBT @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_432_all__set__conv__all__nth,axiom,
    ! [Xs: list_o,P: $o > $o] :
      ( ( ! [X: $o] :
            ( ( member_o @ X @ ( set_o2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs ) )
           => ( P @ ( nth_o @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_433_all__set__conv__all__nth,axiom,
    ! [Xs: list_nat,P: nat > $o] :
      ( ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs ) )
           => ( P @ ( nth_nat @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_434_all__set__conv__all__nth,axiom,
    ! [Xs: list_int,P: int > $o] :
      ( ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs ) )
           => ( P @ X ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs ) )
           => ( P @ ( nth_int @ Xs @ I3 ) ) ) ) ) ).

% all_set_conv_all_nth
thf(fact_435_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_436_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_437_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_438_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_439_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_440_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_441_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_442_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_443_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_444_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_445_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_446_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_447_real__arch__pow,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ? [N3: nat] : ( ord_less_real @ Y @ ( power_power_real @ X4 @ N3 ) ) ) ).

% real_arch_pow
thf(fact_448_less__eq__real__def,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y5: real] :
          ( ( ord_less_real @ X @ Y5 )
          | ( X = Y5 ) ) ) ) ).

% less_eq_real_def
thf(fact_449_complete__real,axiom,
    ! [S2: set_real] :
      ( ? [X2: real] : ( member_real @ X2 @ S2 )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( member_real @ X5 @ S2 )
           => ( ord_less_eq_real @ X5 @ Z3 ) )
       => ? [Y3: real] :
            ( ! [X2: real] :
                ( ( member_real @ X2 @ S2 )
               => ( ord_less_eq_real @ X2 @ Y3 ) )
            & ! [Z3: real] :
                ( ! [X5: real] :
                    ( ( member_real @ X5 @ S2 )
                   => ( ord_less_eq_real @ X5 @ Z3 ) )
               => ( ord_less_eq_real @ Y3 @ Z3 ) ) ) ) ) ).

% complete_real
thf(fact_450_linorder__neqE__linordered__idom,axiom,
    ! [X4: real,Y: real] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_real @ X4 @ Y )
       => ( ord_less_real @ Y @ X4 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_451_linorder__neqE__linordered__idom,axiom,
    ! [X4: rat,Y: rat] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_rat @ X4 @ Y )
       => ( ord_less_rat @ Y @ X4 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_452_linorder__neqE__linordered__idom,axiom,
    ! [X4: int,Y: int] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_int @ X4 @ Y )
       => ( ord_less_int @ Y @ X4 ) ) ) ).

% linorder_neqE_linordered_idom
thf(fact_453_linordered__field__no__ub,axiom,
    ! [X2: real] :
    ? [X_12: real] : ( ord_less_real @ X2 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_454_linordered__field__no__ub,axiom,
    ! [X2: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X2 @ X_12 ) ).

% linordered_field_no_ub
thf(fact_455_linordered__field__no__lb,axiom,
    ! [X2: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X2 ) ).

% linordered_field_no_lb
thf(fact_456_linordered__field__no__lb,axiom,
    ! [X2: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X2 ) ).

% linordered_field_no_lb
thf(fact_457_one__reorient,axiom,
    ! [X4: complex] :
      ( ( one_one_complex = X4 )
      = ( X4 = one_one_complex ) ) ).

% one_reorient
thf(fact_458_one__reorient,axiom,
    ! [X4: real] :
      ( ( one_one_real = X4 )
      = ( X4 = one_one_real ) ) ).

% one_reorient
thf(fact_459_one__reorient,axiom,
    ! [X4: rat] :
      ( ( one_one_rat = X4 )
      = ( X4 = one_one_rat ) ) ).

% one_reorient
thf(fact_460_one__reorient,axiom,
    ! [X4: nat] :
      ( ( one_one_nat = X4 )
      = ( X4 = one_one_nat ) ) ).

% one_reorient
thf(fact_461_one__reorient,axiom,
    ! [X4: int] :
      ( ( one_one_int = X4 )
      = ( X4 = one_one_int ) ) ).

% one_reorient
thf(fact_462_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_463_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_464_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_465_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_466_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_467_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_468_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_469_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_470_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_471_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_472_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_473_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_474_add_Ocommute,axiom,
    ( plus_plus_real
    = ( ^ [A3: real,B2: real] : ( plus_plus_real @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_475_add_Ocommute,axiom,
    ( plus_plus_rat
    = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_476_add_Ocommute,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B2: nat] : ( plus_plus_nat @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_477_add_Ocommute,axiom,
    ( plus_plus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ B2 @ A3 ) ) ) ).

% add.commute
thf(fact_478_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_479_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_480_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_481_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_482_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_483_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_484_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_485_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_486_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_487_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_488_group__cancel_Oadd2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( plus_plus_real @ A @ B3 )
        = ( plus_plus_real @ K @ ( plus_plus_real @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_489_group__cancel_Oadd2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( plus_plus_rat @ A @ B3 )
        = ( plus_plus_rat @ K @ ( plus_plus_rat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_490_group__cancel_Oadd2,axiom,
    ! [B3: nat,K: nat,B: nat,A: nat] :
      ( ( B3
        = ( plus_plus_nat @ K @ B ) )
     => ( ( plus_plus_nat @ A @ B3 )
        = ( plus_plus_nat @ K @ ( plus_plus_nat @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_491_group__cancel_Oadd2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( plus_plus_int @ A @ B3 )
        = ( plus_plus_int @ K @ ( plus_plus_int @ A @ B ) ) ) ) ).

% group_cancel.add2
thf(fact_492_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_493_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_494_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_495_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_496_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_real @ I2 @ K )
        = ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_497_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_rat @ I2 @ K )
        = ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_498_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_nat @ I2 @ K )
        = ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_499_add__mono__thms__linordered__semiring_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( K = L ) )
     => ( ( plus_plus_int @ I2 @ K )
        = ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(4)
thf(fact_500_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_501_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_502_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_503_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_504_subset__code_I1_J,axiom,
    ! [Xs: list_real,B3: set_real] :
      ( ( ord_less_eq_set_real @ ( set_real2 @ Xs ) @ B3 )
      = ( ! [X: real] :
            ( ( member_real @ X @ ( set_real2 @ Xs ) )
           => ( member_real @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_505_subset__code_I1_J,axiom,
    ! [Xs: list_complex,B3: set_complex] :
      ( ( ord_le211207098394363844omplex @ ( set_complex2 @ Xs ) @ B3 )
      = ( ! [X: complex] :
            ( ( member_complex @ X @ ( set_complex2 @ Xs ) )
           => ( member_complex @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_506_subset__code_I1_J,axiom,
    ! [Xs: list_P6011104703257516679at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le3146513528884898305at_nat @ ( set_Pr5648618587558075414at_nat @ Xs ) @ B3 )
      = ( ! [X: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ X @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( member8440522571783428010at_nat @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_507_subset__code_I1_J,axiom,
    ! [Xs: list_VEBT_VEBT,B3: set_VEBT_VEBT] :
      ( ( ord_le4337996190870823476T_VEBT @ ( set_VEBT_VEBT2 @ Xs ) @ B3 )
      = ( ! [X: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( member_VEBT_VEBT @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_508_subset__code_I1_J,axiom,
    ! [Xs: list_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ ( set_nat2 @ Xs ) @ B3 )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_nat2 @ Xs ) )
           => ( member_nat @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_509_subset__code_I1_J,axiom,
    ! [Xs: list_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( set_int2 @ Xs ) @ B3 )
      = ( ! [X: int] :
            ( ( member_int @ X @ ( set_int2 @ Xs ) )
           => ( member_int @ X @ B3 ) ) ) ) ).

% subset_code(1)
thf(fact_510_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_511_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_512_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_513_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_514_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_VEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_515_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_o] :
      ( ( size_size_list_o @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_516_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_nat] :
      ( ( size_size_list_nat @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_517_Ex__list__of__length,axiom,
    ! [N2: nat] :
    ? [Xs2: list_int] :
      ( ( size_size_list_int @ Xs2 )
      = N2 ) ).

% Ex_list_of_length
thf(fact_518_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_519_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_520_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_521_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_522_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_523_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_524_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_525_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_526_le__iff__add,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] :
        ? [C2: nat] :
          ( B2
          = ( plus_plus_nat @ A3 @ C2 ) ) ) ) ).

% le_iff_add
thf(fact_527_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_528_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_529_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_530_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_531_less__eqE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( B
           != ( plus_plus_nat @ A @ C3 ) ) ) ).

% less_eqE
thf(fact_532_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_533_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_534_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_535_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_536_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_537_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_538_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_539_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_540_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I2 @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_541_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_542_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_543_add__mono__thms__linordered__semiring_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(1)
thf(fact_544_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( I2 = J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_545_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_546_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_547_add__mono__thms__linordered__semiring_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(2)
thf(fact_548_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_549_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_550_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_551_add__mono__thms__linordered__semiring_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_semiring(3)
thf(fact_552_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_553_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_554_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_555_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_556_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_557_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_558_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_559_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_560_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_561_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_562_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_563_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_564_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_565_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_566_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_567_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_568_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_569_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_570_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_571_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_572_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_573_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_574_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_575_add__mono__thms__linordered__field_I1_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( K = L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(1)
thf(fact_576_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( I2 = J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_577_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( I2 = J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_578_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( I2 = J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_579_add__mono__thms__linordered__field_I2_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( I2 = J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(2)
thf(fact_580_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I2 @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_581_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_582_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_583_add__mono__thms__linordered__field_I5_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(5)
thf(fact_584_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_585_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_586_add__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ B ) @ C )
      = ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% add_divide_distrib
thf(fact_587_length__induct,axiom,
    ! [P: list_VEBT_VEBT > $o,Xs: list_VEBT_VEBT] :
      ( ! [Xs2: list_VEBT_VEBT] :
          ( ! [Ys2: list_VEBT_VEBT] :
              ( ( ord_less_nat @ ( size_s6755466524823107622T_VEBT @ Ys2 ) @ ( size_s6755466524823107622T_VEBT @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_588_length__induct,axiom,
    ! [P: list_o > $o,Xs: list_o] :
      ( ! [Xs2: list_o] :
          ( ! [Ys2: list_o] :
              ( ( ord_less_nat @ ( size_size_list_o @ Ys2 ) @ ( size_size_list_o @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_589_length__induct,axiom,
    ! [P: list_nat > $o,Xs: list_nat] :
      ( ! [Xs2: list_nat] :
          ( ! [Ys2: list_nat] :
              ( ( ord_less_nat @ ( size_size_list_nat @ Ys2 ) @ ( size_size_list_nat @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_590_length__induct,axiom,
    ! [P: list_int > $o,Xs: list_int] :
      ( ! [Xs2: list_int] :
          ( ! [Ys2: list_int] :
              ( ( ord_less_nat @ ( size_size_list_int @ Ys2 ) @ ( size_size_list_int @ Xs2 ) )
             => ( P @ Ys2 ) )
         => ( P @ Xs2 ) )
     => ( P @ Xs ) ) ).

% length_induct
thf(fact_591_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_592_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_593_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_594_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_595_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_596_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_597_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_598_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_599_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( ord_less_real @ I2 @ J )
        & ( ord_less_eq_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_600_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_rat @ I2 @ J )
        & ( ord_less_eq_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_601_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_nat @ I2 @ J )
        & ( ord_less_eq_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_602_add__mono__thms__linordered__field_I3_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_int @ I2 @ J )
        & ( ord_less_eq_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(3)
thf(fact_603_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: real,J: real,K: real,L: real] :
      ( ( ( ord_less_eq_real @ I2 @ J )
        & ( ord_less_real @ K @ L ) )
     => ( ord_less_real @ ( plus_plus_real @ I2 @ K ) @ ( plus_plus_real @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_604_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: rat,J: rat,K: rat,L: rat] :
      ( ( ( ord_less_eq_rat @ I2 @ J )
        & ( ord_less_rat @ K @ L ) )
     => ( ord_less_rat @ ( plus_plus_rat @ I2 @ K ) @ ( plus_plus_rat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_605_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
        & ( ord_less_nat @ K @ L ) )
     => ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ ( plus_plus_nat @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_606_add__mono__thms__linordered__field_I4_J,axiom,
    ! [I2: int,J: int,K: int,L: int] :
      ( ( ( ord_less_eq_int @ I2 @ J )
        & ( ord_less_int @ K @ L ) )
     => ( ord_less_int @ ( plus_plus_int @ I2 @ K ) @ ( plus_plus_int @ J @ L ) ) ) ).

% add_mono_thms_linordered_field(4)
thf(fact_607_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_608_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_609_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_610_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_611_less__add__one,axiom,
    ! [A: real] : ( ord_less_real @ A @ ( plus_plus_real @ A @ one_one_real ) ) ).

% less_add_one
thf(fact_612_less__add__one,axiom,
    ! [A: rat] : ( ord_less_rat @ A @ ( plus_plus_rat @ A @ one_one_rat ) ) ).

% less_add_one
thf(fact_613_less__add__one,axiom,
    ! [A: nat] : ( ord_less_nat @ A @ ( plus_plus_nat @ A @ one_one_nat ) ) ).

% less_add_one
thf(fact_614_less__add__one,axiom,
    ! [A: int] : ( ord_less_int @ A @ ( plus_plus_int @ A @ one_one_int ) ) ).

% less_add_one
thf(fact_615_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_VEBT_VEBT,Z4: list_VEBT_VEBT] : Y6 = Z4 )
    = ( ^ [Xs3: list_VEBT_VEBT,Ys3: list_VEBT_VEBT] :
          ( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
            = ( size_s6755466524823107622T_VEBT @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) )
             => ( ( nth_VEBT_VEBT @ Xs3 @ I3 )
                = ( nth_VEBT_VEBT @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_616_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_o,Z4: list_o] : Y6 = Z4 )
    = ( ^ [Xs3: list_o,Ys3: list_o] :
          ( ( ( size_size_list_o @ Xs3 )
            = ( size_size_list_o @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs3 ) )
             => ( ( nth_o @ Xs3 @ I3 )
                = ( nth_o @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_617_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_nat,Z4: list_nat] : Y6 = Z4 )
    = ( ^ [Xs3: list_nat,Ys3: list_nat] :
          ( ( ( size_size_list_nat @ Xs3 )
            = ( size_size_list_nat @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) )
             => ( ( nth_nat @ Xs3 @ I3 )
                = ( nth_nat @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_618_list__eq__iff__nth__eq,axiom,
    ( ( ^ [Y6: list_int,Z4: list_int] : Y6 = Z4 )
    = ( ^ [Xs3: list_int,Ys3: list_int] :
          ( ( ( size_size_list_int @ Xs3 )
            = ( size_size_list_int @ Ys3 ) )
          & ! [I3: nat] :
              ( ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs3 ) )
             => ( ( nth_int @ Xs3 @ I3 )
                = ( nth_int @ Ys3 @ I3 ) ) ) ) ) ) ).

% list_eq_iff_nth_eq
thf(fact_619_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > vEBT_VEBT > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X3: vEBT_VEBT] : ( P @ I3 @ X3 ) ) )
      = ( ? [Xs3: list_VEBT_VEBT] :
            ( ( ( size_s6755466524823107622T_VEBT @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_VEBT_VEBT @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_620_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > $o > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X3: $o] : ( P @ I3 @ X3 ) ) )
      = ( ? [Xs3: list_o] :
            ( ( ( size_size_list_o @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_o @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_621_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X3: nat] : ( P @ I3 @ X3 ) ) )
      = ( ? [Xs3: list_nat] :
            ( ( ( size_size_list_nat @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_nat @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_622_Skolem__list__nth,axiom,
    ! [K: nat,P: nat > int > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ K )
           => ? [X3: int] : ( P @ I3 @ X3 ) ) )
      = ( ? [Xs3: list_int] :
            ( ( ( size_size_list_int @ Xs3 )
              = K )
            & ! [I3: nat] :
                ( ( ord_less_nat @ I3 @ K )
               => ( P @ I3 @ ( nth_int @ Xs3 @ I3 ) ) ) ) ) ) ).

% Skolem_list_nth
thf(fact_623_nth__equalityI,axiom,
    ! [Xs: list_VEBT_VEBT,Ys: list_VEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = ( size_s6755466524823107622T_VEBT @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_s6755466524823107622T_VEBT @ Xs ) )
           => ( ( nth_VEBT_VEBT @ Xs @ I4 )
              = ( nth_VEBT_VEBT @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_624_nth__equalityI,axiom,
    ! [Xs: list_o,Ys: list_o] :
      ( ( ( size_size_list_o @ Xs )
        = ( size_size_list_o @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_o @ Xs ) )
           => ( ( nth_o @ Xs @ I4 )
              = ( nth_o @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_625_nth__equalityI,axiom,
    ! [Xs: list_nat,Ys: list_nat] :
      ( ( ( size_size_list_nat @ Xs )
        = ( size_size_list_nat @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_nat @ Xs ) )
           => ( ( nth_nat @ Xs @ I4 )
              = ( nth_nat @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_626_nth__equalityI,axiom,
    ! [Xs: list_int,Ys: list_int] :
      ( ( ( size_size_list_int @ Xs )
        = ( size_size_list_int @ Ys ) )
     => ( ! [I4: nat] :
            ( ( ord_less_nat @ I4 @ ( size_size_list_int @ Xs ) )
           => ( ( nth_int @ Xs @ I4 )
              = ( nth_int @ Ys @ I4 ) ) )
       => ( Xs = Ys ) ) ) ).

% nth_equalityI
thf(fact_627_discrete,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ A3 @ one_one_nat ) ) ) ) ).

% discrete
thf(fact_628_discrete,axiom,
    ( ord_less_int
    = ( ^ [A3: int] : ( ord_less_eq_int @ ( plus_plus_int @ A3 @ one_one_int ) ) ) ) ).

% discrete
thf(fact_629_buildup__nothing__in__leaf,axiom,
    ! [N2: nat,X4: nat] :
      ~ ( vEBT_V5719532721284313246member @ ( vEBT_vebt_buildup @ N2 ) @ X4 ) ).

% buildup_nothing_in_leaf
thf(fact_630_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_631_buildup__nothing__in__min__max,axiom,
    ! [N2: nat,X4: nat] :
      ~ ( vEBT_VEBT_membermima @ ( vEBT_vebt_buildup @ N2 ) @ X4 ) ).

% buildup_nothing_in_min_max
thf(fact_632_invar__vebt_Ointros_I3_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ 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_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(3)
thf(fact_633_invar__vebt_Ointros_I2_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ 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_12: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X_12 )
               => ( ! [X5: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) )
                 => ( vEBT_invar_vebt @ ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(2)
thf(fact_634_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_635_dbl__simps_I3_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_636_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_637_dbl__simps_I3_J,axiom,
    ( ( neg_numeral_dbl_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% dbl_simps(3)
thf(fact_638_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera1916890842035813515d_enat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_639_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_complex @ ( numera6690914467698888265omplex @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numera6690914467698888265omplex @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_640_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_real @ ( numeral_numeral_real @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_real @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_641_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_nat @ ( numeral_numeral_nat @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_nat @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_642_power__numeral,axiom,
    ! [K: num,L: num] :
      ( ( power_power_int @ ( numeral_numeral_int @ K ) @ ( numeral_numeral_nat @ L ) )
      = ( numeral_numeral_int @ ( pow @ K @ L ) ) ) ).

% power_numeral
thf(fact_643_valid__eq2,axiom,
    ! [T2: vEBT_VEBT,D: nat] :
      ( ( vEBT_VEBT_valid @ T2 @ D )
     => ( vEBT_invar_vebt @ T2 @ D ) ) ).

% valid_eq2
thf(fact_644_valid__eq,axiom,
    vEBT_VEBT_valid = vEBT_invar_vebt ).

% valid_eq
thf(fact_645_valid__eq1,axiom,
    ! [T2: vEBT_VEBT,D: nat] :
      ( ( vEBT_invar_vebt @ T2 @ D )
     => ( vEBT_VEBT_valid @ T2 @ D ) ) ).

% valid_eq1
thf(fact_646_zdiv__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit0
thf(fact_647_mod__mod__trivial,axiom,
    ! [A: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ B )
      = ( modulo_modulo_nat @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_648_mod__mod__trivial,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_649_mod__mod__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_mod_trivial
thf(fact_650_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_651_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_652_mod__add__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self1
thf(fact_653_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_654_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_655_mod__add__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% mod_add_self2
thf(fact_656_mod__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( modulo_modulo_nat @ M @ N2 )
        = M ) ) ).

% mod_less
thf(fact_657_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_658_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_659_dbl__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit0 @ K ) ) ) ).

% dbl_simps(5)
thf(fact_660_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% bits_one_mod_two_eq_one
thf(fact_661_bits__one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_one_mod_two_eq_one
thf(fact_662_bits__one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_one_mod_two_eq_one
thf(fact_663_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = one_one_nat ) ).

% one_mod_two_eq_one
thf(fact_664_one__mod__two__eq__one,axiom,
    ( ( modulo_modulo_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% one_mod_two_eq_one
thf(fact_665_one__mod__two__eq__one,axiom,
    ( ( modulo364778990260209775nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% one_mod_two_eq_one
thf(fact_666_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_667_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_668_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_669_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_670_mod__add__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( plus_plus_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( plus_plus_nat @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_671_mod__add__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( plus_plus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( plus_plus_int @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_672_mod__add__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( plus_p5714425477246183910nteger @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_add_cong
thf(fact_673_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_674_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_675_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_676_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_677_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_678_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_679_power__mod,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( modulo_modulo_nat @ ( power_power_nat @ ( modulo_modulo_nat @ A @ B ) @ N2 ) @ B )
      = ( modulo_modulo_nat @ ( power_power_nat @ A @ N2 ) @ B ) ) ).

% power_mod
thf(fact_680_power__mod,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( modulo_modulo_int @ ( power_power_int @ ( modulo_modulo_int @ A @ B ) @ N2 ) @ B )
      = ( modulo_modulo_int @ ( power_power_int @ A @ N2 ) @ B ) ) ).

% power_mod
thf(fact_681_power__mod,axiom,
    ! [A: code_integer,B: code_integer,N2: nat] :
      ( ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ ( modulo364778990260209775nteger @ A @ B ) @ N2 ) @ B )
      = ( modulo364778990260209775nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ B ) ) ).

% power_mod
thf(fact_682_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_683_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_684_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_685_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_686_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_687_cong__exp__iff__simps_I9_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(9)
thf(fact_688_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ one ) )
      = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_689_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ one ) )
      = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_690_cong__exp__iff__simps_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ one ) )
      = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) ) ) ).

% cong_exp_iff_simps(4)
thf(fact_691_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_692_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_693_cong__exp__iff__simps_I8_J,axiom,
    ! [M: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(8)
thf(fact_694_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_695_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_696_cong__exp__iff__simps_I6_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(6)
thf(fact_697_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_698_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_699_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_700_mod__induct,axiom,
    ! [P: nat > $o,N2: nat,P2: nat,M: nat] :
      ( ( P @ N2 )
     => ( ( ord_less_nat @ N2 @ P2 )
       => ( ( ord_less_nat @ M @ P2 )
         => ( ! [N3: nat] :
                ( ( ord_less_nat @ N3 @ P2 )
               => ( ( P @ N3 )
                 => ( P @ ( modulo_modulo_nat @ ( suc @ N3 ) @ P2 ) ) ) )
           => ( P @ M ) ) ) ) ) ).

% mod_induct
thf(fact_701_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_702_dbl__def,axiom,
    ( neg_numeral_dbl_real
    = ( ^ [X: real] : ( plus_plus_real @ X @ X ) ) ) ).

% dbl_def
thf(fact_703_dbl__def,axiom,
    ( neg_numeral_dbl_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ X @ X ) ) ) ).

% dbl_def
thf(fact_704_dbl__def,axiom,
    ( neg_numeral_dbl_int
    = ( ^ [X: int] : ( plus_plus_int @ X @ X ) ) ) ).

% dbl_def
thf(fact_705_pow_Osimps_I1_J,axiom,
    ! [X4: num] :
      ( ( pow @ X4 @ one )
      = X4 ) ).

% pow.simps(1)
thf(fact_706_bounded__Max__nat,axiom,
    ! [P: nat > $o,X4: nat,M7: nat] :
      ( ( P @ X4 )
     => ( ! [X5: nat] :
            ( ( P @ X5 )
           => ( ord_less_eq_nat @ X5 @ M7 ) )
       => ~ ! [M5: nat] :
              ( ( P @ M5 )
             => ~ ! [X2: nat] :
                    ( ( P @ X2 )
                   => ( ord_less_eq_nat @ X2 @ M5 ) ) ) ) ) ).

% bounded_Max_nat
thf(fact_707_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_708_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_709_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_710_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_711_psubsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( A2 != B3 )
       => ( ord_less_set_int @ A2 @ B3 ) ) ) ).

% psubsetI
thf(fact_712_subset__antisym,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ A2 )
       => ( A2 = B3 ) ) ) ).

% subset_antisym
thf(fact_713_subsetI,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( member_real @ X5 @ B3 ) )
     => ( ord_less_eq_set_real @ A2 @ B3 ) ) ).

% subsetI
thf(fact_714_subsetI,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( member_nat @ X5 @ B3 ) )
     => ( ord_less_eq_set_nat @ A2 @ B3 ) ) ).

% subsetI
thf(fact_715_subsetI,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( member_complex @ X5 @ B3 ) )
     => ( ord_le211207098394363844omplex @ A2 @ B3 ) ) ).

% subsetI
thf(fact_716_subsetI,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ A2 )
         => ( member8440522571783428010at_nat @ X5 @ B3 ) )
     => ( ord_le3146513528884898305at_nat @ A2 @ B3 ) ) ).

% subsetI
thf(fact_717_subsetI,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( member_int @ X5 @ B3 ) )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% subsetI
thf(fact_718_verit__eq__simplify_I8_J,axiom,
    ! [X22: num,Y2: num] :
      ( ( ( bit0 @ X22 )
        = ( bit0 @ Y2 ) )
      = ( X22 = Y2 ) ) ).

% verit_eq_simplify(8)
thf(fact_719_order__refl,axiom,
    ! [X4: set_int] : ( ord_less_eq_set_int @ X4 @ X4 ) ).

% order_refl
thf(fact_720_order__refl,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ X4 @ X4 ) ).

% order_refl
thf(fact_721_order__refl,axiom,
    ! [X4: num] : ( ord_less_eq_num @ X4 @ X4 ) ).

% order_refl
thf(fact_722_order__refl,axiom,
    ! [X4: nat] : ( ord_less_eq_nat @ X4 @ X4 ) ).

% order_refl
thf(fact_723_order__refl,axiom,
    ! [X4: int] : ( ord_less_eq_int @ X4 @ X4 ) ).

% order_refl
thf(fact_724_dual__order_Orefl,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% dual_order.refl
thf(fact_725_dual__order_Orefl,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% dual_order.refl
thf(fact_726_dual__order_Orefl,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% dual_order.refl
thf(fact_727_dual__order_Orefl,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% dual_order.refl
thf(fact_728_dual__order_Orefl,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% dual_order.refl
thf(fact_729_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_730_length__subseqs,axiom,
    ! [Xs: list_VEBT_VEBT] :
      ( ( size_s8217280938318005548T_VEBT @ ( subseqs_VEBT_VEBT @ Xs ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_subseqs
thf(fact_731_length__subseqs,axiom,
    ! [Xs: list_o] :
      ( ( size_s2710708370519433104list_o @ ( subseqs_o @ Xs ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_o @ Xs ) ) ) ).

% length_subseqs
thf(fact_732_length__subseqs,axiom,
    ! [Xs: list_nat] :
      ( ( size_s3023201423986296836st_nat @ ( subseqs_nat @ Xs ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_nat @ Xs ) ) ) ).

% length_subseqs
thf(fact_733_length__subseqs,axiom,
    ! [Xs: list_int] :
      ( ( size_s533118279054570080st_int @ ( subseqs_int @ Xs ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( size_size_list_int @ Xs ) ) ) ).

% length_subseqs
thf(fact_734_invar__vebt_Ointros_I5_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M
              = ( suc @ N2 ) )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X3 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I4 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
                                        = I4 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(5)
thf(fact_735_deg__not__0,axiom,
    ! [T2: vEBT_VEBT,N2: nat] :
      ( ( vEBT_invar_vebt @ T2 @ N2 )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% deg_not_0
thf(fact_736__C5_Ohyps_C_I12_J,axiom,
    ( ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ mi @ ma ) )
    = info ) ).

% "5.hyps"(12)
thf(fact_737_le__zero__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_zero_eq
thf(fact_738_not__gr__zero,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr_zero
thf(fact_739_add__0,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add_0
thf(fact_740_add__0,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add_0
thf(fact_741_add__0,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add_0
thf(fact_742_add__0,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ A )
      = A ) ).

% add_0
thf(fact_743_add__0,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add_0
thf(fact_744_zero__eq__add__iff__both__eq__0,axiom,
    ! [X4: nat,Y: nat] :
      ( ( zero_zero_nat
        = ( plus_plus_nat @ X4 @ Y ) )
      = ( ( X4 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% zero_eq_add_iff_both_eq_0
thf(fact_745_add__eq__0__iff__both__eq__0,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( plus_plus_nat @ X4 @ Y )
        = zero_zero_nat )
      = ( ( X4 = zero_zero_nat )
        & ( Y = zero_zero_nat ) ) ) ).

% add_eq_0_iff_both_eq_0
thf(fact_746_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_747_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_748_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_749_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_750_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_751_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_752_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_753_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_754_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_755_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_756_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_757_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_758_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_759_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_760_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_761_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_762_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_763_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_764_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_765_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_766_double__zero__sym,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( plus_plus_real @ A @ A ) )
      = ( A = zero_zero_real ) ) ).

% double_zero_sym
thf(fact_767_double__zero__sym,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( plus_plus_rat @ A @ A ) )
      = ( A = zero_zero_rat ) ) ).

% double_zero_sym
thf(fact_768_double__zero__sym,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( plus_plus_int @ A @ A ) )
      = ( A = zero_zero_int ) ) ).

% double_zero_sym
thf(fact_769_add_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.right_neutral
thf(fact_770_add_Oright__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.right_neutral
thf(fact_771_add_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.right_neutral
thf(fact_772_add_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.right_neutral
thf(fact_773_add_Oright__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.right_neutral
thf(fact_774_division__ring__divide__zero,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% division_ring_divide_zero
thf(fact_775_division__ring__divide__zero,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% division_ring_divide_zero
thf(fact_776_division__ring__divide__zero,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% division_ring_divide_zero
thf(fact_777_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_778_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_779_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_780_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_781_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_782_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_783_div__by__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% div_by_0
thf(fact_784_div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% div_by_0
thf(fact_785_div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% div_by_0
thf(fact_786_div__by__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% div_by_0
thf(fact_787_div__by__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% div_by_0
thf(fact_788_div__by__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% div_by_0
thf(fact_789_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_790_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_791_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_792_div__0,axiom,
    ! [A: rat] :
      ( ( divide_divide_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% div_0
thf(fact_793_div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% div_0
thf(fact_794_div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% div_0
thf(fact_795_div__0,axiom,
    ! [A: real] :
      ( ( divide_divide_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% div_0
thf(fact_796_div__0,axiom,
    ! [A: complex] :
      ( ( divide1717551699836669952omplex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% div_0
thf(fact_797_div__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% div_0
thf(fact_798_bits__div__by__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% bits_div_by_0
thf(fact_799_bits__div__by__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% bits_div_by_0
thf(fact_800_bits__div__by__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ zero_z3403309356797280102nteger )
      = zero_z3403309356797280102nteger ) ).

% bits_div_by_0
thf(fact_801_bits__div__0,axiom,
    ! [A: nat] :
      ( ( divide_divide_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_div_0
thf(fact_802_bits__div__0,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_div_0
thf(fact_803_bits__div__0,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_div_0
thf(fact_804_bits__mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% bits_mod_0
thf(fact_805_bits__mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% bits_mod_0
thf(fact_806_bits__mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_0
thf(fact_807_mod__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mod_0
thf(fact_808_mod__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mod_0
thf(fact_809_mod__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ zero_z3403309356797280102nteger @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_0
thf(fact_810_mod__by__0,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ zero_zero_nat )
      = A ) ).

% mod_by_0
thf(fact_811_mod__by__0,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ zero_zero_int )
      = A ) ).

% mod_by_0
thf(fact_812_mod__by__0,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ zero_z3403309356797280102nteger )
      = A ) ).

% mod_by_0
thf(fact_813_mod__self,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ A )
      = zero_zero_nat ) ).

% mod_self
thf(fact_814_mod__self,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ A )
      = zero_zero_int ) ).

% mod_self
thf(fact_815_mod__self,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ A )
      = zero_z3403309356797280102nteger ) ).

% mod_self
thf(fact_816_less__nat__zero__code,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_nat_zero_code
thf(fact_817_neq0__conv,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
      = ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% neq0_conv
thf(fact_818_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_819_le0,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% le0
thf(fact_820_bot__nat__0_Oextremum,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ zero_zero_nat @ A ) ).

% bot_nat_0.extremum
thf(fact_821_Nat_Oadd__0__right,axiom,
    ! [M: nat] :
      ( ( plus_plus_nat @ M @ zero_zero_nat )
      = M ) ).

% Nat.add_0_right
thf(fact_822_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_823_dbl__simps_I2_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% dbl_simps(2)
thf(fact_824_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_real @ zero_zero_real )
    = zero_zero_real ) ).

% dbl_simps(2)
thf(fact_825_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% dbl_simps(2)
thf(fact_826_dbl__simps_I2_J,axiom,
    ( ( neg_numeral_dbl_int @ zero_zero_int )
    = zero_zero_int ) ).

% dbl_simps(2)
thf(fact_827_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_828_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_829_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_830_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_831_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_832_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_833_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_834_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_835_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_836_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_837_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_838_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_839_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_840_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_841_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_842_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_843_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_844_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_845_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_846_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_847_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_848_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_849_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_850_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_851_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_852_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_853_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_854_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_855_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_856_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_857_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_858_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_859_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_860_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_861_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_862_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_863_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_864_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_865_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_866_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_867_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_868_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_869_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_870_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_871_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_872_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_873_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_874_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_875_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_876_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_877_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_878_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_879_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_880_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_881_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_882_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_883_divide__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% divide_self
thf(fact_884_divide__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% divide_self
thf(fact_885_divide__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% divide_self
thf(fact_886_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_887_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_888_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_889_div__self,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( divide_divide_rat @ A @ A )
        = one_one_rat ) ) ).

% div_self
thf(fact_890_div__self,axiom,
    ! [A: nat] :
      ( ( A != zero_zero_nat )
     => ( ( divide_divide_nat @ A @ A )
        = one_one_nat ) ) ).

% div_self
thf(fact_891_div__self,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( divide_divide_int @ A @ A )
        = one_one_int ) ) ).

% div_self
thf(fact_892_div__self,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( divide_divide_real @ A @ A )
        = one_one_real ) ) ).

% div_self
thf(fact_893_div__self,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( divide1717551699836669952omplex @ A @ A )
        = one_one_complex ) ) ).

% div_self
thf(fact_894_div__self,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( divide6298287555418463151nteger @ A @ A )
        = one_one_Code_integer ) ) ).

% div_self
thf(fact_895_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_896_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_897_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_898_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ zero_zero_rat @ ( suc @ N2 ) )
      = zero_zero_rat ) ).

% power_0_Suc
thf(fact_899_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ zero_zero_nat @ ( suc @ N2 ) )
      = zero_zero_nat ) ).

% power_0_Suc
thf(fact_900_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ zero_zero_real @ ( suc @ N2 ) )
      = zero_zero_real ) ).

% power_0_Suc
thf(fact_901_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ zero_zero_int @ ( suc @ N2 ) )
      = zero_zero_int ) ).

% power_0_Suc
thf(fact_902_power__0__Suc,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ zero_zero_complex @ ( suc @ N2 ) )
      = zero_zero_complex ) ).

% power_0_Suc
thf(fact_903_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_rat ) ).

% power_zero_numeral
thf(fact_904_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ K ) )
      = zero_zero_nat ) ).

% power_zero_numeral
thf(fact_905_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ K ) )
      = zero_zero_real ) ).

% power_zero_numeral
thf(fact_906_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ K ) )
      = zero_zero_int ) ).

% power_zero_numeral
thf(fact_907_power__zero__numeral,axiom,
    ! [K: num] :
      ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ K ) )
      = zero_zero_complex ) ).

% power_zero_numeral
thf(fact_908_mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% mod_by_1
thf(fact_909_mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% mod_by_1
thf(fact_910_mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% mod_by_1
thf(fact_911_bits__mod__by__1,axiom,
    ! [A: nat] :
      ( ( modulo_modulo_nat @ A @ one_one_nat )
      = zero_zero_nat ) ).

% bits_mod_by_1
thf(fact_912_bits__mod__by__1,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ one_one_int )
      = zero_zero_int ) ).

% bits_mod_by_1
thf(fact_913_bits__mod__by__1,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ one_one_Code_integer )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_by_1
thf(fact_914_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_915_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_916_bits__mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% bits_mod_div_trivial
thf(fact_917_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_918_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_919_mod__div__trivial,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( modulo364778990260209775nteger @ A @ B ) @ B )
      = zero_z3403309356797280102nteger ) ).

% mod_div_trivial
thf(fact_920_power__Suc0__right,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_921_power__Suc0__right,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_922_power__Suc0__right,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_923_power__Suc0__right,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( suc @ zero_zero_nat ) )
      = A ) ).

% power_Suc0_right
thf(fact_924_less__Suc0,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ ( suc @ zero_zero_nat ) )
      = ( N2 = zero_zero_nat ) ) ).

% less_Suc0
thf(fact_925_zero__less__Suc,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( suc @ N2 ) ) ).

% zero_less_Suc
thf(fact_926_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_927_less__one,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ N2 @ one_one_nat )
      = ( N2 = zero_zero_nat ) ) ).

% less_one
thf(fact_928_div__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( divide_divide_nat @ M @ ( suc @ zero_zero_nat ) )
      = M ) ).

% div_by_Suc_0
thf(fact_929_div__less,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( ( divide_divide_nat @ M @ N2 )
        = zero_zero_nat ) ) ).

% div_less
thf(fact_930_power__Suc__0,axiom,
    ! [N2: nat] :
      ( ( power_power_nat @ ( suc @ zero_zero_nat ) @ N2 )
      = ( suc @ zero_zero_nat ) ) ).

% power_Suc_0
thf(fact_931_nat__power__eq__Suc__0__iff,axiom,
    ! [X4: nat,M: nat] :
      ( ( ( power_power_nat @ X4 @ M )
        = ( suc @ zero_zero_nat ) )
      = ( ( M = zero_zero_nat )
        | ( X4
          = ( suc @ zero_zero_nat ) ) ) ) ).

% nat_power_eq_Suc_0_iff
thf(fact_932_nat__zero__less__power__iff,axiom,
    ! [X4: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ X4 @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N2 = zero_zero_nat ) ) ) ).

% nat_zero_less_power_iff
thf(fact_933_mod__by__Suc__0,axiom,
    ! [M: nat] :
      ( ( modulo_modulo_nat @ M @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% mod_by_Suc_0
thf(fact_934_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_935_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_936_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_937_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_938_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_939_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_940_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_941_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_942_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_943_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_944_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_945_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_946_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_947_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_948_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_949_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_950_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_951_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_952_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_953_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_954_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_955_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_956_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_957_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_958_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_959_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_960_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_961_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_962_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_963_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_964_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_965_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_966_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_967_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_968_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_969_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_970_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_971_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_972_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_973_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_974_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_975_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_976_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_977_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_978_one__div__two__eq__zero,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% one_div_two_eq_zero
thf(fact_979_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_980_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_981_bits__1__div__2,axiom,
    ( ( divide6298287555418463151nteger @ one_one_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = zero_z3403309356797280102nteger ) ).

% bits_1_div_2
thf(fact_982_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_983_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_984_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_985_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_986_power2__eq__iff__nonneg,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X4 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_987_power2__eq__iff__nonneg,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X4 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_988_power2__eq__iff__nonneg,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X4 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_989_power2__eq__iff__nonneg,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = ( X4 = Y ) ) ) ) ).

% power2_eq_iff_nonneg
thf(fact_990_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_991_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_992_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_993_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_994_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_995_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_996_sum__power2__eq__zero__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_997_sum__power2__eq__zero__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_998_sum__power2__eq__zero__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_eq_zero_iff
thf(fact_999_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_1000_psubsetD,axiom,
    ! [A2: set_real,B3: set_real,C: real] :
      ( ( ord_less_set_real @ A2 @ B3 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_1001_psubsetD,axiom,
    ! [A2: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_set_nat @ A2 @ B3 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_1002_psubsetD,axiom,
    ! [A2: set_complex,B3: set_complex,C: complex] :
      ( ( ord_less_set_complex @ A2 @ B3 )
     => ( ( member_complex @ C @ A2 )
       => ( member_complex @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_1003_psubsetD,axiom,
    ! [A2: set_int,B3: set_int,C: int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_1004_psubsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
     => ( ( member8440522571783428010at_nat @ C @ A2 )
       => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% psubsetD
thf(fact_1005_zero__reorient,axiom,
    ! [X4: complex] :
      ( ( zero_zero_complex = X4 )
      = ( X4 = zero_zero_complex ) ) ).

% zero_reorient
thf(fact_1006_zero__reorient,axiom,
    ! [X4: real] :
      ( ( zero_zero_real = X4 )
      = ( X4 = zero_zero_real ) ) ).

% zero_reorient
thf(fact_1007_zero__reorient,axiom,
    ! [X4: rat] :
      ( ( zero_zero_rat = X4 )
      = ( X4 = zero_zero_rat ) ) ).

% zero_reorient
thf(fact_1008_zero__reorient,axiom,
    ! [X4: nat] :
      ( ( zero_zero_nat = X4 )
      = ( X4 = zero_zero_nat ) ) ).

% zero_reorient
thf(fact_1009_zero__reorient,axiom,
    ! [X4: int] :
      ( ( zero_zero_int = X4 )
      = ( X4 = zero_zero_int ) ) ).

% zero_reorient
thf(fact_1010_verit__sum__simplify,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% verit_sum_simplify
thf(fact_1011_verit__sum__simplify,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% verit_sum_simplify
thf(fact_1012_verit__sum__simplify,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% verit_sum_simplify
thf(fact_1013_verit__sum__simplify,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% verit_sum_simplify
thf(fact_1014_verit__sum__simplify,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% verit_sum_simplify
thf(fact_1015_VEBT__internal_Omembermima_Osimps_I3_J,axiom,
    ! [Mi: nat,Ma: nat,Va: list_VEBT_VEBT,Vb: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ zero_zero_nat @ Va @ Vb ) @ X4 )
      = ( ( X4 = Mi )
        | ( X4 = Ma ) ) ) ).

% VEBT_internal.membermima.simps(3)
thf(fact_1016_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_1017_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_1018_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_1019_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_1020_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_1021_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_1022_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_1023_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_1024_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_1025_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_1026_zero__le,axiom,
    ! [X4: nat] : ( ord_less_eq_nat @ zero_zero_nat @ X4 ) ).

% zero_le
thf(fact_1027_le__numeral__extra_I3_J,axiom,
    ord_less_eq_real @ zero_zero_real @ zero_zero_real ).

% le_numeral_extra(3)
thf(fact_1028_le__numeral__extra_I3_J,axiom,
    ord_less_eq_rat @ zero_zero_rat @ zero_zero_rat ).

% le_numeral_extra(3)
thf(fact_1029_le__numeral__extra_I3_J,axiom,
    ord_less_eq_nat @ zero_zero_nat @ zero_zero_nat ).

% le_numeral_extra(3)
thf(fact_1030_le__numeral__extra_I3_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% le_numeral_extra(3)
thf(fact_1031_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_1032_gr__implies__not__zero,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not_zero
thf(fact_1033_not__less__zero,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less_zero
thf(fact_1034_gr__zeroI,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr_zeroI
thf(fact_1035_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_real @ zero_zero_real @ zero_zero_real ) ).

% less_numeral_extra(3)
thf(fact_1036_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_rat @ zero_zero_rat @ zero_zero_rat ) ).

% less_numeral_extra(3)
thf(fact_1037_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_nat @ zero_zero_nat @ zero_zero_nat ) ).

% less_numeral_extra(3)
thf(fact_1038_less__numeral__extra_I3_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_numeral_extra(3)
thf(fact_1039_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_1040_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_1041_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( numeral_numeral_rat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1042_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_z5237406670263579293d_enat
     != ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1043_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1044_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( numeral_numeral_real @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1045_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_nat
     != ( numeral_numeral_nat @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1046_zero__neq__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( numeral_numeral_int @ N2 ) ) ).

% zero_neq_numeral
thf(fact_1047_zero__neq__one,axiom,
    zero_zero_complex != one_one_complex ).

% zero_neq_one
thf(fact_1048_zero__neq__one,axiom,
    zero_zero_real != one_one_real ).

% zero_neq_one
thf(fact_1049_zero__neq__one,axiom,
    zero_zero_rat != one_one_rat ).

% zero_neq_one
thf(fact_1050_zero__neq__one,axiom,
    zero_zero_nat != one_one_nat ).

% zero_neq_one
thf(fact_1051_zero__neq__one,axiom,
    zero_zero_int != one_one_int ).

% zero_neq_one
thf(fact_1052_add_Ogroup__left__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ zero_zero_complex @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1053_add_Ogroup__left__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ zero_zero_real @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1054_add_Ogroup__left__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ zero_zero_rat @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1055_add_Ogroup__left__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ zero_zero_int @ A )
      = A ) ).

% add.group_left_neutral
thf(fact_1056_add_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ zero_zero_complex )
      = A ) ).

% add.comm_neutral
thf(fact_1057_add_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ zero_zero_real )
      = A ) ).

% add.comm_neutral
thf(fact_1058_add_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ zero_zero_rat )
      = A ) ).

% add.comm_neutral
thf(fact_1059_add_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( plus_plus_nat @ A @ zero_zero_nat )
      = A ) ).

% add.comm_neutral
thf(fact_1060_add_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ zero_zero_int )
      = A ) ).

% add.comm_neutral
thf(fact_1061_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_1062_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_1063_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_1064_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_1065_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_1066_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_1067_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_1068_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_1069_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_1070_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_1071_num_Osize_I4_J,axiom,
    ( ( size_size_num @ one )
    = zero_zero_nat ) ).

% num.size(4)
thf(fact_1072_vebt__buildup_Ocases,axiom,
    ! [X4: nat] :
      ( ( X4 != zero_zero_nat )
     => ( ( X4
         != ( suc @ zero_zero_nat ) )
       => ~ ! [Va2: nat] :
              ( X4
             != ( suc @ ( suc @ Va2 ) ) ) ) ) ).

% vebt_buildup.cases
thf(fact_1073_not0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% not0_implies_Suc
thf(fact_1074_Zero__not__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_not_Suc
thf(fact_1075_Zero__neq__Suc,axiom,
    ! [M: nat] :
      ( zero_zero_nat
     != ( suc @ M ) ) ).

% Zero_neq_Suc
thf(fact_1076_Suc__neq__Zero,axiom,
    ! [M: nat] :
      ( ( suc @ M )
     != zero_zero_nat ) ).

% Suc_neq_Zero
thf(fact_1077_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_1078_diff__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [X5: nat] : ( P @ X5 @ zero_zero_nat )
     => ( ! [Y3: nat] : ( P @ zero_zero_nat @ ( suc @ Y3 ) )
       => ( ! [X5: nat,Y3: nat] :
              ( ( P @ X5 @ Y3 )
             => ( P @ ( suc @ X5 ) @ ( suc @ Y3 ) ) )
         => ( P @ M @ N2 ) ) ) ) ).

% diff_induct
thf(fact_1079_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_1080_old_Onat_Oexhaust,axiom,
    ! [Y: nat] :
      ( ( Y != zero_zero_nat )
     => ~ ! [Nat3: nat] :
            ( Y
           != ( suc @ Nat3 ) ) ) ).

% old.nat.exhaust
thf(fact_1081_nat_OdiscI,axiom,
    ! [Nat: nat,X22: nat] :
      ( ( Nat
        = ( suc @ X22 ) )
     => ( Nat != zero_zero_nat ) ) ).

% nat.discI
thf(fact_1082_old_Onat_Odistinct_I1_J,axiom,
    ! [Nat2: nat] :
      ( zero_zero_nat
     != ( suc @ Nat2 ) ) ).

% old.nat.distinct(1)
thf(fact_1083_old_Onat_Odistinct_I2_J,axiom,
    ! [Nat2: nat] :
      ( ( suc @ Nat2 )
     != zero_zero_nat ) ).

% old.nat.distinct(2)
thf(fact_1084_nat_Odistinct_I1_J,axiom,
    ! [X22: nat] :
      ( zero_zero_nat
     != ( suc @ X22 ) ) ).

% nat.distinct(1)
thf(fact_1085_infinite__descent0,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ zero_zero_nat )
     => ( ! [N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ~ ( P @ N3 )
             => ? [M2: nat] :
                  ( ( ord_less_nat @ M2 @ N3 )
                  & ~ ( P @ M2 ) ) ) )
       => ( P @ N2 ) ) ) ).

% infinite_descent0
thf(fact_1086_gr__implies__not0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_nat @ M @ N2 )
     => ( N2 != zero_zero_nat ) ) ).

% gr_implies_not0
thf(fact_1087_less__zeroE,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% less_zeroE
thf(fact_1088_not__less0,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ N2 @ zero_zero_nat ) ).

% not_less0
thf(fact_1089_not__gr0,axiom,
    ! [N2: nat] :
      ( ( ~ ( ord_less_nat @ zero_zero_nat @ N2 ) )
      = ( N2 = zero_zero_nat ) ) ).

% not_gr0
thf(fact_1090_gr0I,axiom,
    ! [N2: nat] :
      ( ( N2 != zero_zero_nat )
     => ( ord_less_nat @ zero_zero_nat @ N2 ) ) ).

% gr0I
thf(fact_1091_bot__nat__0_Oextremum__strict,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ zero_zero_nat ) ).

% bot_nat_0.extremum_strict
thf(fact_1092_le__0__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_eq_nat @ N2 @ zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% le_0_eq
thf(fact_1093_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_1094_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_1095_less__eq__nat_Osimps_I1_J,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ N2 ) ).

% less_eq_nat.simps(1)
thf(fact_1096_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_1097_plus__nat_Oadd__0,axiom,
    ! [N2: nat] :
      ( ( plus_plus_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% plus_nat.add_0
thf(fact_1098_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_1099_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_1100_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_1101_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_1102_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_1103_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_1104_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_1105_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_1106_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_1107_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_1108_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_1109_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_1110_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1111_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_le_numeral
thf(fact_1112_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1113_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_le_numeral
thf(fact_1114_zero__le__numeral,axiom,
    ! [N2: num] : ( ord_less_eq_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_le_numeral
thf(fact_1115_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le2932123472753598470d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_le_zero
thf(fact_1116_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_le_zero
thf(fact_1117_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_le_zero
thf(fact_1118_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_le_zero
thf(fact_1119_not__numeral__le__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_le_zero
thf(fact_1120_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ zero_zero_rat ) ).

% not_numeral_less_zero
thf(fact_1121_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_le72135733267957522d_enat @ ( numera1916890842035813515d_enat @ N2 ) @ zero_z5237406670263579293d_enat ) ).

% not_numeral_less_zero
thf(fact_1122_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ zero_zero_real ) ).

% not_numeral_less_zero
thf(fact_1123_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_nat @ ( numeral_numeral_nat @ N2 ) @ zero_zero_nat ) ).

% not_numeral_less_zero
thf(fact_1124_not__numeral__less__zero,axiom,
    ! [N2: num] :
      ~ ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ zero_zero_int ) ).

% not_numeral_less_zero
thf(fact_1125_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1126_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ ( numera1916890842035813515d_enat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1127_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ N2 ) ) ).

% zero_less_numeral
thf(fact_1128_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% zero_less_numeral
thf(fact_1129_zero__less__numeral,axiom,
    ! [N2: num] : ( ord_less_int @ zero_zero_int @ ( numeral_numeral_int @ N2 ) ) ).

% zero_less_numeral
thf(fact_1130_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_1131_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_1132_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_1133_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_1134_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_1135_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_1136_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_1137_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_1138_not__one__le__zero,axiom,
    ~ ( ord_less_eq_real @ one_one_real @ zero_zero_real ) ).

% not_one_le_zero
thf(fact_1139_not__one__le__zero,axiom,
    ~ ( ord_less_eq_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_le_zero
thf(fact_1140_not__one__le__zero,axiom,
    ~ ( ord_less_eq_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_le_zero
thf(fact_1141_not__one__le__zero,axiom,
    ~ ( ord_less_eq_int @ one_one_int @ zero_zero_int ) ).

% not_one_le_zero
thf(fact_1142_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_1143_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_1144_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_1145_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_1146_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_1147_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_1148_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_1149_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_1150_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_1151_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_1152_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_1153_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_1154_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_1155_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_1156_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_1157_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_1158_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_1159_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_1160_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_1161_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_1162_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_1163_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_1164_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_1165_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_1166_add__nonneg__eq__0__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ X4 @ Y )
            = zero_zero_real )
          = ( ( X4 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1167_add__nonneg__eq__0__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ( plus_plus_rat @ X4 @ Y )
            = zero_zero_rat )
          = ( ( X4 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1168_add__nonneg__eq__0__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ( ( plus_plus_nat @ X4 @ Y )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1169_add__nonneg__eq__0__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ( plus_plus_int @ X4 @ Y )
            = zero_zero_int )
          = ( ( X4 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonneg_eq_0_iff
thf(fact_1170_add__nonpos__eq__0__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ( plus_plus_real @ X4 @ Y )
            = zero_zero_real )
          = ( ( X4 = zero_zero_real )
            & ( Y = zero_zero_real ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1171_add__nonpos__eq__0__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ( ( plus_plus_rat @ X4 @ Y )
            = zero_zero_rat )
          = ( ( X4 = zero_zero_rat )
            & ( Y = zero_zero_rat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1172_add__nonpos__eq__0__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ zero_zero_nat )
     => ( ( ord_less_eq_nat @ Y @ zero_zero_nat )
       => ( ( ( plus_plus_nat @ X4 @ Y )
            = zero_zero_nat )
          = ( ( X4 = zero_zero_nat )
            & ( Y = zero_zero_nat ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1173_add__nonpos__eq__0__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ zero_zero_int )
     => ( ( ord_less_eq_int @ Y @ zero_zero_int )
       => ( ( ( plus_plus_int @ X4 @ Y )
            = zero_zero_int )
          = ( ( X4 = zero_zero_int )
            & ( Y = zero_zero_int ) ) ) ) ) ).

% add_nonpos_eq_0_iff
thf(fact_1174_not__one__less__zero,axiom,
    ~ ( ord_less_real @ one_one_real @ zero_zero_real ) ).

% not_one_less_zero
thf(fact_1175_not__one__less__zero,axiom,
    ~ ( ord_less_rat @ one_one_rat @ zero_zero_rat ) ).

% not_one_less_zero
thf(fact_1176_not__one__less__zero,axiom,
    ~ ( ord_less_nat @ one_one_nat @ zero_zero_nat ) ).

% not_one_less_zero
thf(fact_1177_not__one__less__zero,axiom,
    ~ ( ord_less_int @ one_one_int @ zero_zero_int ) ).

% not_one_less_zero
thf(fact_1178_zero__less__one,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% zero_less_one
thf(fact_1179_zero__less__one,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% zero_less_one
thf(fact_1180_zero__less__one,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% zero_less_one
thf(fact_1181_zero__less__one,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% zero_less_one
thf(fact_1182_less__numeral__extra_I1_J,axiom,
    ord_less_real @ zero_zero_real @ one_one_real ).

% less_numeral_extra(1)
thf(fact_1183_less__numeral__extra_I1_J,axiom,
    ord_less_rat @ zero_zero_rat @ one_one_rat ).

% less_numeral_extra(1)
thf(fact_1184_less__numeral__extra_I1_J,axiom,
    ord_less_nat @ zero_zero_nat @ one_one_nat ).

% less_numeral_extra(1)
thf(fact_1185_less__numeral__extra_I1_J,axiom,
    ord_less_int @ zero_zero_int @ one_one_int ).

% less_numeral_extra(1)
thf(fact_1186_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_1187_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_1188_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_1189_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_1190_canonically__ordered__monoid__add__class_OlessE,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ! [C3: nat] :
            ( ( B
              = ( plus_plus_nat @ A @ C3 ) )
           => ( C3 = zero_zero_nat ) ) ) ).

% canonically_ordered_monoid_add_class.lessE
thf(fact_1191_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_1192_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_1193_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_1194_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_1195_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_1196_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_1197_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_1198_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_1199_add__less__zeroD,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y ) @ zero_zero_real )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
        | ( ord_less_real @ Y @ zero_zero_real ) ) ) ).

% add_less_zeroD
thf(fact_1200_add__less__zeroD,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ ( plus_plus_rat @ X4 @ Y ) @ zero_zero_rat )
     => ( ( ord_less_rat @ X4 @ zero_zero_rat )
        | ( ord_less_rat @ Y @ zero_zero_rat ) ) ) ).

% add_less_zeroD
thf(fact_1201_add__less__zeroD,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ ( plus_plus_int @ X4 @ Y ) @ zero_zero_int )
     => ( ( ord_less_int @ X4 @ zero_zero_int )
        | ( ord_less_int @ Y @ zero_zero_int ) ) ) ).

% add_less_zeroD
thf(fact_1202_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_1203_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_1204_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_1205_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_1206_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_1207_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_1208_divide__nonneg__nonneg,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_1209_divide__nonneg__nonneg,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y ) ) ) ) ).

% divide_nonneg_nonneg
thf(fact_1210_divide__nonneg__nonpos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_nonpos
thf(fact_1211_divide__nonneg__nonpos,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_nonpos
thf(fact_1212_divide__nonpos__nonneg,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_nonneg
thf(fact_1213_divide__nonpos__nonneg,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_nonneg
thf(fact_1214_divide__nonpos__nonpos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_1215_divide__nonpos__nonpos,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_eq_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y ) ) ) ) ).

% divide_nonpos_nonpos
thf(fact_1216_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_1217_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_1218_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_1219_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_1220_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_1221_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_1222_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_1223_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_1224_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_1225_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_1226_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_1227_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_1228_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_1229_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_1230_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_1231_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_1232_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_1233_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_1234_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_1235_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_1236_divide__pos__pos,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_1237_divide__pos__pos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y ) ) ) ) ).

% divide_pos_pos
thf(fact_1238_divide__pos__neg,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_pos_neg
thf(fact_1239_divide__pos__neg,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ ( divide_divide_real @ X4 @ Y ) @ zero_zero_real ) ) ) ).

% divide_pos_neg
thf(fact_1240_divide__neg__pos,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_neg_pos
thf(fact_1241_divide__neg__pos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ ( divide_divide_real @ X4 @ Y ) @ zero_zero_real ) ) ) ).

% divide_neg_pos
thf(fact_1242_divide__neg__neg,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_1243_divide__neg__neg,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y ) ) ) ) ).

% divide_neg_neg
thf(fact_1244_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_1245_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_1246_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_1247_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_1248_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_1249_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_1250_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_1251_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_1252_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_1253_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_1254_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_1255_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_1256_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_1257_power__0,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% power_0
thf(fact_1258_power__0,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% power_0
thf(fact_1259_power__0,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% power_0
thf(fact_1260_power__0,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% power_0
thf(fact_1261_power__0,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% power_0
thf(fact_1262_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_1263_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_1264_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_1265_Ex__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
            & ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        | ? [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
            & ( P @ ( suc @ I3 ) ) ) ) ) ).

% Ex_less_Suc2
thf(fact_1266_gr0__conv__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
      = ( ? [M6: nat] :
            ( N2
            = ( suc @ M6 ) ) ) ) ).

% gr0_conv_Suc
thf(fact_1267_All__less__Suc2,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ ( suc @ N2 ) )
           => ( P @ I3 ) ) )
      = ( ( P @ zero_zero_nat )
        & ! [I3: nat] :
            ( ( ord_less_nat @ I3 @ N2 )
           => ( P @ ( suc @ I3 ) ) ) ) ) ).

% All_less_Suc2
thf(fact_1268_gr0__implies__Suc,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ? [M5: nat] :
          ( N2
          = ( suc @ M5 ) ) ) ).

% gr0_implies_Suc
thf(fact_1269_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_1270_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_1271_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_1272_ex__least__nat__le,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_eq_nat @ K2 @ N2 )
            & ! [I: nat] :
                ( ( ord_less_nat @ I @ K2 )
               => ~ ( P @ I ) )
            & ( P @ K2 ) ) ) ) ).

% ex_least_nat_le
thf(fact_1273_less__imp__add__positive,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ? [K2: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ K2 )
          & ( ( plus_plus_nat @ I2 @ K2 )
            = J ) ) ) ).

% less_imp_add_positive
thf(fact_1274_div__less__mono,axiom,
    ! [A2: nat,B3: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( modulo_modulo_nat @ A2 @ N2 )
            = zero_zero_nat )
         => ( ( ( modulo_modulo_nat @ B3 @ N2 )
              = zero_zero_nat )
           => ( ord_less_nat @ ( divide_divide_nat @ A2 @ N2 ) @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ) ) ).

% div_less_mono
thf(fact_1275_One__nat__def,axiom,
    ( one_one_nat
    = ( suc @ zero_zero_nat ) ) ).

% One_nat_def
thf(fact_1276_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_1277_nat__power__less__imp__less,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ I2 )
     => ( ( ord_less_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N2 ) )
       => ( ord_less_nat @ M @ N2 ) ) ) ).

% nat_power_less_imp_less
thf(fact_1278_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_1279_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_1280_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_1281_order__antisym__conv,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X4 )
     => ( ( ord_less_eq_set_int @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1282_order__antisym__conv,axiom,
    ! [Y: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1283_order__antisym__conv,axiom,
    ! [Y: num,X4: num] :
      ( ( ord_less_eq_num @ Y @ X4 )
     => ( ( ord_less_eq_num @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1284_order__antisym__conv,axiom,
    ! [Y: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y @ X4 )
     => ( ( ord_less_eq_nat @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1285_order__antisym__conv,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ Y @ X4 )
     => ( ( ord_less_eq_int @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% order_antisym_conv
thf(fact_1286_linorder__le__cases,axiom,
    ! [X4: rat,Y: rat] :
      ( ~ ( ord_less_eq_rat @ X4 @ Y )
     => ( ord_less_eq_rat @ Y @ X4 ) ) ).

% linorder_le_cases
thf(fact_1287_linorder__le__cases,axiom,
    ! [X4: num,Y: num] :
      ( ~ ( ord_less_eq_num @ X4 @ Y )
     => ( ord_less_eq_num @ Y @ X4 ) ) ).

% linorder_le_cases
thf(fact_1288_linorder__le__cases,axiom,
    ! [X4: nat,Y: nat] :
      ( ~ ( ord_less_eq_nat @ X4 @ Y )
     => ( ord_less_eq_nat @ Y @ X4 ) ) ).

% linorder_le_cases
thf(fact_1289_linorder__le__cases,axiom,
    ! [X4: int,Y: int] :
      ( ~ ( ord_less_eq_int @ X4 @ Y )
     => ( ord_less_eq_int @ Y @ X4 ) ) ).

% linorder_le_cases
thf(fact_1290_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1291_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1292_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1293_ord__le__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1294_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > rat,C: rat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1295_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > num,C: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1296_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > nat,C: nat] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1297_ord__le__eq__subst,axiom,
    ! [A: num,B: num,F: num > int,C: int] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1298_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > rat,C: rat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1299_ord__le__eq__subst,axiom,
    ! [A: nat,B: nat,F: nat > num,C: num] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_le_eq_subst
thf(fact_1300_ord__eq__le__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1301_ord__eq__le__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1302_ord__eq__le__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1303_ord__eq__le__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1304_ord__eq__le__subst,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1305_ord__eq__le__subst,axiom,
    ! [A: num,F: num > num,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1306_ord__eq__le__subst,axiom,
    ! [A: nat,F: num > nat,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1307_ord__eq__le__subst,axiom,
    ! [A: int,F: num > int,B: num,C: num] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_num @ B @ C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1308_ord__eq__le__subst,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1309_ord__eq__le__subst,axiom,
    ! [A: num,F: nat > num,B: nat,C: nat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_eq_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_le_subst
thf(fact_1310_linorder__linear,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
      | ( ord_less_eq_rat @ Y @ X4 ) ) ).

% linorder_linear
thf(fact_1311_linorder__linear,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
      | ( ord_less_eq_num @ Y @ X4 ) ) ).

% linorder_linear
thf(fact_1312_linorder__linear,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
      | ( ord_less_eq_nat @ Y @ X4 ) ) ).

% linorder_linear
thf(fact_1313_linorder__linear,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
      | ( ord_less_eq_int @ Y @ X4 ) ) ).

% linorder_linear
thf(fact_1314_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_1315_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_1316_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_1317_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_1318_order__eq__refl,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( X4 = Y )
     => ( ord_less_eq_set_int @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1319_order__eq__refl,axiom,
    ! [X4: rat,Y: rat] :
      ( ( X4 = Y )
     => ( ord_less_eq_rat @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1320_order__eq__refl,axiom,
    ! [X4: num,Y: num] :
      ( ( X4 = Y )
     => ( ord_less_eq_num @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1321_order__eq__refl,axiom,
    ! [X4: nat,Y: nat] :
      ( ( X4 = Y )
     => ( ord_less_eq_nat @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1322_order__eq__refl,axiom,
    ! [X4: int,Y: int] :
      ( ( X4 = Y )
     => ( ord_less_eq_int @ X4 @ Y ) ) ).

% order_eq_refl
thf(fact_1323_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1324_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1325_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1326_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1327_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1328_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1329_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1330_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_int @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1331_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 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1332_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 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ ( F @ A ) @ C ) ) ) ) ).

% order_subst2
thf(fact_1333_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1334_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1335_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 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1336_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 )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1337_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1338_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1339_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 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_eq_nat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1340_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 )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_eq_int @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_num @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1341_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1342_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_eq_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_subst1
thf(fact_1343_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ A3 @ B2 )
          & ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1344_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A3 @ B2 )
          & ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1345_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
          & ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1346_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
          & ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1347_Orderings_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
          & ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).

% Orderings.order_eq_iff
thf(fact_1348_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_1349_antisym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_eq_rat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1350_antisym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_num @ A @ B )
     => ( ( ord_less_eq_num @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1351_antisym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_nat @ A @ B )
     => ( ( ord_less_eq_nat @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1352_antisym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ B )
     => ( ( ord_less_eq_int @ B @ A )
       => ( A = B ) ) ) ).

% antisym
thf(fact_1353_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_1354_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_1355_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_1356_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_1357_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_1358_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_1359_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_1360_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_1361_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_1362_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_1363_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ B2 @ A3 )
          & ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1364_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_eq_rat @ B2 @ A3 )
          & ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1365_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ B2 @ A3 )
          & ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1366_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ B2 @ A3 )
          & ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1367_dual__order_Oeq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ B2 @ A3 )
          & ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).

% dual_order.eq_iff
thf(fact_1368_linorder__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A5: rat,B5: rat] :
          ( ( ord_less_eq_rat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: rat,B5: rat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1369_linorder__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A5: num,B5: num] :
          ( ( ord_less_eq_num @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: num,B5: num] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1370_linorder__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_eq_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat,B5: nat] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1371_linorder__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_eq_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int,B5: int] :
            ( ( P @ B5 @ A5 )
           => ( P @ A5 @ B5 ) )
       => ( P @ A @ B ) ) ) ).

% linorder_wlog
thf(fact_1372_order__trans,axiom,
    ! [X4: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_eq_set_int @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1373_order__trans,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_eq_rat @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1374_order__trans,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_eq_num @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1375_order__trans,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_eq_nat @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1376_order__trans,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ X4 @ Z ) ) ) ).

% order_trans
thf(fact_1377_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_1378_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_1379_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_1380_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_1381_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_1382_order__antisym,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ord_less_eq_set_int @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1383_order__antisym,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ord_less_eq_rat @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1384_order__antisym,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ord_less_eq_num @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1385_order__antisym,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ord_less_eq_nat @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1386_order__antisym,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ord_less_eq_int @ Y @ X4 )
       => ( X4 = Y ) ) ) ).

% order_antisym
thf(fact_1387_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_1388_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_1389_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_1390_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_1391_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_1392_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_1393_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_1394_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_1395_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_1396_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_1397_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
    = ( ^ [X: set_int,Y5: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y5 )
          & ( ord_less_eq_set_int @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1398_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [X: rat,Y5: rat] :
          ( ( ord_less_eq_rat @ X @ Y5 )
          & ( ord_less_eq_rat @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1399_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: num,Z4: num] : Y6 = Z4 )
    = ( ^ [X: num,Y5: num] :
          ( ( ord_less_eq_num @ X @ Y5 )
          & ( ord_less_eq_num @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1400_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [X: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X @ Y5 )
          & ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1401_order__class_Oorder__eq__iff,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [X: int,Y5: int] :
          ( ( ord_less_eq_int @ X @ Y5 )
          & ( ord_less_eq_int @ Y5 @ X ) ) ) ) ).

% order_class.order_eq_iff
thf(fact_1402_le__cases3,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ( ord_less_eq_rat @ X4 @ Y )
       => ~ ( ord_less_eq_rat @ Y @ Z ) )
     => ( ( ( ord_less_eq_rat @ Y @ X4 )
         => ~ ( ord_less_eq_rat @ X4 @ Z ) )
       => ( ( ( ord_less_eq_rat @ X4 @ Z )
           => ~ ( ord_less_eq_rat @ Z @ Y ) )
         => ( ( ( ord_less_eq_rat @ Z @ Y )
             => ~ ( ord_less_eq_rat @ Y @ X4 ) )
           => ( ( ( ord_less_eq_rat @ Y @ Z )
               => ~ ( ord_less_eq_rat @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_rat @ Z @ X4 )
                 => ~ ( ord_less_eq_rat @ X4 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1403_le__cases3,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ( ord_less_eq_num @ X4 @ Y )
       => ~ ( ord_less_eq_num @ Y @ Z ) )
     => ( ( ( ord_less_eq_num @ Y @ X4 )
         => ~ ( ord_less_eq_num @ X4 @ Z ) )
       => ( ( ( ord_less_eq_num @ X4 @ Z )
           => ~ ( ord_less_eq_num @ Z @ Y ) )
         => ( ( ( ord_less_eq_num @ Z @ Y )
             => ~ ( ord_less_eq_num @ Y @ X4 ) )
           => ( ( ( ord_less_eq_num @ Y @ Z )
               => ~ ( ord_less_eq_num @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_num @ Z @ X4 )
                 => ~ ( ord_less_eq_num @ X4 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1404_le__cases3,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ( ord_less_eq_nat @ X4 @ Y )
       => ~ ( ord_less_eq_nat @ Y @ Z ) )
     => ( ( ( ord_less_eq_nat @ Y @ X4 )
         => ~ ( ord_less_eq_nat @ X4 @ Z ) )
       => ( ( ( ord_less_eq_nat @ X4 @ Z )
           => ~ ( ord_less_eq_nat @ Z @ Y ) )
         => ( ( ( ord_less_eq_nat @ Z @ Y )
             => ~ ( ord_less_eq_nat @ Y @ X4 ) )
           => ( ( ( ord_less_eq_nat @ Y @ Z )
               => ~ ( ord_less_eq_nat @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_nat @ Z @ X4 )
                 => ~ ( ord_less_eq_nat @ X4 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1405_le__cases3,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ( ord_less_eq_int @ X4 @ Y )
       => ~ ( ord_less_eq_int @ Y @ Z ) )
     => ( ( ( ord_less_eq_int @ Y @ X4 )
         => ~ ( ord_less_eq_int @ X4 @ Z ) )
       => ( ( ( ord_less_eq_int @ X4 @ Z )
           => ~ ( ord_less_eq_int @ Z @ Y ) )
         => ( ( ( ord_less_eq_int @ Z @ Y )
             => ~ ( ord_less_eq_int @ Y @ X4 ) )
           => ( ( ( ord_less_eq_int @ Y @ Z )
               => ~ ( ord_less_eq_int @ Z @ X4 ) )
             => ~ ( ( ord_less_eq_int @ Z @ X4 )
                 => ~ ( ord_less_eq_int @ X4 @ Y ) ) ) ) ) ) ) ).

% le_cases3
thf(fact_1406_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_1407_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_1408_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_1409_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_1410_verit__comp__simplify1_I2_J,axiom,
    ! [A: set_int] : ( ord_less_eq_set_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1411_verit__comp__simplify1_I2_J,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1412_verit__comp__simplify1_I2_J,axiom,
    ! [A: num] : ( ord_less_eq_num @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1413_verit__comp__simplify1_I2_J,axiom,
    ! [A: nat] : ( ord_less_eq_nat @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1414_verit__comp__simplify1_I2_J,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ A ) ).

% verit_comp_simplify1(2)
thf(fact_1415_order__less__imp__not__less,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ~ ( ord_less_real @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1416_order__less__imp__not__less,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ~ ( ord_less_rat @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1417_order__less__imp__not__less,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ~ ( ord_less_num @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1418_order__less__imp__not__less,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ~ ( ord_less_nat @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1419_order__less__imp__not__less,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ~ ( ord_less_int @ Y @ X4 ) ) ).

% order_less_imp_not_less
thf(fact_1420_order__less__imp__not__eq2,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1421_order__less__imp__not__eq2,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1422_order__less__imp__not__eq2,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1423_order__less__imp__not__eq2,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1424_order__less__imp__not__eq2,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( Y != X4 ) ) ).

% order_less_imp_not_eq2
thf(fact_1425_order__less__imp__not__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1426_order__less__imp__not__eq,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1427_order__less__imp__not__eq,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1428_order__less__imp__not__eq,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1429_order__less__imp__not__eq,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( X4 != Y ) ) ).

% order_less_imp_not_eq
thf(fact_1430_linorder__less__linear,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_real @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1431_linorder__less__linear,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_rat @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1432_linorder__less__linear,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_num @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1433_linorder__less__linear,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_nat @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1434_linorder__less__linear,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
      | ( X4 = Y )
      | ( ord_less_int @ Y @ X4 ) ) ).

% linorder_less_linear
thf(fact_1435_order__less__imp__triv,axiom,
    ! [X4: real,Y: real,P: $o] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ( ord_less_real @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1436_order__less__imp__triv,axiom,
    ! [X4: rat,Y: rat,P: $o] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ( ord_less_rat @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1437_order__less__imp__triv,axiom,
    ! [X4: num,Y: num,P: $o] :
      ( ( ord_less_num @ X4 @ Y )
     => ( ( ord_less_num @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1438_order__less__imp__triv,axiom,
    ! [X4: nat,Y: nat,P: $o] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( ( ord_less_nat @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1439_order__less__imp__triv,axiom,
    ! [X4: int,Y: int,P: $o] :
      ( ( ord_less_int @ X4 @ Y )
     => ( ( ord_less_int @ Y @ X4 )
       => P ) ) ).

% order_less_imp_triv
thf(fact_1440_order__less__not__sym,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ~ ( ord_less_real @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1441_order__less__not__sym,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ~ ( ord_less_rat @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1442_order__less__not__sym,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ~ ( ord_less_num @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1443_order__less__not__sym,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ~ ( ord_less_nat @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1444_order__less__not__sym,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ~ ( ord_less_int @ Y @ X4 ) ) ).

% order_less_not_sym
thf(fact_1445_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1446_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1447_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1448_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1449_order__less__subst2,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1450_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_real @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1451_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1452_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_num @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1453_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_nat @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1454_order__less__subst2,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_int @ ( F @ B ) @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_less_subst2
thf(fact_1455_order__less__subst1,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1456_order__less__subst1,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1457_order__less__subst1,axiom,
    ! [A: real,F: num > real,B: num,C: num] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_num @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1458_order__less__subst1,axiom,
    ! [A: real,F: nat > real,B: nat,C: nat] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1459_order__less__subst1,axiom,
    ! [A: real,F: int > real,B: int,C: int] :
      ( ( ord_less_real @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1460_order__less__subst1,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1461_order__less__subst1,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1462_order__less__subst1,axiom,
    ! [A: rat,F: num > rat,B: num,C: num] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_num @ B @ C )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_num @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1463_order__less__subst1,axiom,
    ! [A: rat,F: nat > rat,B: nat,C: nat] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_nat @ B @ C )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1464_order__less__subst1,axiom,
    ! [A: rat,F: int > rat,B: int,C: int] :
      ( ( ord_less_rat @ A @ ( F @ B ) )
     => ( ( ord_less_int @ B @ C )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_subst1
thf(fact_1465_order__less__irrefl,axiom,
    ! [X4: real] :
      ~ ( ord_less_real @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1466_order__less__irrefl,axiom,
    ! [X4: rat] :
      ~ ( ord_less_rat @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1467_order__less__irrefl,axiom,
    ! [X4: num] :
      ~ ( ord_less_num @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1468_order__less__irrefl,axiom,
    ! [X4: nat] :
      ~ ( ord_less_nat @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1469_order__less__irrefl,axiom,
    ! [X4: int] :
      ~ ( ord_less_int @ X4 @ X4 ) ).

% order_less_irrefl
thf(fact_1470_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > real,C: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1471_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > rat,C: rat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1472_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > num,C: num] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1473_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > nat,C: nat] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1474_ord__less__eq__subst,axiom,
    ! [A: real,B: real,F: real > int,C: int] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1475_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > real,C: real] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1476_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > rat,C: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1477_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > num,C: num] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1478_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > nat,C: nat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1479_ord__less__eq__subst,axiom,
    ! [A: rat,B: rat,F: rat > int,C: int] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ( F @ B )
          = C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% ord_less_eq_subst
thf(fact_1480_ord__eq__less__subst,axiom,
    ! [A: real,F: real > real,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1481_ord__eq__less__subst,axiom,
    ! [A: rat,F: real > rat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1482_ord__eq__less__subst,axiom,
    ! [A: num,F: real > num,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1483_ord__eq__less__subst,axiom,
    ! [A: nat,F: real > nat,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1484_ord__eq__less__subst,axiom,
    ! [A: int,F: real > int,B: real,C: real] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_real @ B @ C )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1485_ord__eq__less__subst,axiom,
    ! [A: real,F: rat > real,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1486_ord__eq__less__subst,axiom,
    ! [A: rat,F: rat > rat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1487_ord__eq__less__subst,axiom,
    ! [A: num,F: rat > num,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1488_ord__eq__less__subst,axiom,
    ! [A: nat,F: rat > nat,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1489_ord__eq__less__subst,axiom,
    ! [A: int,F: rat > int,B: rat,C: rat] :
      ( ( A
        = ( F @ B ) )
     => ( ( ord_less_rat @ B @ C )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% ord_eq_less_subst
thf(fact_1490_order__less__trans,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1491_order__less__trans,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1492_order__less__trans,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1493_order__less__trans,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1494_order__less__trans,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X4 @ Z ) ) ) ).

% order_less_trans
thf(fact_1495_order__less__asym_H,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order_less_asym'
thf(fact_1496_order__less__asym_H,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1497_order__less__asym_H,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order_less_asym'
thf(fact_1498_order__less__asym_H,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order_less_asym'
thf(fact_1499_order__less__asym_H,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order_less_asym'
thf(fact_1500_linorder__neq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( X4 != Y )
      = ( ( ord_less_real @ X4 @ Y )
        | ( ord_less_real @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1501_linorder__neq__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( X4 != Y )
      = ( ( ord_less_rat @ X4 @ Y )
        | ( ord_less_rat @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1502_linorder__neq__iff,axiom,
    ! [X4: num,Y: num] :
      ( ( X4 != Y )
      = ( ( ord_less_num @ X4 @ Y )
        | ( ord_less_num @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1503_linorder__neq__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( X4 != Y )
      = ( ( ord_less_nat @ X4 @ Y )
        | ( ord_less_nat @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1504_linorder__neq__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( X4 != Y )
      = ( ( ord_less_int @ X4 @ Y )
        | ( ord_less_int @ Y @ X4 ) ) ) ).

% linorder_neq_iff
thf(fact_1505_order__less__asym,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ~ ( ord_less_real @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1506_order__less__asym,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ~ ( ord_less_rat @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1507_order__less__asym,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ~ ( ord_less_num @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1508_order__less__asym,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ~ ( ord_less_nat @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1509_order__less__asym,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ~ ( ord_less_int @ Y @ X4 ) ) ).

% order_less_asym
thf(fact_1510_linorder__neqE,axiom,
    ! [X4: real,Y: real] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_real @ X4 @ Y )
       => ( ord_less_real @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1511_linorder__neqE,axiom,
    ! [X4: rat,Y: rat] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_rat @ X4 @ Y )
       => ( ord_less_rat @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1512_linorder__neqE,axiom,
    ! [X4: num,Y: num] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_num @ X4 @ Y )
       => ( ord_less_num @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1513_linorder__neqE,axiom,
    ! [X4: nat,Y: nat] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_nat @ X4 @ Y )
       => ( ord_less_nat @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1514_linorder__neqE,axiom,
    ! [X4: int,Y: int] :
      ( ( X4 != Y )
     => ( ~ ( ord_less_int @ X4 @ Y )
       => ( ord_less_int @ Y @ X4 ) ) ) ).

% linorder_neqE
thf(fact_1515_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_1516_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_1517_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_1518_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_1519_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_1520_order_Ostrict__implies__not__eq,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1521_order_Ostrict__implies__not__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1522_order_Ostrict__implies__not__eq,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1523_order_Ostrict__implies__not__eq,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1524_order_Ostrict__implies__not__eq,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ( A != B ) ) ).

% order.strict_implies_not_eq
thf(fact_1525_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_1526_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_1527_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_1528_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_1529_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_1530_not__less__iff__gr__or__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ~ ( ord_less_real @ X4 @ Y ) )
      = ( ( ord_less_real @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1531_not__less__iff__gr__or__eq,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X4 @ Y ) )
      = ( ( ord_less_rat @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1532_not__less__iff__gr__or__eq,axiom,
    ! [X4: num,Y: num] :
      ( ( ~ ( ord_less_num @ X4 @ Y ) )
      = ( ( ord_less_num @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1533_not__less__iff__gr__or__eq,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X4 @ Y ) )
      = ( ( ord_less_nat @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1534_not__less__iff__gr__or__eq,axiom,
    ! [X4: int,Y: int] :
      ( ( ~ ( ord_less_int @ X4 @ Y ) )
      = ( ( ord_less_int @ Y @ X4 )
        | ( X4 = Y ) ) ) ).

% not_less_iff_gr_or_eq
thf(fact_1535_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_1536_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_1537_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_1538_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_1539_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_1540_linorder__less__wlog,axiom,
    ! [P: real > real > $o,A: real,B: real] :
      ( ! [A5: real,B5: real] :
          ( ( ord_less_real @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: real] : ( P @ A5 @ A5 )
       => ( ! [A5: real,B5: real] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1541_linorder__less__wlog,axiom,
    ! [P: rat > rat > $o,A: rat,B: rat] :
      ( ! [A5: rat,B5: rat] :
          ( ( ord_less_rat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: rat] : ( P @ A5 @ A5 )
       => ( ! [A5: rat,B5: rat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1542_linorder__less__wlog,axiom,
    ! [P: num > num > $o,A: num,B: num] :
      ( ! [A5: num,B5: num] :
          ( ( ord_less_num @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: num] : ( P @ A5 @ A5 )
       => ( ! [A5: num,B5: num] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1543_linorder__less__wlog,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( ord_less_nat @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ A5 )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1544_linorder__less__wlog,axiom,
    ! [P: int > int > $o,A: int,B: int] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_int @ A5 @ B5 )
         => ( P @ A5 @ B5 ) )
     => ( ! [A5: int] : ( P @ A5 @ A5 )
       => ( ! [A5: int,B5: int] :
              ( ( P @ B5 @ A5 )
             => ( P @ A5 @ B5 ) )
         => ( P @ A @ B ) ) ) ) ).

% linorder_less_wlog
thf(fact_1545_exists__least__iff,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ? [N: nat] :
          ( ( P4 @ N )
          & ! [M6: nat] :
              ( ( ord_less_nat @ M6 @ N )
             => ~ ( P4 @ M6 ) ) ) ) ) ).

% exists_least_iff
thf(fact_1546_dual__order_Oirrefl,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% dual_order.irrefl
thf(fact_1547_dual__order_Oirrefl,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1548_dual__order_Oirrefl,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% dual_order.irrefl
thf(fact_1549_dual__order_Oirrefl,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% dual_order.irrefl
thf(fact_1550_dual__order_Oirrefl,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% dual_order.irrefl
thf(fact_1551_dual__order_Oasym,axiom,
    ! [B: real,A: real] :
      ( ( ord_less_real @ B @ A )
     => ~ ( ord_less_real @ A @ B ) ) ).

% dual_order.asym
thf(fact_1552_dual__order_Oasym,axiom,
    ! [B: rat,A: rat] :
      ( ( ord_less_rat @ B @ A )
     => ~ ( ord_less_rat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1553_dual__order_Oasym,axiom,
    ! [B: num,A: num] :
      ( ( ord_less_num @ B @ A )
     => ~ ( ord_less_num @ A @ B ) ) ).

% dual_order.asym
thf(fact_1554_dual__order_Oasym,axiom,
    ! [B: nat,A: nat] :
      ( ( ord_less_nat @ B @ A )
     => ~ ( ord_less_nat @ A @ B ) ) ).

% dual_order.asym
thf(fact_1555_dual__order_Oasym,axiom,
    ! [B: int,A: int] :
      ( ( ord_less_int @ B @ A )
     => ~ ( ord_less_int @ A @ B ) ) ).

% dual_order.asym
thf(fact_1556_linorder__cases,axiom,
    ! [X4: real,Y: real] :
      ( ~ ( ord_less_real @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_real @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1557_linorder__cases,axiom,
    ! [X4: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_rat @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1558_linorder__cases,axiom,
    ! [X4: num,Y: num] :
      ( ~ ( ord_less_num @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_num @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1559_linorder__cases,axiom,
    ! [X4: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_nat @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1560_linorder__cases,axiom,
    ! [X4: int,Y: int] :
      ( ~ ( ord_less_int @ X4 @ Y )
     => ( ( X4 != Y )
       => ( ord_less_int @ Y @ X4 ) ) ) ).

% linorder_cases
thf(fact_1561_antisym__conv3,axiom,
    ! [Y: real,X4: real] :
      ( ~ ( ord_less_real @ Y @ X4 )
     => ( ( ~ ( ord_less_real @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1562_antisym__conv3,axiom,
    ! [Y: rat,X4: rat] :
      ( ~ ( ord_less_rat @ Y @ X4 )
     => ( ( ~ ( ord_less_rat @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1563_antisym__conv3,axiom,
    ! [Y: num,X4: num] :
      ( ~ ( ord_less_num @ Y @ X4 )
     => ( ( ~ ( ord_less_num @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1564_antisym__conv3,axiom,
    ! [Y: nat,X4: nat] :
      ( ~ ( ord_less_nat @ Y @ X4 )
     => ( ( ~ ( ord_less_nat @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1565_antisym__conv3,axiom,
    ! [Y: int,X4: int] :
      ( ~ ( ord_less_int @ Y @ X4 )
     => ( ( ~ ( ord_less_int @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv3
thf(fact_1566_less__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [X5: nat] :
          ( ! [Y4: nat] :
              ( ( ord_less_nat @ Y4 @ X5 )
             => ( P @ Y4 ) )
         => ( P @ X5 ) )
     => ( P @ A ) ) ).

% less_induct
thf(fact_1567_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_1568_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_1569_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_1570_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_1571_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_1572_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_1573_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_1574_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_1575_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_1576_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_1577_order_Oasym,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ~ ( ord_less_real @ B @ A ) ) ).

% order.asym
thf(fact_1578_order_Oasym,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ~ ( ord_less_rat @ B @ A ) ) ).

% order.asym
thf(fact_1579_order_Oasym,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_num @ A @ B )
     => ~ ( ord_less_num @ B @ A ) ) ).

% order.asym
thf(fact_1580_order_Oasym,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_nat @ A @ B )
     => ~ ( ord_less_nat @ B @ A ) ) ).

% order.asym
thf(fact_1581_order_Oasym,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_int @ A @ B )
     => ~ ( ord_less_int @ B @ A ) ) ).

% order.asym
thf(fact_1582_less__imp__neq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1583_less__imp__neq,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1584_less__imp__neq,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1585_less__imp__neq,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1586_less__imp__neq,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( X4 != Y ) ) ).

% less_imp_neq
thf(fact_1587_dense,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ? [Z2: real] :
          ( ( ord_less_real @ X4 @ Z2 )
          & ( ord_less_real @ Z2 @ Y ) ) ) ).

% dense
thf(fact_1588_dense,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ? [Z2: rat] :
          ( ( ord_less_rat @ X4 @ Z2 )
          & ( ord_less_rat @ Z2 @ Y ) ) ) ).

% dense
thf(fact_1589_gt__ex,axiom,
    ! [X4: real] :
    ? [X_12: real] : ( ord_less_real @ X4 @ X_12 ) ).

% gt_ex
thf(fact_1590_gt__ex,axiom,
    ! [X4: rat] :
    ? [X_12: rat] : ( ord_less_rat @ X4 @ X_12 ) ).

% gt_ex
thf(fact_1591_gt__ex,axiom,
    ! [X4: nat] :
    ? [X_12: nat] : ( ord_less_nat @ X4 @ X_12 ) ).

% gt_ex
thf(fact_1592_gt__ex,axiom,
    ! [X4: int] :
    ? [X_12: int] : ( ord_less_int @ X4 @ X_12 ) ).

% gt_ex
thf(fact_1593_lt__ex,axiom,
    ! [X4: real] :
    ? [Y3: real] : ( ord_less_real @ Y3 @ X4 ) ).

% lt_ex
thf(fact_1594_lt__ex,axiom,
    ! [X4: rat] :
    ? [Y3: rat] : ( ord_less_rat @ Y3 @ X4 ) ).

% lt_ex
thf(fact_1595_lt__ex,axiom,
    ! [X4: int] :
    ? [Y3: int] : ( ord_less_int @ Y3 @ X4 ) ).

% lt_ex
thf(fact_1596_verit__comp__simplify1_I1_J,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1597_verit__comp__simplify1_I1_J,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1598_verit__comp__simplify1_I1_J,axiom,
    ! [A: num] :
      ~ ( ord_less_num @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1599_verit__comp__simplify1_I1_J,axiom,
    ! [A: nat] :
      ~ ( ord_less_nat @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1600_verit__comp__simplify1_I1_J,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ A @ A ) ).

% verit_comp_simplify1(1)
thf(fact_1601_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_1602_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_1603_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_1604_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_1605_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_1606_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_1607_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_1608_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_1609_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_1610_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_1611_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_1612_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_1613_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_1614_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_1615_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_1616_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_1617_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_1618_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_1619_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_1620_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_1621_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_1622_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_1623_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_1624_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_1625_field__le__epsilon,axiom,
    ! [X4: real,Y: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ X4 @ ( plus_plus_real @ Y @ E ) ) )
     => ( ord_less_eq_real @ X4 @ Y ) ) ).

% field_le_epsilon
thf(fact_1626_field__le__epsilon,axiom,
    ! [X4: rat,Y: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ Y @ E ) ) )
     => ( ord_less_eq_rat @ X4 @ Y ) ) ).

% field_le_epsilon
thf(fact_1627_divide__nonpos__pos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonpos_pos
thf(fact_1628_divide__nonpos__pos,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonpos_pos
thf(fact_1629_divide__nonpos__neg,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_1630_divide__nonpos__neg,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y ) ) ) ) ).

% divide_nonpos_neg
thf(fact_1631_divide__nonneg__pos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ X4 @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_1632_divide__nonneg__pos,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ zero_zero_rat @ ( divide_divide_rat @ X4 @ Y ) ) ) ) ).

% divide_nonneg_pos
thf(fact_1633_divide__nonneg__neg,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ Y @ zero_zero_real )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y ) @ zero_zero_real ) ) ) ).

% divide_nonneg_neg
thf(fact_1634_divide__nonneg__neg,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ Y @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y ) @ zero_zero_rat ) ) ) ).

% divide_nonneg_neg
thf(fact_1635_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_1636_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_1637_frac__less2,axiom,
    ! [X4: real,Y: real,W: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_1638_frac__less2,axiom,
    ! [X4: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less2
thf(fact_1639_frac__less,axiom,
    ! [X4: real,Y: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_1640_frac__less,axiom,
    ! [X4: rat,Y: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_rat @ X4 @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_less
thf(fact_1641_frac__le,axiom,
    ! [Y: real,X4: real,W: real,Z: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ( ord_less_real @ zero_zero_real @ W )
         => ( ( ord_less_eq_real @ W @ Z )
           => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Z ) @ ( divide_divide_real @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_1642_frac__le,axiom,
    ! [Y: rat,X4: rat,W: rat,Z: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X4 @ Y )
       => ( ( ord_less_rat @ zero_zero_rat @ W )
         => ( ( ord_less_eq_rat @ W @ Z )
           => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Z ) @ ( divide_divide_rat @ Y @ W ) ) ) ) ) ) ).

% frac_le
thf(fact_1643_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_1644_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_1645_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_1646_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_1647_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_1648_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_1649_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_1650_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_1651_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_1652_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_1653_zero__less__two,axiom,
    ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ one_one_real ) ).

% zero_less_two
thf(fact_1654_zero__less__two,axiom,
    ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ one_one_rat ) ).

% zero_less_two
thf(fact_1655_zero__less__two,axiom,
    ord_less_nat @ zero_zero_nat @ ( plus_plus_nat @ one_one_nat @ one_one_nat ) ).

% zero_less_two
thf(fact_1656_zero__less__two,axiom,
    ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ one_one_int ) ).

% zero_less_two
thf(fact_1657_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_1658_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_1659_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_1660_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_1661_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_1662_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_1663_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_1664_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_1665_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_1666_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_1667_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_1668_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_1669_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_1670_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_1671_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_1672_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_1673_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_1674_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_1675_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_1676_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_1677_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_1678_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_1679_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_1680_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_1681_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_1682_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_1683_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_1684_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_1685_in__mono,axiom,
    ! [A2: set_real,B3: set_real,X4: real] :
      ( ( ord_less_eq_set_real @ A2 @ B3 )
     => ( ( member_real @ X4 @ A2 )
       => ( member_real @ X4 @ B3 ) ) ) ).

% in_mono
thf(fact_1686_in__mono,axiom,
    ! [A2: set_nat,B3: set_nat,X4: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( member_nat @ X4 @ A2 )
       => ( member_nat @ X4 @ B3 ) ) ) ).

% in_mono
thf(fact_1687_in__mono,axiom,
    ! [A2: set_complex,B3: set_complex,X4: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B3 )
     => ( ( member_complex @ X4 @ A2 )
       => ( member_complex @ X4 @ B3 ) ) ) ).

% in_mono
thf(fact_1688_in__mono,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,X4: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
     => ( ( member8440522571783428010at_nat @ X4 @ A2 )
       => ( member8440522571783428010at_nat @ X4 @ B3 ) ) ) ).

% in_mono
thf(fact_1689_in__mono,axiom,
    ! [A2: set_int,B3: set_int,X4: int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( member_int @ X4 @ A2 )
       => ( member_int @ X4 @ B3 ) ) ) ).

% in_mono
thf(fact_1690_subsetD,axiom,
    ! [A2: set_real,B3: set_real,C: real] :
      ( ( ord_less_eq_set_real @ A2 @ B3 )
     => ( ( member_real @ C @ A2 )
       => ( member_real @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1691_subsetD,axiom,
    ! [A2: set_nat,B3: set_nat,C: nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( member_nat @ C @ A2 )
       => ( member_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1692_subsetD,axiom,
    ! [A2: set_complex,B3: set_complex,C: complex] :
      ( ( ord_le211207098394363844omplex @ A2 @ B3 )
     => ( ( member_complex @ C @ A2 )
       => ( member_complex @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1693_subsetD,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat,C: product_prod_nat_nat] :
      ( ( ord_le3146513528884898305at_nat @ A2 @ B3 )
     => ( ( member8440522571783428010at_nat @ C @ A2 )
       => ( member8440522571783428010at_nat @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1694_subsetD,axiom,
    ! [A2: set_int,B3: set_int,C: int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( member_int @ C @ A2 )
       => ( member_int @ C @ B3 ) ) ) ).

% subsetD
thf(fact_1695_equalityE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
         => ~ ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).

% equalityE
thf(fact_1696_subset__eq,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [X: real] :
          ( ( member_real @ X @ A6 )
         => ( member_real @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1697_subset__eq,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ A6 )
         => ( member_nat @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1698_subset__eq,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
        ! [X: complex] :
          ( ( member_complex @ X @ A6 )
         => ( member_complex @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1699_subset__eq,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
        ! [X: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X @ A6 )
         => ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1700_subset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [X: int] :
          ( ( member_int @ X @ A6 )
         => ( member_int @ X @ B6 ) ) ) ) ).

% subset_eq
thf(fact_1701_equalityD1,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% equalityD1
thf(fact_1702_equalityD2,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( A2 = B3 )
     => ( ord_less_eq_set_int @ B3 @ A2 ) ) ).

% equalityD2
thf(fact_1703_subset__iff,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
        ! [T: real] :
          ( ( member_real @ T @ A6 )
         => ( member_real @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1704_subset__iff,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
        ! [T: nat] :
          ( ( member_nat @ T @ A6 )
         => ( member_nat @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1705_subset__iff,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
        ! [T: complex] :
          ( ( member_complex @ T @ A6 )
         => ( member_complex @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1706_subset__iff,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
        ! [T: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ T @ A6 )
         => ( member8440522571783428010at_nat @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1707_subset__iff,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
        ! [T: int] :
          ( ( member_int @ T @ A6 )
         => ( member_int @ T @ B6 ) ) ) ) ).

% subset_iff
thf(fact_1708_subset__refl,axiom,
    ! [A2: set_int] : ( ord_less_eq_set_int @ A2 @ A2 ) ).

% subset_refl
thf(fact_1709_Collect__mono,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ! [X5: complex] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) ) ) ).

% Collect_mono
thf(fact_1710_Collect__mono,axiom,
    ! [P: real > $o,Q: real > $o] :
      ( ! [X5: real] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_real @ ( collect_real @ P ) @ ( collect_real @ Q ) ) ) ).

% Collect_mono
thf(fact_1711_Collect__mono,axiom,
    ! [P: list_nat > $o,Q: list_nat > $o] :
      ( ! [X5: list_nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_le6045566169113846134st_nat @ ( collect_list_nat @ P ) @ ( collect_list_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1712_Collect__mono,axiom,
    ! [P: nat > $o,Q: nat > $o] :
      ( ! [X5: nat] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_nat @ ( collect_nat @ P ) @ ( collect_nat @ Q ) ) ) ).

% Collect_mono
thf(fact_1713_Collect__mono,axiom,
    ! [P: int > $o,Q: int > $o] :
      ( ! [X5: int] :
          ( ( P @ X5 )
         => ( Q @ X5 ) )
     => ( ord_less_eq_set_int @ ( collect_int @ P ) @ ( collect_int @ Q ) ) ) ).

% Collect_mono
thf(fact_1714_subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C4 )
       => ( ord_less_eq_set_int @ A2 @ C4 ) ) ) ).

% subset_trans
thf(fact_1715_set__eq__subset,axiom,
    ( ( ^ [Y6: set_int,Z4: set_int] : Y6 = Z4 )
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% set_eq_subset
thf(fact_1716_Collect__mono__iff,axiom,
    ! [P: complex > $o,Q: complex > $o] :
      ( ( ord_le211207098394363844omplex @ ( collect_complex @ P ) @ ( collect_complex @ Q ) )
      = ( ! [X: complex] :
            ( ( P @ X )
           => ( Q @ X ) ) ) ) ).

% Collect_mono_iff
thf(fact_1717_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_1718_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_1719_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_1720_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_1721_subset__iff__psubset__eq,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_set_int @ A6 @ B6 )
          | ( A6 = B6 ) ) ) ) ).

% subset_iff_psubset_eq
thf(fact_1722_subset__psubset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_set_int @ B3 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% subset_psubset_trans
thf(fact_1723_subset__not__subset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ~ ( ord_less_eq_set_int @ B6 @ A6 ) ) ) ) ).

% subset_not_subset_eq
thf(fact_1724_psubset__subset__trans,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C4 )
       => ( ord_less_set_int @ A2 @ C4 ) ) ) ).

% psubset_subset_trans
thf(fact_1725_psubset__imp__subset,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ( ord_less_eq_set_int @ A2 @ B3 ) ) ).

% psubset_imp_subset
thf(fact_1726_psubset__eq,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ( ord_less_eq_set_int @ A6 @ B6 )
          & ( A6 != B6 ) ) ) ) ).

% psubset_eq
thf(fact_1727_psubsetE,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ~ ( ( ord_less_eq_set_int @ A2 @ B3 )
         => ( ord_less_eq_set_int @ B3 @ A2 ) ) ) ).

% psubsetE
thf(fact_1728_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = zero_zero_nat )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1729_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = zero_zero_int )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1730_cong__exp__iff__simps_I2_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = zero_z3403309356797280102nteger )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(2)
thf(fact_1731_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_1732_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_1733_cong__exp__iff__simps_I1_J,axiom,
    ! [N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ one ) )
      = zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(1)
thf(fact_1734_numeral__1__eq__Suc__0,axiom,
    ( ( numeral_numeral_nat @ one )
    = ( suc @ zero_zero_nat ) ) ).

% numeral_1_eq_Suc_0
thf(fact_1735_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_1736_ex__least__nat__less,axiom,
    ! [P: nat > $o,N2: nat] :
      ( ( P @ N2 )
     => ( ~ ( P @ zero_zero_nat )
       => ? [K2: nat] :
            ( ( ord_less_nat @ K2 @ N2 )
            & ! [I: nat] :
                ( ( ord_less_eq_nat @ I @ K2 )
               => ~ ( P @ I ) )
            & ( P @ ( suc @ K2 ) ) ) ) ) ).

% ex_least_nat_less
thf(fact_1737_length__pos__if__in__set,axiom,
    ! [X4: real,Xs: list_real] :
      ( ( member_real @ X4 @ ( set_real2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_real @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1738_length__pos__if__in__set,axiom,
    ! [X4: complex,Xs: list_complex] :
      ( ( member_complex @ X4 @ ( set_complex2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s3451745648224563538omplex @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1739_length__pos__if__in__set,axiom,
    ! [X4: product_prod_nat_nat,Xs: list_P6011104703257516679at_nat] :
      ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s5460976970255530739at_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1740_length__pos__if__in__set,axiom,
    ! [X4: vEBT_VEBT,Xs: list_VEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_s6755466524823107622T_VEBT @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1741_length__pos__if__in__set,axiom,
    ! [X4: $o,Xs: list_o] :
      ( ( member_o @ X4 @ ( set_o2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_o @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1742_length__pos__if__in__set,axiom,
    ! [X4: nat,Xs: list_nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_nat @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1743_length__pos__if__in__set,axiom,
    ! [X4: int,Xs: list_int] :
      ( ( member_int @ X4 @ ( set_int2 @ Xs ) )
     => ( ord_less_nat @ zero_zero_nat @ ( size_size_list_int @ Xs ) ) ) ).

% length_pos_if_in_set
thf(fact_1744_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_1745_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_1746_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_1747_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_1748_nat__one__le__power,axiom,
    ! [I2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ I2 )
     => ( ord_less_eq_nat @ ( suc @ zero_zero_nat ) @ ( power_power_nat @ I2 @ N2 ) ) ) ).

% nat_one_le_power
thf(fact_1749_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_1750_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_1751_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_1752_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_1753_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_1754_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_1755_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_1756_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_1757_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_1758_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_1759_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_1760_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_1761_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_1762_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_1763_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_1764_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_1765_power__strict__decreasing,axiom,
    ! [N2: nat,N4: nat,A: real] :
      ( ( ord_less_nat @ N2 @ N4 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ( ord_less_real @ A @ one_one_real )
         => ( ord_less_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1766_power__strict__decreasing,axiom,
    ! [N2: nat,N4: nat,A: rat] :
      ( ( ord_less_nat @ N2 @ N4 )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ( ord_less_rat @ A @ one_one_rat )
         => ( ord_less_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1767_power__strict__decreasing,axiom,
    ! [N2: nat,N4: nat,A: nat] :
      ( ( ord_less_nat @ N2 @ N4 )
     => ( ( ord_less_nat @ zero_zero_nat @ A )
       => ( ( ord_less_nat @ A @ one_one_nat )
         => ( ord_less_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1768_power__strict__decreasing,axiom,
    ! [N2: nat,N4: nat,A: int] :
      ( ( ord_less_nat @ N2 @ N4 )
     => ( ( ord_less_int @ zero_zero_int @ A )
       => ( ( ord_less_int @ A @ one_one_int )
         => ( ord_less_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_strict_decreasing
thf(fact_1769_power__decreasing,axiom,
    ! [N2: nat,N4: nat,A: real] :
      ( ( ord_less_eq_nat @ N2 @ N4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ( ord_less_eq_real @ A @ one_one_real )
         => ( ord_less_eq_real @ ( power_power_real @ A @ N4 ) @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_1770_power__decreasing,axiom,
    ! [N2: nat,N4: nat,A: rat] :
      ( ( ord_less_eq_nat @ N2 @ N4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ A )
       => ( ( ord_less_eq_rat @ A @ one_one_rat )
         => ( ord_less_eq_rat @ ( power_power_rat @ A @ N4 ) @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_1771_power__decreasing,axiom,
    ! [N2: nat,N4: nat,A: nat] :
      ( ( ord_less_eq_nat @ N2 @ N4 )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ A )
       => ( ( ord_less_eq_nat @ A @ one_one_nat )
         => ( ord_less_eq_nat @ ( power_power_nat @ A @ N4 ) @ ( power_power_nat @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_1772_power__decreasing,axiom,
    ! [N2: nat,N4: nat,A: int] :
      ( ( ord_less_eq_nat @ N2 @ N4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ A )
       => ( ( ord_less_eq_int @ A @ one_one_int )
         => ( ord_less_eq_int @ ( power_power_int @ A @ N4 ) @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% power_decreasing
thf(fact_1773_zero__power2,axiom,
    ( ( power_power_rat @ zero_zero_rat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_rat ) ).

% zero_power2
thf(fact_1774_zero__power2,axiom,
    ( ( power_power_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_nat ) ).

% zero_power2
thf(fact_1775_zero__power2,axiom,
    ( ( power_power_real @ zero_zero_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_real ) ).

% zero_power2
thf(fact_1776_zero__power2,axiom,
    ( ( power_power_int @ zero_zero_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_int ) ).

% zero_power2
thf(fact_1777_zero__power2,axiom,
    ( ( power_power_complex @ zero_zero_complex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = zero_zero_complex ) ).

% zero_power2
thf(fact_1778_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_1779_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_1780_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_1781_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_1782_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_1783_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_1784_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_1785_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_1786_numeral__2__eq__2,axiom,
    ( ( numeral_numeral_nat @ ( bit0 @ one ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% numeral_2_eq_2
thf(fact_1787_verit__le__mono__div,axiom,
    ! [A2: nat,B3: nat,N2: nat] :
      ( ( ord_less_nat @ A2 @ B3 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_nat
          @ ( plus_plus_nat @ ( divide_divide_nat @ A2 @ N2 )
            @ ( if_nat
              @ ( ( modulo_modulo_nat @ B3 @ N2 )
                = zero_zero_nat )
              @ one_one_nat
              @ zero_zero_nat ) )
          @ ( divide_divide_nat @ B3 @ N2 ) ) ) ) ).

% verit_le_mono_div
thf(fact_1788_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_1789_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_1790_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_1791_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_1792_power2__le__imp__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ X4 @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_1793_power2__le__imp__le,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_eq_rat @ X4 @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_1794_power2__le__imp__le,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_eq_nat @ X4 @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_1795_power2__le__imp__le,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ X4 @ Y ) ) ) ).

% power2_le_imp_le
thf(fact_1796_power2__eq__imp__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( X4 = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_1797_power2__eq__imp__eq,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( X4 = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_1798_power2__eq__imp__eq,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
       => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
         => ( X4 = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_1799_power2__eq__imp__eq,axiom,
    ! [X4: int,Y: int] :
      ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( X4 = Y ) ) ) ) ).

% power2_eq_imp_eq
thf(fact_1800_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_1801_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_1802_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_1803_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_1804_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_1805_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_1806_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_1807_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_1808_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_1809_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_1810_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_1811_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_1812_power2__less__imp__less,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_real @ X4 @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_1813_power2__less__imp__less,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ord_less_rat @ X4 @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_1814_power2__less__imp__less,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ Y )
       => ( ord_less_nat @ X4 @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_1815_power2__less__imp__less,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_int @ X4 @ Y ) ) ) ).

% power2_less_imp_less
thf(fact_1816_sum__power2__le__zero__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_1817_sum__power2__le__zero__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_1818_sum__power2__le__zero__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_power2_le_zero_iff
thf(fact_1819_sum__power2__ge__zero,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_1820_sum__power2__ge__zero,axiom,
    ! [X4: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_1821_sum__power2__ge__zero,axiom,
    ! [X4: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_power2_ge_zero
thf(fact_1822_sum__power2__gt__zero__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X4 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_1823_sum__power2__gt__zero__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X4 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_1824_sum__power2__gt__zero__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
      = ( ( X4 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_power2_gt_zero_iff
thf(fact_1825_not__sum__power2__lt__zero,axiom,
    ! [X4: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_real ) ).

% not_sum_power2_lt_zero
thf(fact_1826_not__sum__power2__lt__zero,axiom,
    ! [X4: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_rat ) ).

% not_sum_power2_lt_zero
thf(fact_1827_not__sum__power2__lt__zero,axiom,
    ! [X4: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ zero_zero_int ) ).

% not_sum_power2_lt_zero
thf(fact_1828_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_1829_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_1830_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_1831_order__le__imp__less__or__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ( ord_less_real @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1832_order__le__imp__less__or__eq,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ord_less_set_int @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1833_order__le__imp__less__or__eq,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ord_less_rat @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1834_order__le__imp__less__or__eq,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ord_less_num @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1835_order__le__imp__less__or__eq,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ord_less_nat @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1836_order__le__imp__less__or__eq,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ord_less_int @ X4 @ Y )
        | ( X4 = Y ) ) ) ).

% order_le_imp_less_or_eq
thf(fact_1837_linorder__le__less__linear,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
      | ( ord_less_real @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1838_linorder__le__less__linear,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
      | ( ord_less_rat @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1839_linorder__le__less__linear,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
      | ( ord_less_num @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1840_linorder__le__less__linear,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
      | ( ord_less_nat @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1841_linorder__le__less__linear,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
      | ( ord_less_int @ Y @ X4 ) ) ).

% linorder_le_less_linear
thf(fact_1842_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 )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1843_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1844_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_num @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1845_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 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1846_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 )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1847_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 )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1848_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1849_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_num @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1850_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 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1851_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 )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_less_le_subst2
thf(fact_1852_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1853_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1854_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1855_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1856_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1857_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1858_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1859_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1860_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1861_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ A @ ( F @ C ) ) ) ) ) ).

% order_less_le_subst1
thf(fact_1862_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1863_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1864_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1865_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1866_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_eq_rat @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1867_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1868_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1869_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_num @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_num @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1870_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_nat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_nat @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1871_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_eq_num @ X5 @ Y3 )
             => ( ord_less_eq_int @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_int @ ( F @ A ) @ C ) ) ) ) ).

% order_le_less_subst2
thf(fact_1872_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 )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1873_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1874_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_num @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1875_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 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1876_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 )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_real @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_real @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1877_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 )
       => ( ! [X5: real,Y3: real] :
              ( ( ord_less_real @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1878_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 )
       => ( ! [X5: rat,Y3: rat] :
              ( ( ord_less_rat @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1879_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 )
       => ( ! [X5: num,Y3: num] :
              ( ( ord_less_num @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1880_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 )
       => ( ! [X5: nat,Y3: nat] :
              ( ( ord_less_nat @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1881_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 )
       => ( ! [X5: int,Y3: int] :
              ( ( ord_less_int @ X5 @ Y3 )
             => ( ord_less_rat @ ( F @ X5 ) @ ( F @ Y3 ) ) )
         => ( ord_less_rat @ A @ ( F @ C ) ) ) ) ) ).

% order_le_less_subst1
thf(fact_1882_order__less__le__trans,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ( ord_less_eq_real @ Y @ Z )
       => ( ord_less_real @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1883_order__less__le__trans,axiom,
    ! [X4: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_set_int @ X4 @ Y )
     => ( ( ord_less_eq_set_int @ Y @ Z )
       => ( ord_less_set_int @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1884_order__less__le__trans,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ( ord_less_eq_rat @ Y @ Z )
       => ( ord_less_rat @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1885_order__less__le__trans,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( ( ord_less_eq_num @ Y @ Z )
       => ( ord_less_num @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1886_order__less__le__trans,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( ( ord_less_eq_nat @ Y @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1887_order__less__le__trans,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_int @ X4 @ Z ) ) ) ).

% order_less_le_trans
thf(fact_1888_order__le__less__trans,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ( ord_less_real @ Y @ Z )
       => ( ord_less_real @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1889_order__le__less__trans,axiom,
    ! [X4: set_int,Y: set_int,Z: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ord_less_set_int @ Y @ Z )
       => ( ord_less_set_int @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1890_order__le__less__trans,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ord_less_rat @ Y @ Z )
       => ( ord_less_rat @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1891_order__le__less__trans,axiom,
    ! [X4: num,Y: num,Z: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ord_less_num @ Y @ Z )
       => ( ord_less_num @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1892_order__le__less__trans,axiom,
    ! [X4: nat,Y: nat,Z: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ord_less_nat @ Y @ Z )
       => ( ord_less_nat @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1893_order__le__less__trans,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ X4 @ Z ) ) ) ).

% order_le_less_trans
thf(fact_1894_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_1895_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_1896_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_1897_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_1898_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_1899_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_1900_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_1901_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_1902_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_1903_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_1904_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_1905_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_1906_order__less__imp__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ord_less_eq_real @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1907_order__less__imp__le,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_set_int @ X4 @ Y )
     => ( ord_less_eq_set_int @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1908_order__less__imp__le,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ord_less_eq_rat @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1909_order__less__imp__le,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_num @ X4 @ Y )
     => ( ord_less_eq_num @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1910_order__less__imp__le,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_nat @ X4 @ Y )
     => ( ord_less_eq_nat @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1911_order__less__imp__le,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ X4 @ Y )
     => ( ord_less_eq_int @ X4 @ Y ) ) ).

% order_less_imp_le
thf(fact_1912_linorder__not__less,axiom,
    ! [X4: real,Y: real] :
      ( ( ~ ( ord_less_real @ X4 @ Y ) )
      = ( ord_less_eq_real @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1913_linorder__not__less,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ~ ( ord_less_rat @ X4 @ Y ) )
      = ( ord_less_eq_rat @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1914_linorder__not__less,axiom,
    ! [X4: num,Y: num] :
      ( ( ~ ( ord_less_num @ X4 @ Y ) )
      = ( ord_less_eq_num @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1915_linorder__not__less,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ~ ( ord_less_nat @ X4 @ Y ) )
      = ( ord_less_eq_nat @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1916_linorder__not__less,axiom,
    ! [X4: int,Y: int] :
      ( ( ~ ( ord_less_int @ X4 @ Y ) )
      = ( ord_less_eq_int @ Y @ X4 ) ) ).

% linorder_not_less
thf(fact_1917_linorder__not__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ~ ( ord_less_eq_real @ X4 @ Y ) )
      = ( ord_less_real @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1918_linorder__not__le,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ~ ( ord_less_eq_rat @ X4 @ Y ) )
      = ( ord_less_rat @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1919_linorder__not__le,axiom,
    ! [X4: num,Y: num] :
      ( ( ~ ( ord_less_eq_num @ X4 @ Y ) )
      = ( ord_less_num @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1920_linorder__not__le,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ~ ( ord_less_eq_nat @ X4 @ Y ) )
      = ( ord_less_nat @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1921_linorder__not__le,axiom,
    ! [X4: int,Y: int] :
      ( ( ~ ( ord_less_eq_int @ X4 @ Y ) )
      = ( ord_less_int @ Y @ X4 ) ) ).

% linorder_not_le
thf(fact_1922_order__less__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y5: real] :
          ( ( ord_less_eq_real @ X @ Y5 )
          & ( X != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1923_order__less__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y5: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y5 )
          & ( X != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1924_order__less__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y5: rat] :
          ( ( ord_less_eq_rat @ X @ Y5 )
          & ( X != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1925_order__less__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y5: num] :
          ( ( ord_less_eq_num @ X @ Y5 )
          & ( X != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1926_order__less__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X @ Y5 )
          & ( X != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1927_order__less__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y5: int] :
          ( ( ord_less_eq_int @ X @ Y5 )
          & ( X != Y5 ) ) ) ) ).

% order_less_le
thf(fact_1928_order__le__less,axiom,
    ( ord_less_eq_real
    = ( ^ [X: real,Y5: real] :
          ( ( ord_less_real @ X @ Y5 )
          | ( X = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1929_order__le__less,axiom,
    ( ord_less_eq_set_int
    = ( ^ [X: set_int,Y5: set_int] :
          ( ( ord_less_set_int @ X @ Y5 )
          | ( X = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1930_order__le__less,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y5: rat] :
          ( ( ord_less_rat @ X @ Y5 )
          | ( X = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1931_order__le__less,axiom,
    ( ord_less_eq_num
    = ( ^ [X: num,Y5: num] :
          ( ( ord_less_num @ X @ Y5 )
          | ( X = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1932_order__le__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [X: nat,Y5: nat] :
          ( ( ord_less_nat @ X @ Y5 )
          | ( X = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1933_order__le__less,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Y5: int] :
          ( ( ord_less_int @ X @ Y5 )
          | ( X = Y5 ) ) ) ) ).

% order_le_less
thf(fact_1934_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_1935_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_1936_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_1937_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_1938_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_1939_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_1940_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_1941_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_1942_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_1943_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_1944_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_1945_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_1946_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A3: real] :
          ( ( ord_less_eq_real @ B2 @ A3 )
          & ~ ( ord_less_eq_real @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1947_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [B2: set_int,A3: set_int] :
          ( ( ord_less_eq_set_int @ B2 @ A3 )
          & ~ ( ord_less_eq_set_int @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1948_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( ord_less_eq_rat @ B2 @ A3 )
          & ~ ( ord_less_eq_rat @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1949_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_less_eq_num @ B2 @ A3 )
          & ~ ( ord_less_eq_num @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1950_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B2 @ A3 )
          & ~ ( ord_less_eq_nat @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1951_dual__order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_less_eq_int @ B2 @ A3 )
          & ~ ( ord_less_eq_int @ A3 @ B2 ) ) ) ) ).

% dual_order.strict_iff_not
thf(fact_1952_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_1953_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_1954_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_1955_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_1956_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_1957_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_1958_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_1959_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_1960_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_1961_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_1962_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_1963_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_1964_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [B2: real,A3: real] :
          ( ( ord_less_eq_real @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1965_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [B2: set_int,A3: set_int] :
          ( ( ord_less_eq_set_int @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1966_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( ord_less_eq_rat @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1967_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_less_eq_num @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1968_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_less_eq_nat @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1969_dual__order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_less_eq_int @ B2 @ A3 )
          & ( A3 != B2 ) ) ) ) ).

% dual_order.strict_iff_order
thf(fact_1970_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [B2: real,A3: real] :
          ( ( ord_less_real @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1971_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [B2: set_int,A3: set_int] :
          ( ( ord_less_set_int @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1972_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [B2: rat,A3: rat] :
          ( ( ord_less_rat @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1973_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [B2: num,A3: num] :
          ( ( ord_less_num @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1974_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [B2: nat,A3: nat] :
          ( ( ord_less_nat @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1975_dual__order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [B2: int,A3: int] :
          ( ( ord_less_int @ B2 @ A3 )
          | ( A3 = B2 ) ) ) ) ).

% dual_order.order_iff_strict
thf(fact_1976_dense__le__bounded,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ! [W2: real] :
            ( ( ord_less_real @ X4 @ W2 )
           => ( ( ord_less_real @ W2 @ Y )
             => ( ord_less_eq_real @ W2 @ Z ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_1977_dense__le__bounded,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( ord_less_rat @ X4 @ Y )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ X4 @ W2 )
           => ( ( ord_less_rat @ W2 @ Y )
             => ( ord_less_eq_rat @ W2 @ Z ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_le_bounded
thf(fact_1978_dense__ge__bounded,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( ord_less_real @ Z @ X4 )
     => ( ! [W2: real] :
            ( ( ord_less_real @ Z @ W2 )
           => ( ( ord_less_real @ W2 @ X4 )
             => ( ord_less_eq_real @ Y @ W2 ) ) )
       => ( ord_less_eq_real @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_1979_dense__ge__bounded,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_rat @ Z @ X4 )
     => ( ! [W2: rat] :
            ( ( ord_less_rat @ Z @ W2 )
           => ( ( ord_less_rat @ W2 @ X4 )
             => ( ord_less_eq_rat @ Y @ W2 ) ) )
       => ( ord_less_eq_rat @ Y @ Z ) ) ) ).

% dense_ge_bounded
thf(fact_1980_order_Ostrict__iff__not,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_eq_real @ A3 @ B2 )
          & ~ ( ord_less_eq_real @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1981_order_Ostrict__iff__not,axiom,
    ( ord_less_set_int
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ A3 @ B2 )
          & ~ ( ord_less_eq_set_int @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1982_order_Ostrict__iff__not,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A3 @ B2 )
          & ~ ( ord_less_eq_rat @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1983_order_Ostrict__iff__not,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
          & ~ ( ord_less_eq_num @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1984_order_Ostrict__iff__not,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
          & ~ ( ord_less_eq_nat @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1985_order_Ostrict__iff__not,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
          & ~ ( ord_less_eq_int @ B2 @ A3 ) ) ) ) ).

% order.strict_iff_not
thf(fact_1986_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_1987_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_1988_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_1989_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_1990_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_1991_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_1992_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_1993_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_1994_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_1995_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_1996_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_1997_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_1998_order_Ostrict__iff__order,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_eq_real @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_1999_order_Ostrict__iff__order,axiom,
    ( ord_less_set_int
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_eq_set_int @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2000_order_Ostrict__iff__order,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_eq_rat @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2001_order_Ostrict__iff__order,axiom,
    ( ord_less_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_eq_num @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2002_order_Ostrict__iff__order,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_eq_nat @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2003_order_Ostrict__iff__order,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_eq_int @ A3 @ B2 )
          & ( A3 != B2 ) ) ) ) ).

% order.strict_iff_order
thf(fact_2004_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B2: real] :
          ( ( ord_less_real @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2005_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A3: set_int,B2: set_int] :
          ( ( ord_less_set_int @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2006_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( ord_less_rat @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2007_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_num
    = ( ^ [A3: num,B2: num] :
          ( ( ord_less_num @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2008_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( ord_less_nat @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2009_order_Oorder__iff__strict,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B2: int] :
          ( ( ord_less_int @ A3 @ B2 )
          | ( A3 = B2 ) ) ) ) ).

% order.order_iff_strict
thf(fact_2010_not__le__imp__less,axiom,
    ! [Y: real,X4: real] :
      ( ~ ( ord_less_eq_real @ Y @ X4 )
     => ( ord_less_real @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_2011_not__le__imp__less,axiom,
    ! [Y: rat,X4: rat] :
      ( ~ ( ord_less_eq_rat @ Y @ X4 )
     => ( ord_less_rat @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_2012_not__le__imp__less,axiom,
    ! [Y: num,X4: num] :
      ( ~ ( ord_less_eq_num @ Y @ X4 )
     => ( ord_less_num @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_2013_not__le__imp__less,axiom,
    ! [Y: nat,X4: nat] :
      ( ~ ( ord_less_eq_nat @ Y @ X4 )
     => ( ord_less_nat @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_2014_not__le__imp__less,axiom,
    ! [Y: int,X4: int] :
      ( ~ ( ord_less_eq_int @ Y @ X4 )
     => ( ord_less_int @ X4 @ Y ) ) ).

% not_le_imp_less
thf(fact_2015_less__le__not__le,axiom,
    ( ord_less_real
    = ( ^ [X: real,Y5: real] :
          ( ( ord_less_eq_real @ X @ Y5 )
          & ~ ( ord_less_eq_real @ Y5 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_2016_less__le__not__le,axiom,
    ( ord_less_set_int
    = ( ^ [X: set_int,Y5: set_int] :
          ( ( ord_less_eq_set_int @ X @ Y5 )
          & ~ ( ord_less_eq_set_int @ Y5 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_2017_less__le__not__le,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y5: rat] :
          ( ( ord_less_eq_rat @ X @ Y5 )
          & ~ ( ord_less_eq_rat @ Y5 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_2018_less__le__not__le,axiom,
    ( ord_less_num
    = ( ^ [X: num,Y5: num] :
          ( ( ord_less_eq_num @ X @ Y5 )
          & ~ ( ord_less_eq_num @ Y5 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_2019_less__le__not__le,axiom,
    ( ord_less_nat
    = ( ^ [X: nat,Y5: nat] :
          ( ( ord_less_eq_nat @ X @ Y5 )
          & ~ ( ord_less_eq_nat @ Y5 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_2020_less__le__not__le,axiom,
    ( ord_less_int
    = ( ^ [X: int,Y5: int] :
          ( ( ord_less_eq_int @ X @ Y5 )
          & ~ ( ord_less_eq_int @ Y5 @ X ) ) ) ) ).

% less_le_not_le
thf(fact_2021_dense__le,axiom,
    ! [Y: real,Z: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ X5 @ Y )
         => ( ord_less_eq_real @ X5 @ Z ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_le
thf(fact_2022_dense__le,axiom,
    ! [Y: rat,Z: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Y )
         => ( ord_less_eq_rat @ X5 @ Z ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_le
thf(fact_2023_dense__ge,axiom,
    ! [Z: real,Y: real] :
      ( ! [X5: real] :
          ( ( ord_less_real @ Z @ X5 )
         => ( ord_less_eq_real @ Y @ X5 ) )
     => ( ord_less_eq_real @ Y @ Z ) ) ).

% dense_ge
thf(fact_2024_dense__ge,axiom,
    ! [Z: rat,Y: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_rat @ Z @ X5 )
         => ( ord_less_eq_rat @ Y @ X5 ) )
     => ( ord_less_eq_rat @ Y @ Z ) ) ).

% dense_ge
thf(fact_2025_antisym__conv2,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ( ~ ( ord_less_real @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_2026_antisym__conv2,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ( ~ ( ord_less_set_int @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_2027_antisym__conv2,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ( ~ ( ord_less_rat @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_2028_antisym__conv2,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_num @ X4 @ Y )
     => ( ( ~ ( ord_less_num @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_2029_antisym__conv2,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_nat @ X4 @ Y )
     => ( ( ~ ( ord_less_nat @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_2030_antisym__conv2,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ( ~ ( ord_less_int @ X4 @ Y ) )
        = ( X4 = Y ) ) ) ).

% antisym_conv2
thf(fact_2031_antisym__conv1,axiom,
    ! [X4: real,Y: real] :
      ( ~ ( ord_less_real @ X4 @ Y )
     => ( ( ord_less_eq_real @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_2032_antisym__conv1,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ~ ( ord_less_set_int @ X4 @ Y )
     => ( ( ord_less_eq_set_int @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_2033_antisym__conv1,axiom,
    ! [X4: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X4 @ Y )
     => ( ( ord_less_eq_rat @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_2034_antisym__conv1,axiom,
    ! [X4: num,Y: num] :
      ( ~ ( ord_less_num @ X4 @ Y )
     => ( ( ord_less_eq_num @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_2035_antisym__conv1,axiom,
    ! [X4: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y )
     => ( ( ord_less_eq_nat @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_2036_antisym__conv1,axiom,
    ! [X4: int,Y: int] :
      ( ~ ( ord_less_int @ X4 @ Y )
     => ( ( ord_less_eq_int @ X4 @ Y )
        = ( X4 = Y ) ) ) ).

% antisym_conv1
thf(fact_2037_nless__le,axiom,
    ! [A: real,B: real] :
      ( ( ~ ( ord_less_real @ A @ B ) )
      = ( ~ ( ord_less_eq_real @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2038_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_2039_nless__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ~ ( ord_less_rat @ A @ B ) )
      = ( ~ ( ord_less_eq_rat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2040_nless__le,axiom,
    ! [A: num,B: num] :
      ( ( ~ ( ord_less_num @ A @ B ) )
      = ( ~ ( ord_less_eq_num @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2041_nless__le,axiom,
    ! [A: nat,B: nat] :
      ( ( ~ ( ord_less_nat @ A @ B ) )
      = ( ~ ( ord_less_eq_nat @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2042_nless__le,axiom,
    ! [A: int,B: int] :
      ( ( ~ ( ord_less_int @ A @ B ) )
      = ( ~ ( ord_less_eq_int @ A @ B )
        | ( A = B ) ) ) ).

% nless_le
thf(fact_2043_leI,axiom,
    ! [X4: real,Y: real] :
      ( ~ ( ord_less_real @ X4 @ Y )
     => ( ord_less_eq_real @ Y @ X4 ) ) ).

% leI
thf(fact_2044_leI,axiom,
    ! [X4: rat,Y: rat] :
      ( ~ ( ord_less_rat @ X4 @ Y )
     => ( ord_less_eq_rat @ Y @ X4 ) ) ).

% leI
thf(fact_2045_leI,axiom,
    ! [X4: num,Y: num] :
      ( ~ ( ord_less_num @ X4 @ Y )
     => ( ord_less_eq_num @ Y @ X4 ) ) ).

% leI
thf(fact_2046_leI,axiom,
    ! [X4: nat,Y: nat] :
      ( ~ ( ord_less_nat @ X4 @ Y )
     => ( ord_less_eq_nat @ Y @ X4 ) ) ).

% leI
thf(fact_2047_leI,axiom,
    ! [X4: int,Y: int] :
      ( ~ ( ord_less_int @ X4 @ Y )
     => ( ord_less_eq_int @ Y @ X4 ) ) ).

% leI
thf(fact_2048_leD,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ Y @ X4 )
     => ~ ( ord_less_real @ X4 @ Y ) ) ).

% leD
thf(fact_2049_leD,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ Y @ X4 )
     => ~ ( ord_less_set_int @ X4 @ Y ) ) ).

% leD
thf(fact_2050_leD,axiom,
    ! [Y: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y @ X4 )
     => ~ ( ord_less_rat @ X4 @ Y ) ) ).

% leD
thf(fact_2051_leD,axiom,
    ! [Y: num,X4: num] :
      ( ( ord_less_eq_num @ Y @ X4 )
     => ~ ( ord_less_num @ X4 @ Y ) ) ).

% leD
thf(fact_2052_leD,axiom,
    ! [Y: nat,X4: nat] :
      ( ( ord_less_eq_nat @ Y @ X4 )
     => ~ ( ord_less_nat @ X4 @ Y ) ) ).

% leD
thf(fact_2053_leD,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ Y @ X4 )
     => ~ ( ord_less_int @ X4 @ Y ) ) ).

% leD
thf(fact_2054_verit__comp__simplify1_I3_J,axiom,
    ! [B4: real,A4: real] :
      ( ( ~ ( ord_less_eq_real @ B4 @ A4 ) )
      = ( ord_less_real @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2055_verit__comp__simplify1_I3_J,axiom,
    ! [B4: rat,A4: rat] :
      ( ( ~ ( ord_less_eq_rat @ B4 @ A4 ) )
      = ( ord_less_rat @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2056_verit__comp__simplify1_I3_J,axiom,
    ! [B4: num,A4: num] :
      ( ( ~ ( ord_less_eq_num @ B4 @ A4 ) )
      = ( ord_less_num @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2057_verit__comp__simplify1_I3_J,axiom,
    ! [B4: nat,A4: nat] :
      ( ( ~ ( ord_less_eq_nat @ B4 @ A4 ) )
      = ( ord_less_nat @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2058_verit__comp__simplify1_I3_J,axiom,
    ! [B4: int,A4: int] :
      ( ( ~ ( ord_less_eq_int @ B4 @ A4 ) )
      = ( ord_less_int @ A4 @ B4 ) ) ).

% verit_comp_simplify1(3)
thf(fact_2059_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_2060_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_2061_verit__eq__simplify_I10_J,axiom,
    ! [X22: num] :
      ( one
     != ( bit0 @ X22 ) ) ).

% verit_eq_simplify(10)
thf(fact_2062_invar__vebt_Ointros_I4_J,axiom,
    ! [TreeList2: list_VEBT_VEBT,N2: nat,Summary: vEBT_VEBT,M: nat,Deg: nat,Mi: nat,Ma: nat] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
         => ( vEBT_invar_vebt @ X5 @ N2 ) )
     => ( ( vEBT_invar_vebt @ Summary @ M )
       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList2 )
            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
         => ( ( M = N2 )
           => ( ( Deg
                = ( plus_plus_nat @ N2 @ M ) )
             => ( ! [I4: nat] :
                    ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                   => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ X3 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I4 ) ) )
               => ( ( ( Mi = Ma )
                   => ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                       => ~ ? [X_12: nat] : ( vEBT_V8194947554948674370ptions @ X5 @ X_12 ) ) )
                 => ( ( ord_less_eq_nat @ Mi @ Ma )
                   => ( ( ord_less_nat @ Ma @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                     => ( ( ( Mi != Ma )
                         => ! [I4: nat] :
                              ( ( ord_less_nat @ I4 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
                             => ( ( ( ( vEBT_VEBT_high @ Ma @ N2 )
                                    = I4 )
                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ Ma @ N2 ) ) )
                                & ! [X5: nat] :
                                    ( ( ( ( vEBT_VEBT_high @ X5 @ N2 )
                                        = I4 )
                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I4 ) @ ( vEBT_VEBT_low @ X5 @ N2 ) ) )
                                   => ( ( ord_less_nat @ Mi @ X5 )
                                      & ( ord_less_eq_nat @ X5 @ Ma ) ) ) ) ) )
                       => ( vEBT_invar_vebt @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ Deg @ TreeList2 @ Summary ) @ Deg ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.intros(4)
thf(fact_2063_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_2064_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_2065_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_2066_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_2067_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_2068_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_2069_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_2070_invar__vebt_Osimps,axiom,
    ( vEBT_invar_vebt
    = ( ^ [A1: vEBT_VEBT,A22: nat] :
          ( ( ? [A3: $o,B2: $o] :
                ( A1
                = ( vEBT_Leaf @ A3 @ B2 ) )
            & ( A22
              = ( suc @ zero_zero_nat ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary2 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary2 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT] :
              ( ( A1
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ A22 @ TreeList @ Summary2 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary2 @ X3 )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ A22 @ TreeList @ Summary2 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary2 @ N )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              & ( A22
                = ( plus_plus_nat @ N @ N ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X3 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
              & ( ( Mi2 = Ma2 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
              & ( ord_less_eq_nat @ Mi2 @ Ma2 )
              & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi2 != Ma2 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi2 @ X )
                            & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) )
          | ? [TreeList: list_VEBT_VEBT,N: nat,Summary2: vEBT_VEBT,Mi2: nat,Ma2: nat] :
              ( ( A1
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi2 @ Ma2 ) ) @ A22 @ TreeList @ Summary2 ) )
              & ! [X: vEBT_VEBT] :
                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                 => ( vEBT_invar_vebt @ X @ N ) )
              & ( vEBT_invar_vebt @ Summary2 @ ( suc @ N ) )
              & ( ( size_s6755466524823107622T_VEBT @ TreeList )
                = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
              & ( A22
                = ( plus_plus_nat @ N @ ( suc @ N ) ) )
              & ! [I3: nat] :
                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ X3 ) )
                    = ( vEBT_V8194947554948674370ptions @ Summary2 @ I3 ) ) )
              & ( ( Mi2 = Ma2 )
               => ! [X: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList ) )
                   => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
              & ( ord_less_eq_nat @ Mi2 @ Ma2 )
              & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A22 ) )
              & ( ( Mi2 != Ma2 )
               => ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ N ) ) )
                   => ( ( ( ( vEBT_VEBT_high @ Ma2 @ N )
                          = I3 )
                       => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ Ma2 @ N ) ) )
                      & ! [X: nat] :
                          ( ( ( ( vEBT_VEBT_high @ X @ N )
                              = I3 )
                            & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList @ I3 ) @ ( vEBT_VEBT_low @ X @ N ) ) )
                         => ( ( ord_less_nat @ Mi2 @ X )
                            & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.simps
thf(fact_2071_invar__vebt_Ocases,axiom,
    ! [A12: vEBT_VEBT,A23: nat] :
      ( ( vEBT_invar_vebt @ A12 @ A23 )
     => ( ( ? [A5: $o,B5: $o] :
              ( A12
              = ( vEBT_Leaf @ A5 @ B5 ) )
         => ( A23
           != ( suc @ zero_zero_nat ) ) )
       => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
              ( ( A12
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
             => ( ( A23 = Deg2 )
               => ( ! [X2: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                     => ( vEBT_invar_vebt @ X2 @ N3 ) )
                 => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
                   => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                     => ( ( M5 = N3 )
                       => ( ( Deg2
                            = ( plus_plus_nat @ N3 @ M5 ) )
                         => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_1 )
                           => ~ ! [X2: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
         => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat] :
                ( ( A12
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ Deg2 @ TreeList3 @ Summary3 ) )
               => ( ( A23 = Deg2 )
                 => ( ! [X2: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_invar_vebt @ X2 @ N3 ) )
                   => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
                     => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                          = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                       => ( ( M5
                            = ( suc @ N3 ) )
                         => ( ( Deg2
                              = ( plus_plus_nat @ N3 @ M5 ) )
                           => ( ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X_1 )
                             => ~ ! [X2: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) ) ) ) ) ) ) ) )
           => ( ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                  ( ( A12
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList3 @ Summary3 ) )
                 => ( ( A23 = Deg2 )
                   => ( ! [X2: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_invar_vebt @ X2 @ N3 ) )
                     => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
                       => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                            = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                         => ( ( M5 = N3 )
                           => ( ( Deg2
                                = ( plus_plus_nat @ N3 @ M5 ) )
                             => ( ! [I: nat] :
                                    ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                   => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X3 ) )
                                      = ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
                               => ( ( ( Mi3 = Ma3 )
                                   => ! [X2: vEBT_VEBT] :
                                        ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                       => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
                                 => ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                                   => ( ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                     => ~ ( ( Mi3 != Ma3 )
                                         => ! [I: nat] :
                                              ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                             => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                                                    = I )
                                                 => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                                                & ! [X2: nat] :
                                                    ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
                                                        = I )
                                                      & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
                                                   => ( ( ord_less_nat @ Mi3 @ X2 )
                                                      & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) )
             => ~ ! [TreeList3: list_VEBT_VEBT,N3: nat,Summary3: vEBT_VEBT,M5: nat,Deg2: nat,Mi3: nat,Ma3: nat] :
                    ( ( A12
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ Deg2 @ TreeList3 @ Summary3 ) )
                   => ( ( A23 = Deg2 )
                     => ( ! [X2: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ( vEBT_invar_vebt @ X2 @ N3 ) )
                       => ( ( vEBT_invar_vebt @ Summary3 @ M5 )
                         => ( ( ( size_s6755466524823107622T_VEBT @ TreeList3 )
                              = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                           => ( ( M5
                                = ( suc @ N3 ) )
                             => ( ( Deg2
                                  = ( plus_plus_nat @ N3 @ M5 ) )
                               => ( ! [I: nat] :
                                      ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                     => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ X3 ) )
                                        = ( vEBT_V8194947554948674370ptions @ Summary3 @ I ) ) )
                                 => ( ( ( Mi3 = Ma3 )
                                     => ! [X2: vEBT_VEBT] :
                                          ( ( member_VEBT_VEBT @ X2 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                         => ~ ? [X_1: nat] : ( vEBT_V8194947554948674370ptions @ X2 @ X_1 ) ) )
                                   => ( ( ord_less_eq_nat @ Mi3 @ Ma3 )
                                     => ( ( ord_less_nat @ Ma3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                                       => ~ ( ( Mi3 != Ma3 )
                                           => ! [I: nat] :
                                                ( ( ord_less_nat @ I @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M5 ) )
                                               => ( ( ( ( vEBT_VEBT_high @ Ma3 @ N3 )
                                                      = I )
                                                   => ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ Ma3 @ N3 ) ) )
                                                  & ! [X2: nat] :
                                                      ( ( ( ( vEBT_VEBT_high @ X2 @ N3 )
                                                          = I )
                                                        & ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I ) @ ( vEBT_VEBT_low @ X2 @ N3 ) ) )
                                                     => ( ( ord_less_nat @ Mi3 @ X2 )
                                                        & ( ord_less_eq_nat @ X2 @ Ma3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% invar_vebt.cases
thf(fact_2072_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_2073_not__Some__eq,axiom,
    ! [X4: option4927543243414619207at_nat] :
      ( ( ! [Y5: product_prod_nat_nat] :
            ( X4
           != ( some_P7363390416028606310at_nat @ Y5 ) ) )
      = ( X4 = none_P5556105721700978146at_nat ) ) ).

% not_Some_eq
thf(fact_2074_not__Some__eq,axiom,
    ! [X4: option_num] :
      ( ( ! [Y5: num] :
            ( X4
           != ( some_num @ Y5 ) ) )
      = ( X4 = none_num ) ) ).

% not_Some_eq
thf(fact_2075_not__None__eq,axiom,
    ! [X4: option4927543243414619207at_nat] :
      ( ( X4 != none_P5556105721700978146at_nat )
      = ( ? [Y5: product_prod_nat_nat] :
            ( X4
            = ( some_P7363390416028606310at_nat @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_2076_not__None__eq,axiom,
    ! [X4: option_num] :
      ( ( X4 != none_num )
      = ( ? [Y5: num] :
            ( X4
            = ( some_num @ Y5 ) ) ) ) ).

% not_None_eq
thf(fact_2077_pos2,axiom,
    ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ).

% pos2
thf(fact_2078_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_2079_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_2080_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_2081_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_2082_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_2083_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_2084_i0__less,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le72135733267957522d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = ( N2 != zero_z5237406670263579293d_enat ) ) ).

% i0_less
thf(fact_2085_half__nonnegative__int__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% half_nonnegative_int_iff
thf(fact_2086_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_2087_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_2088_VEBT_Oexhaust,axiom,
    ! [Y: vEBT_VEBT] :
      ( ! [X112: option4927543243414619207at_nat,X122: nat,X132: list_VEBT_VEBT,X142: vEBT_VEBT] :
          ( Y
         != ( vEBT_Node @ X112 @ X122 @ X132 @ X142 ) )
     => ~ ! [X212: $o,X223: $o] :
            ( Y
           != ( vEBT_Leaf @ X212 @ X223 ) ) ) ).

% VEBT.exhaust
thf(fact_2089_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_2090_zdiv__mono__strict,axiom,
    ! [A2: int,B3: int,N2: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ( ( modulo_modulo_int @ A2 @ N2 )
            = zero_zero_int )
         => ( ( ( modulo_modulo_int @ B3 @ N2 )
              = zero_zero_int )
           => ( ord_less_int @ ( divide_divide_int @ A2 @ N2 ) @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ) ) ).

% zdiv_mono_strict
thf(fact_2091_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_2092_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_2093_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_2094_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_2095_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_2096_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_2097_pos__imp__zdiv__pos__iff,axiom,
    ! [K: int,I2: int] :
      ( ( ord_less_int @ zero_zero_int @ K )
     => ( ( ord_less_int @ zero_zero_int @ ( divide_divide_int @ I2 @ K ) )
        = ( ord_less_eq_int @ K @ I2 ) ) ) ).

% pos_imp_zdiv_pos_iff
thf(fact_2098_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_2099_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_2100_verit__le__mono__div__int,axiom,
    ! [A2: int,B3: int,N2: int] :
      ( ( ord_less_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int
          @ ( plus_plus_int @ ( divide_divide_int @ A2 @ N2 )
            @ ( if_int
              @ ( ( modulo_modulo_int @ B3 @ N2 )
                = zero_zero_int )
              @ one_one_int
              @ zero_zero_int ) )
          @ ( divide_divide_int @ B3 @ N2 ) ) ) ) ).

% verit_le_mono_div_int
thf(fact_2101_int__div__less__self,axiom,
    ! [X4: int,K: int] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ( ord_less_int @ one_one_int @ K )
       => ( ord_less_int @ ( divide_divide_int @ X4 @ K ) @ X4 ) ) ) ).

% int_div_less_self
thf(fact_2102_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_2103_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_2104_zdiv__mono2__neg,axiom,
    ! [A: int,B4: int,B: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B4 ) @ ( divide_divide_int @ A @ B ) ) ) ) ) ).

% zdiv_mono2_neg
thf(fact_2105_zdiv__mono1__neg,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ B @ zero_zero_int )
       => ( ord_less_eq_int @ ( divide_divide_int @ A4 @ B ) @ ( divide_divide_int @ A @ B ) ) ) ) ).

% zdiv_mono1_neg
thf(fact_2106_zdiv__eq__0__iff,axiom,
    ! [I2: int,K: int] :
      ( ( ( divide_divide_int @ I2 @ K )
        = zero_zero_int )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K @ I2 ) ) ) ) ).

% zdiv_eq_0_iff
thf(fact_2107_zdiv__mono2,axiom,
    ! [A: int,B4: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( ord_less_int @ zero_zero_int @ B4 )
       => ( ( ord_less_eq_int @ B4 @ B )
         => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A @ B4 ) ) ) ) ) ).

% zdiv_mono2
thf(fact_2108_zdiv__mono1,axiom,
    ! [A: int,A4: int,B: int] :
      ( ( ord_less_eq_int @ A @ A4 )
     => ( ( ord_less_int @ zero_zero_int @ B )
       => ( ord_less_eq_int @ ( divide_divide_int @ A @ B ) @ ( divide_divide_int @ A4 @ B ) ) ) ) ).

% zdiv_mono1
thf(fact_2109_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_2110_not__iless0,axiom,
    ! [N2: extended_enat] :
      ~ ( ord_le72135733267957522d_enat @ N2 @ zero_z5237406670263579293d_enat ) ).

% not_iless0
thf(fact_2111_i0__lb,axiom,
    ! [N2: extended_enat] : ( ord_le2932123472753598470d_enat @ zero_z5237406670263579293d_enat @ N2 ) ).

% i0_lb
thf(fact_2112_ile0__eq,axiom,
    ! [N2: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = ( N2 = zero_z5237406670263579293d_enat ) ) ).

% ile0_eq
thf(fact_2113_realpow__pos__nth2,axiom,
    ! [A: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ? [R2: real] :
          ( ( ord_less_real @ zero_zero_real @ R2 )
          & ( ( power_power_real @ R2 @ ( suc @ N2 ) )
            = A ) ) ) ).

% realpow_pos_nth2
thf(fact_2114_vebt__buildup_Osimps_I1_J,axiom,
    ( ( vEBT_vebt_buildup @ zero_zero_nat )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(1)
thf(fact_2115_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_2116_realpow__pos__nth__unique,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [X5: real] :
            ( ( ord_less_real @ zero_zero_real @ X5 )
            & ( ( power_power_real @ X5 @ N2 )
              = A )
            & ! [Y4: real] :
                ( ( ( ord_less_real @ zero_zero_real @ Y4 )
                  & ( ( power_power_real @ Y4 @ N2 )
                    = A ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% realpow_pos_nth_unique
thf(fact_2117_realpow__pos__nth,axiom,
    ! [N2: nat,A: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ( ( power_power_real @ R2 @ N2 )
              = A ) ) ) ) ).

% realpow_pos_nth
thf(fact_2118_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_2119_not__exp__less__eq__0__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ zero_zero_int ) ).

% not_exp_less_eq_0_int
thf(fact_2120_vebt__buildup_Osimps_I2_J,axiom,
    ( ( vEBT_vebt_buildup @ ( suc @ zero_zero_nat ) )
    = ( vEBT_Leaf @ $false @ $false ) ) ).

% vebt_buildup.simps(2)
thf(fact_2121_VEBT__internal_Onaive__member_Osimps_I1_J,axiom,
    ! [A: $o,B: $o,X4: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Leaf @ A @ B ) @ X4 )
      = ( ( ( X4 = zero_zero_nat )
         => A )
        & ( ( X4 != zero_zero_nat )
         => ( ( ( X4 = one_one_nat )
             => B )
            & ( X4 = one_one_nat ) ) ) ) ) ).

% VEBT_internal.naive_member.simps(1)
thf(fact_2122_real__arch__pow__inv,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ? [N3: nat] : ( ord_less_real @ ( power_power_real @ X4 @ N3 ) @ Y ) ) ) ).

% real_arch_pow_inv
thf(fact_2123_option_Odistinct_I1_J,axiom,
    ! [X22: product_prod_nat_nat] :
      ( none_P5556105721700978146at_nat
     != ( some_P7363390416028606310at_nat @ X22 ) ) ).

% option.distinct(1)
thf(fact_2124_option_Odistinct_I1_J,axiom,
    ! [X22: num] :
      ( none_num
     != ( some_num @ X22 ) ) ).

% option.distinct(1)
thf(fact_2125_option_OdiscI,axiom,
    ! [Option: option4927543243414619207at_nat,X22: product_prod_nat_nat] :
      ( ( Option
        = ( some_P7363390416028606310at_nat @ X22 ) )
     => ( Option != none_P5556105721700978146at_nat ) ) ).

% option.discI
thf(fact_2126_option_OdiscI,axiom,
    ! [Option: option_num,X22: num] :
      ( ( Option
        = ( some_num @ X22 ) )
     => ( Option != none_num ) ) ).

% option.discI
thf(fact_2127_option_Oexhaust,axiom,
    ! [Y: option4927543243414619207at_nat] :
      ( ( Y != none_P5556105721700978146at_nat )
     => ~ ! [X23: product_prod_nat_nat] :
            ( Y
           != ( some_P7363390416028606310at_nat @ X23 ) ) ) ).

% option.exhaust
thf(fact_2128_option_Oexhaust,axiom,
    ! [Y: option_num] :
      ( ( Y != none_num )
     => ~ ! [X23: num] :
            ( Y
           != ( some_num @ X23 ) ) ) ).

% option.exhaust
thf(fact_2129_split__option__ex,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ? [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          | ? [X: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_2130_split__option__ex,axiom,
    ( ( ^ [P3: option_num > $o] :
        ? [X6: option_num] : ( P3 @ X6 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          | ? [X: num] : ( P4 @ ( some_num @ X ) ) ) ) ) ).

% split_option_ex
thf(fact_2131_split__option__all,axiom,
    ( ( ^ [P3: option4927543243414619207at_nat > $o] :
        ! [X6: option4927543243414619207at_nat] : ( P3 @ X6 ) )
    = ( ^ [P4: option4927543243414619207at_nat > $o] :
          ( ( P4 @ none_P5556105721700978146at_nat )
          & ! [X: product_prod_nat_nat] : ( P4 @ ( some_P7363390416028606310at_nat @ X ) ) ) ) ) ).

% split_option_all
thf(fact_2132_split__option__all,axiom,
    ( ( ^ [P3: option_num > $o] :
        ! [X6: option_num] : ( P3 @ X6 ) )
    = ( ^ [P4: option_num > $o] :
          ( ( P4 @ none_num )
          & ! [X: num] : ( P4 @ ( some_num @ X ) ) ) ) ) ).

% split_option_all
thf(fact_2133_combine__options__cases,axiom,
    ! [X4: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X4 = none_P5556105721700978146at_nat )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X4 @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B5: product_prod_nat_nat] :
              ( ( X4
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2134_combine__options__cases,axiom,
    ! [X4: option4927543243414619207at_nat,P: option4927543243414619207at_nat > option_num > $o,Y: option_num] :
      ( ( ( X4 = none_P5556105721700978146at_nat )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X4 @ Y ) )
       => ( ! [A5: product_prod_nat_nat,B5: num] :
              ( ( X4
                = ( some_P7363390416028606310at_nat @ A5 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2135_combine__options__cases,axiom,
    ! [X4: option_num,P: option_num > option4927543243414619207at_nat > $o,Y: option4927543243414619207at_nat] :
      ( ( ( X4 = none_num )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_P5556105721700978146at_nat )
         => ( P @ X4 @ Y ) )
       => ( ! [A5: num,B5: product_prod_nat_nat] :
              ( ( X4
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_P7363390416028606310at_nat @ B5 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2136_combine__options__cases,axiom,
    ! [X4: option_num,P: option_num > option_num > $o,Y: option_num] :
      ( ( ( X4 = none_num )
       => ( P @ X4 @ Y ) )
     => ( ( ( Y = none_num )
         => ( P @ X4 @ Y ) )
       => ( ! [A5: num,B5: num] :
              ( ( X4
                = ( some_num @ A5 ) )
             => ( ( Y
                  = ( some_num @ B5 ) )
               => ( P @ X4 @ Y ) ) )
         => ( P @ X4 @ Y ) ) ) ) ).

% combine_options_cases
thf(fact_2137_option_Osize_I3_J,axiom,
    ( ( size_s170228958280169651at_nat @ none_P5556105721700978146at_nat )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2138_option_Osize_I3_J,axiom,
    ( ( size_size_option_num @ none_num )
    = ( suc @ zero_zero_nat ) ) ).

% option.size(3)
thf(fact_2139_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_2140_option_Osize_I4_J,axiom,
    ! [X22: num] :
      ( ( size_size_option_num @ ( some_num @ X22 ) )
      = ( suc @ zero_zero_nat ) ) ).

% option.size(4)
thf(fact_2141_divides__aux__eq,axiom,
    ! [Q3: nat,R3: nat] :
      ( ( unique6322359934112328802ux_nat @ ( product_Pair_nat_nat @ Q3 @ R3 ) )
      = ( R3 = zero_zero_nat ) ) ).

% divides_aux_eq
thf(fact_2142_divides__aux__eq,axiom,
    ! [Q3: int,R3: int] :
      ( ( unique6319869463603278526ux_int @ ( product_Pair_int_int @ Q3 @ R3 ) )
      = ( R3 = zero_zero_int ) ) ).

% divides_aux_eq
thf(fact_2143_gcd__nat__induct,axiom,
    ! [P: nat > nat > $o,M: nat,N2: nat] :
      ( ! [M5: nat] : ( P @ M5 @ zero_zero_nat )
     => ( ! [M5: nat,N3: nat] :
            ( ( ord_less_nat @ zero_zero_nat @ N3 )
           => ( ( P @ N3 @ ( modulo_modulo_nat @ M5 @ N3 ) )
             => ( P @ M5 @ N3 ) ) )
       => ( P @ M @ N2 ) ) ) ).

% gcd_nat_induct
thf(fact_2144_option_Osize__gen_I2_J,axiom,
    ! [X4: product_prod_nat_nat > nat,X22: product_prod_nat_nat] :
      ( ( size_o8335143837870341156at_nat @ X4 @ ( some_P7363390416028606310at_nat @ X22 ) )
      = ( plus_plus_nat @ ( X4 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_2145_option_Osize__gen_I2_J,axiom,
    ! [X4: num > nat,X22: num] :
      ( ( size_option_num @ X4 @ ( some_num @ X22 ) )
      = ( plus_plus_nat @ ( X4 @ X22 ) @ ( suc @ zero_zero_nat ) ) ) ).

% option.size_gen(2)
thf(fact_2146_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_2147_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_2148_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_2149_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_2150_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_2151_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_2152_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_2153_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_2154_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_2155_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_2156_power__le__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_real @ A @ zero_zero_real ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_real ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_2157_power__le__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_rat @ A @ zero_zero_rat ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_rat ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_2158_power__le__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ( ord_less_nat @ zero_zero_nat @ ( numeral_numeral_nat @ W ) )
        & ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( ord_less_eq_int @ A @ zero_zero_int ) )
          | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
            & ( A = zero_zero_int ) ) ) ) ) ).

% power_le_zero_eq_numeral
thf(fact_2159_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_2160_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_2161_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_2162_arith__geo__mean,axiom,
    ! [U: real,X4: real,Y: real] :
      ( ( ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_real @ X4 @ Y ) )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ord_less_eq_real @ U @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_2163_arith__geo__mean,axiom,
    ! [U: rat,X4: rat,Y: rat] :
      ( ( ( power_power_rat @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( times_times_rat @ X4 @ Y ) )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
         => ( ord_less_eq_rat @ U @ ( divide_divide_rat @ ( plus_plus_rat @ X4 @ Y ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arith_geo_mean
thf(fact_2164_semiring__norm_I13_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ).

% semiring_norm(13)
thf(fact_2165_semiring__norm_I11_J,axiom,
    ! [M: num] :
      ( ( times_times_num @ M @ one )
      = M ) ).

% semiring_norm(11)
thf(fact_2166_semiring__norm_I12_J,axiom,
    ! [N2: num] :
      ( ( times_times_num @ one @ N2 )
      = N2 ) ).

% semiring_norm(12)
thf(fact_2167_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_2168_mult__0__right,axiom,
    ! [M: nat] :
      ( ( times_times_nat @ M @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_0_right
thf(fact_2169_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_2170_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_2171_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_2172_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_2173_mult__zero__left,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ zero_zero_rat @ A )
      = zero_zero_rat ) ).

% mult_zero_left
thf(fact_2174_mult__zero__left,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ zero_zero_complex @ A )
      = zero_zero_complex ) ).

% mult_zero_left
thf(fact_2175_mult__zero__left,axiom,
    ! [A: real] :
      ( ( times_times_real @ zero_zero_real @ A )
      = zero_zero_real ) ).

% mult_zero_left
thf(fact_2176_mult__zero__left,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% mult_zero_left
thf(fact_2177_mult__zero__left,axiom,
    ! [A: int] :
      ( ( times_times_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% mult_zero_left
thf(fact_2178_mult__zero__right,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ zero_zero_rat )
      = zero_zero_rat ) ).

% mult_zero_right
thf(fact_2179_mult__zero__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ zero_zero_complex )
      = zero_zero_complex ) ).

% mult_zero_right
thf(fact_2180_mult__zero__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ zero_zero_real )
      = zero_zero_real ) ).

% mult_zero_right
thf(fact_2181_mult__zero__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% mult_zero_right
thf(fact_2182_mult__zero__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% mult_zero_right
thf(fact_2183_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_2184_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_2185_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_2186_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_2187_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_2188_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_2189_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_2190_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_2191_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_2192_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_2193_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_2194_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_2195_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_2196_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_2197_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_2198_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ M ) @ ( numera1916890842035813515d_enat @ N2 ) )
      = ( numera1916890842035813515d_enat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2199_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2200_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2201_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2202_numeral__times__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% numeral_times_numeral
thf(fact_2203_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ V ) @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ W ) @ Z ) )
      = ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2204_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Z ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2205_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Z ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2206_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ V ) @ ( times_times_nat @ ( numeral_numeral_nat @ W ) @ Z ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2207_mult__numeral__left__semiring__numeral,axiom,
    ! [V: num,W: num,Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Z ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Z ) ) ).

% mult_numeral_left_semiring_numeral
thf(fact_2208_mult__1,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ one_one_rat @ A )
      = A ) ).

% mult_1
thf(fact_2209_mult__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ one_one_complex @ A )
      = A ) ).

% mult_1
thf(fact_2210_mult__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ one_one_real @ A )
      = A ) ).

% mult_1
thf(fact_2211_mult__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ one_one_nat @ A )
      = A ) ).

% mult_1
thf(fact_2212_mult__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ one_one_int @ A )
      = A ) ).

% mult_1
thf(fact_2213_mult_Oright__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.right_neutral
thf(fact_2214_mult_Oright__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.right_neutral
thf(fact_2215_mult_Oright__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.right_neutral
thf(fact_2216_mult_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.right_neutral
thf(fact_2217_mult_Oright__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.right_neutral
thf(fact_2218_num__double,axiom,
    ! [N2: num] :
      ( ( times_times_num @ ( bit0 @ one ) @ N2 )
      = ( bit0 @ N2 ) ) ).

% num_double
thf(fact_2219_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_2220_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_2221_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_2222_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_2223_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_2224_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_2225_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_2226_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_2227_dvd__0__right,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ zero_z3403309356797280102nteger ) ).

% dvd_0_right
thf(fact_2228_dvd__0__right,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ A @ zero_zero_complex ) ).

% dvd_0_right
thf(fact_2229_dvd__0__right,axiom,
    ! [A: real] : ( dvd_dvd_real @ A @ zero_zero_real ) ).

% dvd_0_right
thf(fact_2230_dvd__0__right,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ A @ zero_zero_rat ) ).

% dvd_0_right
thf(fact_2231_dvd__0__right,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ zero_zero_nat ) ).

% dvd_0_right
thf(fact_2232_dvd__0__right,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ zero_zero_int ) ).

% dvd_0_right
thf(fact_2233_dvd__0__left__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left_iff
thf(fact_2234_dvd__0__left__iff,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
      = ( A = zero_zero_complex ) ) ).

% dvd_0_left_iff
thf(fact_2235_dvd__0__left__iff,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
      = ( A = zero_zero_real ) ) ).

% dvd_0_left_iff
thf(fact_2236_dvd__0__left__iff,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
      = ( A = zero_zero_rat ) ) ).

% dvd_0_left_iff
thf(fact_2237_dvd__0__left__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
      = ( A = zero_zero_nat ) ) ).

% dvd_0_left_iff
thf(fact_2238_dvd__0__left__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
      = ( A = zero_zero_int ) ) ).

% dvd_0_left_iff
thf(fact_2239_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_2240_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_2241_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_2242_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_2243_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_2244_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_2245_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_2246_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_2247_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_2248_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_2249_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_2250_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_2251_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_2252_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_2253_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_2254_power__mult__numeral,axiom,
    ! [A: nat,M: num,N2: num] :
      ( ( power_power_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_2255_power__mult__numeral,axiom,
    ! [A: real,M: num,N2: num] :
      ( ( power_power_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_2256_power__mult__numeral,axiom,
    ! [A: int,M: num,N2: num] :
      ( ( power_power_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_2257_power__mult__numeral,axiom,
    ! [A: complex,M: num,N2: num] :
      ( ( power_power_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% power_mult_numeral
thf(fact_2258_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_2259_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_2260_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_2261_not__real__square__gt__zero,axiom,
    ! [X4: real] :
      ( ( ~ ( ord_less_real @ zero_zero_real @ ( times_times_real @ X4 @ X4 ) ) )
      = ( X4 = zero_zero_real ) ) ).

% not_real_square_gt_zero
thf(fact_2262_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_2263_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_2264_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_2265_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_2266_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_2267_of__bool__less__eq__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( P
       => Q ) ) ).

% of_bool_less_eq_iff
thf(fact_2268_of__bool__eq_I1_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $false )
    = zero_zero_complex ) ).

% of_bool_eq(1)
thf(fact_2269_of__bool__eq_I1_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $false )
    = zero_zero_real ) ).

% of_bool_eq(1)
thf(fact_2270_of__bool__eq_I1_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $false )
    = zero_zero_rat ) ).

% of_bool_eq(1)
thf(fact_2271_of__bool__eq_I1_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $false )
    = zero_zero_nat ) ).

% of_bool_eq(1)
thf(fact_2272_of__bool__eq_I1_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $false )
    = zero_zero_int ) ).

% of_bool_eq(1)
thf(fact_2273_of__bool__eq_I1_J,axiom,
    ( ( zero_n356916108424825756nteger @ $false )
    = zero_z3403309356797280102nteger ) ).

% of_bool_eq(1)
thf(fact_2274_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = zero_zero_complex )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2275_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = zero_zero_real )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2276_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = zero_zero_rat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2277_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = zero_zero_nat )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2278_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = zero_zero_int )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2279_of__bool__eq__0__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = zero_z3403309356797280102nteger )
      = ~ P ) ).

% of_bool_eq_0_iff
thf(fact_2280_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_2281_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_2282_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_2283_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_2284_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_2285_of__bool__less__iff,axiom,
    ! [P: $o,Q: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) )
      = ( ~ P
        & Q ) ) ).

% of_bool_less_iff
thf(fact_2286_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n1201886186963655149omplex @ P )
        = one_one_complex )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2287_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n3304061248610475627l_real @ P )
        = one_one_real )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2288_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2052037380579107095ol_rat @ P )
        = one_one_rat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2289_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P )
        = one_one_nat )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2290_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P )
        = one_one_int )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2291_of__bool__eq__1__iff,axiom,
    ! [P: $o] :
      ( ( ( zero_n356916108424825756nteger @ P )
        = one_one_Code_integer )
      = P ) ).

% of_bool_eq_1_iff
thf(fact_2292_of__bool__eq_I2_J,axiom,
    ( ( zero_n1201886186963655149omplex @ $true )
    = one_one_complex ) ).

% of_bool_eq(2)
thf(fact_2293_of__bool__eq_I2_J,axiom,
    ( ( zero_n3304061248610475627l_real @ $true )
    = one_one_real ) ).

% of_bool_eq(2)
thf(fact_2294_of__bool__eq_I2_J,axiom,
    ( ( zero_n2052037380579107095ol_rat @ $true )
    = one_one_rat ) ).

% of_bool_eq(2)
thf(fact_2295_of__bool__eq_I2_J,axiom,
    ( ( zero_n2687167440665602831ol_nat @ $true )
    = one_one_nat ) ).

% of_bool_eq(2)
thf(fact_2296_of__bool__eq_I2_J,axiom,
    ( ( zero_n2684676970156552555ol_int @ $true )
    = one_one_int ) ).

% of_bool_eq(2)
thf(fact_2297_of__bool__eq_I2_J,axiom,
    ( ( zero_n356916108424825756nteger @ $true )
    = one_one_Code_integer ) ).

% of_bool_eq(2)
thf(fact_2298_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_2299_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_2300_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_2301_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_2302_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_2303_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_2304_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_2305_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_2306_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_2307_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_2308_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_2309_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_2310_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_2311_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_2312_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_2313_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_2314_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_2315_sum__squares__eq__zero__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) )
        = zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2316_sum__squares__eq__zero__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) )
        = zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2317_sum__squares__eq__zero__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) )
        = zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_eq_zero_iff
thf(fact_2318_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_2319_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_2320_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_2321_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_2322_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_2323_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_2324_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_2325_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_2326_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_2327_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_2328_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_2329_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_2330_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_2331_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_2332_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_2333_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_2334_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_2335_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_2336_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_2337_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_2338_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_2339_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_2340_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_2341_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_2342_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_2343_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_2344_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_2345_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_2346_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_2347_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_2348_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_2349_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_2350_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_2351_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_2352_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_2353_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_2354_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_2355_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_2356_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_2357_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_2358_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_2359_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_2360_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_2361_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_2362_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_2363_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_2364_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_2365_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_2366_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_2367_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_2368_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_2369_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_2370_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_2371_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_2372_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_2373_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_2374_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_2375_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_2376_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_2377_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_2378_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_2379_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_2380_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_2381_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_2382_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_2383_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_2384_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_2385_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_2386_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_2387_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_2388_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_2389_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_2390_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_2391_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_2392_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_2393_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_2394_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_2395_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_2396_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_2397_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_2398_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_2399_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_2400_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_2401_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_2402_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_2403_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_2404_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_2405_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_2406_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_2407_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_2408_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_2409_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_2410_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_2411_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_2412_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_2413_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_2414_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_2415_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_2416_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_2417_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_2418_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_2419_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_2420_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_2421_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_2422_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_2423_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_2424_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_2425_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_2426_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_2427_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_2428_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_2429_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_2430_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_2431_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_2432_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_2433_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_2434_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_2435_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_2436_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_2437_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_2438_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_2439_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_2440_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_2441_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_2442_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_2443_zero__less__of__bool__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) )
      = P ) ).

% zero_less_of_bool_iff
thf(fact_2444_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_2445_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_2446_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_2447_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_2448_of__bool__less__one__iff,axiom,
    ! [P: $o] :
      ( ( ord_le6747313008572928689nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer )
      = ~ P ) ).

% of_bool_less_one_iff
thf(fact_2449_dvd__1__left,axiom,
    ! [K: nat] : ( dvd_dvd_nat @ ( suc @ zero_zero_nat ) @ K ) ).

% dvd_1_left
thf(fact_2450_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_2451_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_2452_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_2453_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_2454_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_2455_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_2456_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_2457_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_2458_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2459_le__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% le_divide_eq_numeral1(1)
thf(fact_2460_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2461_divide__le__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_le_eq_numeral1(1)
thf(fact_2462_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2463_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( numeral_numeral_real @ W ) )
            = B ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2464_eq__divide__eq__numeral1_I1_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) )
            = B ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(1)
thf(fact_2465_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_rat @ W )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) )
        & ( ( ( numeral_numeral_rat @ W )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2466_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) )
        = A )
      = ( ( ( ( numeral_numeral_real @ W )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) )
        & ( ( ( numeral_numeral_real @ W )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2467_divide__eq__eq__numeral1_I1_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( numera6690914467698888265omplex @ W ) )
        = A )
      = ( ( ( ( numera6690914467698888265omplex @ W )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) ) )
        & ( ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(1)
thf(fact_2468_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2469_less__divide__eq__numeral1_I1_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) )
      = ( ord_less_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) @ B ) ) ).

% less_divide_eq_numeral1(1)
thf(fact_2470_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( numeral_numeral_rat @ W ) ) @ A )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( numeral_numeral_rat @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2471_divide__less__eq__numeral1_I1_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( numeral_numeral_real @ W ) ) @ A )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) ) ) ).

% divide_less_eq_numeral1(1)
thf(fact_2472_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_2473_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_2474_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_2475_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_2476_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_2477_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_2478_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_2479_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_2480_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_2481_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_2482_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_2483_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_2484_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_2485_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_2486_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_2487_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_2488_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_2489_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_2490_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_2491_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_2492_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_2493_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_2494_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_2495_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_2496_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_2497_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_2498_zmod__numeral__Bit0,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ) ).

% zmod_numeral_Bit0
thf(fact_2499_power__add__numeral2,axiom,
    ! [A: complex,M: num,N2: num,B: complex] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2500_power__add__numeral2,axiom,
    ! [A: real,M: num,N2: num,B: real] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2501_power__add__numeral2,axiom,
    ! [A: nat,M: num,N2: num,B: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2502_power__add__numeral2,axiom,
    ! [A: int,M: num,N2: num,B: int] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) @ B ) )
      = ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) @ B ) ) ).

% power_add_numeral2
thf(fact_2503_power__add__numeral,axiom,
    ! [A: complex,M: num,N2: num] :
      ( ( times_times_complex @ ( power_power_complex @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_complex @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2504_power__add__numeral,axiom,
    ! [A: real,M: num,N2: num] :
      ( ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_real @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2505_power__add__numeral,axiom,
    ! [A: nat,M: num,N2: num] :
      ( ( times_times_nat @ ( power_power_nat @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_nat @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_nat @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2506_power__add__numeral,axiom,
    ! [A: int,M: num,N2: num] :
      ( ( times_times_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ M ) ) @ ( power_power_int @ A @ ( numeral_numeral_nat @ N2 ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% power_add_numeral
thf(fact_2507_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_2508_zle__add1__eq__le,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% zle_add1_eq_le
thf(fact_2509_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_2510_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_2511_even__mult__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_2512_even__mult__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( times_times_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_2513_even__mult__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( times_times_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_mult_iff
thf(fact_2514_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_2515_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_2516_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_2517_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_2518_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_2519_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_2520_even__mod__2__iff,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_2521_even__mod__2__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_2522_even__mod__2__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_mod_2_iff
thf(fact_2523_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_2524_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_2525_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( zero_n2687167440665602831ol_nat @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_2526_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( zero_n2684676970156552555ol_int @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_2527_odd__of__bool__self,axiom,
    ! [P2: $o] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( zero_n356916108424825756nteger @ P2 ) ) )
      = P2 ) ).

% odd_of_bool_self
thf(fact_2528_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_2529_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_2530_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_2531_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_2532_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_2533_of__bool__half__eq__0,axiom,
    ! [B: $o] :
      ( ( divide6298287555418463151nteger @ ( zero_n356916108424825756nteger @ B ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = zero_z3403309356797280102nteger ) ).

% of_bool_half_eq_0
thf(fact_2534_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_2535_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_2536_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_2537_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_2538_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_2539_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_2540_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_2541_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_2542_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_2543_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_2544_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_2545_zero__le__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_real @ zero_zero_real @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_2546_zero__le__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_2547_zero__le__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_eq_int @ zero_zero_int @ A ) ) ) ) ).

% zero_le_power_eq_numeral
thf(fact_2548_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_2549_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_2550_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_2551_power__less__zero__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_real )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_2552_power__less__zero__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_rat )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_2553_power__less__zero__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) @ zero_zero_int )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
        & ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% power_less_zero_eq_numeral
thf(fact_2554_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_2555_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_2556_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_2557_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_2558_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_2559_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_2560_zero__less__power__eq__numeral,axiom,
    ! [A: real,W: num] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_real ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_real @ zero_zero_real @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_2561_zero__less__power__eq__numeral,axiom,
    ! [A: rat,W: num] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_rat ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_rat @ zero_zero_rat @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_2562_zero__less__power__eq__numeral,axiom,
    ! [A: int,W: num] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) )
      = ( ( ( numeral_numeral_nat @ W )
          = zero_zero_nat )
        | ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( A != zero_zero_int ) )
        | ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
          & ( ord_less_int @ zero_zero_int @ A ) ) ) ) ).

% zero_less_power_eq_numeral
thf(fact_2563_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_2564_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_2565_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_2566_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_2567_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_2568_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_2569_add1__zle__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z )
      = ( ord_less_int @ W @ Z ) ) ).

% add1_zle_eq
thf(fact_2570_zdvd__imp__le,axiom,
    ! [Z: int,N2: int] :
      ( ( dvd_dvd_int @ Z @ N2 )
     => ( ( ord_less_int @ zero_zero_int @ N2 )
       => ( ord_less_eq_int @ Z @ N2 ) ) ) ).

% zdvd_imp_le
thf(fact_2571_int__gr__induct,axiom,
    ! [K: int,I2: int,P: int > $o] :
      ( ( ord_less_int @ K @ I2 )
     => ( ( P @ ( plus_plus_int @ K @ one_one_int ) )
       => ( ! [I4: int] :
              ( ( ord_less_int @ K @ I4 )
             => ( ( P @ I4 )
               => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_gr_induct
thf(fact_2572_le__imp__0__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ Z ) ) ) ).

% le_imp_0_less
thf(fact_2573_zless__add1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ ( plus_plus_int @ Z @ one_one_int ) )
      = ( ( ord_less_int @ W @ Z )
        | ( W = Z ) ) ) ).

% zless_add1_eq
thf(fact_2574_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 )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ J3 ) ) ) ) ) ).

% split_zmod
thf(fact_2575_odd__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ Z ) @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% odd_less_0_iff
thf(fact_2576_q__pos__lemma,axiom,
    ! [B4: int,Q4: int,R4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R4 ) )
     => ( ( ord_less_int @ R4 @ B4 )
       => ( ( ord_less_int @ zero_zero_int @ B4 )
         => ( ord_less_eq_int @ zero_zero_int @ Q4 ) ) ) ) ).

% q_pos_lemma
thf(fact_2577_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_2578_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_2579_int__mod__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_neg_eq
thf(fact_2580_int__mod__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( modulo_modulo_int @ A @ B )
            = R3 ) ) ) ) ).

% int_mod_pos_eq
thf(fact_2581_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_2582_zless__imp__add1__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ W @ Z )
     => ( ord_less_eq_int @ ( plus_plus_int @ W @ one_one_int ) @ Z ) ) ).

% zless_imp_add1_zle
thf(fact_2583_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_2584_zdiv__mono2__lemma,axiom,
    ! [B: int,Q3: int,R3: int,B4: int,Q4: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R4 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R4 ) )
       => ( ( ord_less_int @ R4 @ B4 )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B )
               => ( ord_less_eq_int @ Q3 @ Q4 ) ) ) ) ) ) ) ).

% zdiv_mono2_lemma
thf(fact_2585_zmod__trivial__iff,axiom,
    ! [I2: int,K: int] :
      ( ( ( modulo_modulo_int @ I2 @ K )
        = I2 )
      = ( ( K = zero_zero_int )
        | ( ( ord_less_eq_int @ zero_zero_int @ I2 )
          & ( ord_less_int @ I2 @ K ) )
        | ( ( ord_less_eq_int @ I2 @ zero_zero_int )
          & ( ord_less_int @ K @ I2 ) ) ) ) ).

% zmod_trivial_iff
thf(fact_2586_zdiv__mono2__neg__lemma,axiom,
    ! [B: int,Q3: int,R3: int,B4: int,Q4: int,R4: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 )
        = ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R4 ) )
     => ( ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ B4 @ Q4 ) @ R4 ) @ zero_zero_int )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
           => ( ( ord_less_int @ zero_zero_int @ B4 )
             => ( ( ord_less_eq_int @ B4 @ B )
               => ( ord_less_eq_int @ Q4 @ Q3 ) ) ) ) ) ) ) ).

% zdiv_mono2_neg_lemma
thf(fact_2587_int__one__le__iff__zero__less,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ Z )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% int_one_le_iff_zero_less
thf(fact_2588_unique__quotient__lemma,axiom,
    ! [B: int,Q4: int,R4: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R4 )
       => ( ( ord_less_int @ R4 @ B )
         => ( ( ord_less_int @ R3 @ B )
           => ( ord_less_eq_int @ Q4 @ Q3 ) ) ) ) ) ).

% unique_quotient_lemma
thf(fact_2589_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_2590_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_2591_unique__quotient__lemma__neg,axiom,
    ! [B: int,Q4: int,R4: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ B @ Q4 ) @ R4 ) @ ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( ord_less_int @ B @ R4 )
           => ( ord_less_eq_int @ Q3 @ Q4 ) ) ) ) ) ).

% unique_quotient_lemma_neg
thf(fact_2592_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_2593_zmod__le__nonneg__dividend,axiom,
    ! [M: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ M )
     => ( ord_less_eq_int @ ( modulo_modulo_int @ M @ K ) @ M ) ) ).

% zmod_le_nonneg_dividend
thf(fact_2594_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_2595_verit__la__generic,axiom,
    ! [A: int,X4: int] :
      ( ( ord_less_eq_int @ A @ X4 )
      | ( A = X4 )
      | ( ord_less_eq_int @ X4 @ A ) ) ).

% verit_la_generic
thf(fact_2596_less__eq__int__code_I1_J,axiom,
    ord_less_eq_int @ zero_zero_int @ zero_zero_int ).

% less_eq_int_code(1)
thf(fact_2597_zmod__eq__0D,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
     => ? [Q2: int] :
          ( M
          = ( times_times_int @ D @ Q2 ) ) ) ).

% zmod_eq_0D
thf(fact_2598_zmod__eq__0__iff,axiom,
    ! [M: int,D: int] :
      ( ( ( modulo_modulo_int @ M @ D )
        = zero_zero_int )
      = ( ? [Q5: int] :
            ( M
            = ( times_times_int @ D @ Q5 ) ) ) ) ).

% zmod_eq_0_iff
thf(fact_2599_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_2600_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_2601_zmult__zless__mono2,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_int @ I2 @ J )
     => ( ( ord_less_int @ zero_zero_int @ K )
       => ( ord_less_int @ ( times_times_int @ K @ I2 ) @ ( times_times_int @ K @ J ) ) ) ) ).

% zmult_zless_mono2
thf(fact_2602_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_2603_less__int__code_I1_J,axiom,
    ~ ( ord_less_int @ zero_zero_int @ zero_zero_int ) ).

% less_int_code(1)
thf(fact_2604_iadd__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( plus_p3455044024723400733d_enat @ M @ N2 )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        & ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% iadd_is_0
thf(fact_2605_imult__is__0,axiom,
    ! [M: extended_enat,N2: extended_enat] :
      ( ( ( times_7803423173614009249d_enat @ M @ N2 )
        = zero_z5237406670263579293d_enat )
      = ( ( M = zero_z5237406670263579293d_enat )
        | ( N2 = zero_z5237406670263579293d_enat ) ) ) ).

% imult_is_0
thf(fact_2606_zero__one__enat__neq_I1_J,axiom,
    zero_z5237406670263579293d_enat != one_on7984719198319812577d_enat ).

% zero_one_enat_neq(1)
thf(fact_2607_int__ge__induct,axiom,
    ! [K: int,I2: int,P: int > $o] :
      ( ( ord_less_eq_int @ K @ I2 )
     => ( ( P @ K )
       => ( ! [I4: int] :
              ( ( ord_less_eq_int @ K @ I4 )
             => ( ( P @ I4 )
               => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_ge_induct
thf(fact_2608_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_2609_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_2610_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_2611_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_2612_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_2613_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_2614_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_2615_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_2616_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_2617_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_2618_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_2619_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_2620_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_2621_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_2622_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_2623_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_2624_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_2625_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_2626_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_2627_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_2628_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_2629_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_2630_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_2631_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_2632_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_2633_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_2634_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_2635_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_2636_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_2637_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_2638_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_2639_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_2640_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_2641_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_2642_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_2643_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_2644_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_2645_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_2646_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_2647_dvd__antisym,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ M @ N2 )
     => ( ( dvd_dvd_nat @ N2 @ M )
       => ( M = N2 ) ) ) ).

% dvd_antisym
thf(fact_2648_bezout__lemma__nat,axiom,
    ! [D: nat,A: nat,B: nat,X4: nat,Y: nat] :
      ( ( dvd_dvd_nat @ D @ A )
     => ( ( dvd_dvd_nat @ D @ B )
       => ( ( ( ( times_times_nat @ A @ X4 )
              = ( plus_plus_nat @ ( times_times_nat @ B @ Y ) @ D ) )
            | ( ( times_times_nat @ B @ X4 )
              = ( plus_plus_nat @ ( times_times_nat @ A @ Y ) @ D ) ) )
         => ? [X5: nat,Y3: nat] :
              ( ( dvd_dvd_nat @ D @ A )
              & ( dvd_dvd_nat @ D @ ( plus_plus_nat @ A @ B ) )
              & ( ( ( times_times_nat @ A @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ Y3 ) @ D ) )
                | ( ( times_times_nat @ ( plus_plus_nat @ A @ B ) @ X5 )
                  = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D ) ) ) ) ) ) ) ).

% bezout_lemma_nat
thf(fact_2649_bezout__add__nat,axiom,
    ! [A: nat,B: nat] :
    ? [D3: nat,X5: nat,Y3: nat] :
      ( ( dvd_dvd_nat @ D3 @ A )
      & ( dvd_dvd_nat @ D3 @ B )
      & ( ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) )
        | ( ( times_times_nat @ B @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ A @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_nat
thf(fact_2650_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_2651_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_2652_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_2653_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_2654_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_2655_dvdE,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ~ ! [K2: complex] :
            ( A
           != ( times_times_complex @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2656_dvdE,axiom,
    ! [B: real,A: real] :
      ( ( dvd_dvd_real @ B @ A )
     => ~ ! [K2: real] :
            ( A
           != ( times_times_real @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2657_dvdE,axiom,
    ! [B: nat,A: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ~ ! [K2: nat] :
            ( A
           != ( times_times_nat @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2658_dvdE,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ~ ! [K2: int] :
            ( A
           != ( times_times_int @ B @ K2 ) ) ) ).

% dvdE
thf(fact_2659_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_2660_dvdI,axiom,
    ! [A: complex,B: complex,K: complex] :
      ( ( A
        = ( times_times_complex @ B @ K ) )
     => ( dvd_dvd_complex @ B @ A ) ) ).

% dvdI
thf(fact_2661_dvdI,axiom,
    ! [A: real,B: real,K: real] :
      ( ( A
        = ( times_times_real @ B @ K ) )
     => ( dvd_dvd_real @ B @ A ) ) ).

% dvdI
thf(fact_2662_dvdI,axiom,
    ! [A: nat,B: nat,K: nat] :
      ( ( A
        = ( times_times_nat @ B @ K ) )
     => ( dvd_dvd_nat @ B @ A ) ) ).

% dvdI
thf(fact_2663_dvdI,axiom,
    ! [A: int,B: int,K: int] :
      ( ( A
        = ( times_times_int @ B @ K ) )
     => ( dvd_dvd_int @ B @ A ) ) ).

% dvdI
thf(fact_2664_dvd__def,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [B2: code_integer,A3: code_integer] :
        ? [K3: code_integer] :
          ( A3
          = ( times_3573771949741848930nteger @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2665_dvd__def,axiom,
    ( dvd_dvd_complex
    = ( ^ [B2: complex,A3: complex] :
        ? [K3: complex] :
          ( A3
          = ( times_times_complex @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2666_dvd__def,axiom,
    ( dvd_dvd_real
    = ( ^ [B2: real,A3: real] :
        ? [K3: real] :
          ( A3
          = ( times_times_real @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2667_dvd__def,axiom,
    ( dvd_dvd_nat
    = ( ^ [B2: nat,A3: nat] :
        ? [K3: nat] :
          ( A3
          = ( times_times_nat @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2668_dvd__def,axiom,
    ( dvd_dvd_int
    = ( ^ [B2: int,A3: int] :
        ? [K3: int] :
          ( A3
          = ( times_times_int @ B2 @ K3 ) ) ) ) ).

% dvd_def
thf(fact_2669_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_2670_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_2671_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_2672_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_2673_of__bool__conj,axiom,
    ! [P: $o,Q: $o] :
      ( ( zero_n356916108424825756nteger
        @ ( P
          & Q ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ P ) @ ( zero_n356916108424825756nteger @ Q ) ) ) ).

% of_bool_conj
thf(fact_2674_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_2675_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_2676_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_2677_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_2678_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_2679_dvd__refl,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ A @ A ) ).

% dvd_refl
thf(fact_2680_dvd__refl,axiom,
    ! [A: int] : ( dvd_dvd_int @ A @ A ) ).

% dvd_refl
thf(fact_2681_dvd__refl,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ A @ A ) ).

% dvd_refl
thf(fact_2682_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_2683_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_2684_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_2685_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_2686_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_2687_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_2688_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_2689_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_2690_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_2691_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_2692_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_2693_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_2694_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( ( zero_n2687167440665602831ol_nat @ P2 )
        = ( zero_n2687167440665602831ol_nat @ Q3 ) )
      = ( P2 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_2695_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( ( zero_n2684676970156552555ol_int @ P2 )
        = ( zero_n2684676970156552555ol_int @ Q3 ) )
      = ( P2 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_2696_of__bool__eq__iff,axiom,
    ! [P2: $o,Q3: $o] :
      ( ( ( zero_n356916108424825756nteger @ P2 )
        = ( zero_n356916108424825756nteger @ Q3 ) )
      = ( P2 = Q3 ) ) ).

% of_bool_eq_iff
thf(fact_2697_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_2698_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_2699_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_2700_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_2701_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_2702_dvd__triv__left,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2703_dvd__triv__left,axiom,
    ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2704_dvd__triv__left,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2705_dvd__triv__left,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2706_dvd__triv__left,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ A @ B ) ) ).

% dvd_triv_left
thf(fact_2707_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_2708_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_2709_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_2710_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_2711_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_2712_mult_Ocommute,axiom,
    ( times_times_complex
    = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_2713_mult_Ocommute,axiom,
    ( times_times_real
    = ( ^ [A3: real,B2: real] : ( times_times_real @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_2714_mult_Ocommute,axiom,
    ( times_times_nat
    = ( ^ [A3: nat,B2: nat] : ( times_times_nat @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_2715_mult_Ocommute,axiom,
    ( times_times_int
    = ( ^ [A3: int,B2: int] : ( times_times_int @ B2 @ A3 ) ) ) ).

% mult.commute
thf(fact_2716_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_2717_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_2718_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_2719_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_2720_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_2721_dvd__triv__right,axiom,
    ! [A: code_integer,B: code_integer] : ( dvd_dvd_Code_integer @ A @ ( times_3573771949741848930nteger @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2722_dvd__triv__right,axiom,
    ! [A: complex,B: complex] : ( dvd_dvd_complex @ A @ ( times_times_complex @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2723_dvd__triv__right,axiom,
    ! [A: real,B: real] : ( dvd_dvd_real @ A @ ( times_times_real @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2724_dvd__triv__right,axiom,
    ! [A: nat,B: nat] : ( dvd_dvd_nat @ A @ ( times_times_nat @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2725_dvd__triv__right,axiom,
    ! [A: int,B: int] : ( dvd_dvd_int @ A @ ( times_times_int @ B @ A ) ) ).

% dvd_triv_right
thf(fact_2726_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_2727_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_2728_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_2729_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_2730_bezout__add__strong__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [D3: nat,X5: nat,Y3: nat] :
          ( ( dvd_dvd_nat @ D3 @ A )
          & ( dvd_dvd_nat @ D3 @ B )
          & ( ( times_times_nat @ A @ X5 )
            = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ D3 ) ) ) ) ).

% bezout_add_strong_nat
thf(fact_2731_unit__dvdE,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [C3: code_integer] :
              ( B
             != ( times_3573771949741848930nteger @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_2732_unit__dvdE,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [C3: nat] :
              ( B
             != ( times_times_nat @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_2733_unit__dvdE,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [C3: int] :
              ( B
             != ( times_times_int @ A @ C3 ) ) ) ) ).

% unit_dvdE
thf(fact_2734_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_2735_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_2736_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_2737_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_2738_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_2739_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_2740_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_2741_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_2742_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_2743_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_2744_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_2745_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_2746_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_2747_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_2748_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_2749_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_2750_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_2751_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_2752_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_2753_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_2754_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_2755_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_2756_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_2757_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_2758_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_2759_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_2760_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_2761_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_2762_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_2763_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_2764_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_2765_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_2766_dvd__field__iff,axiom,
    ( dvd_dvd_complex
    = ( ^ [A3: complex,B2: complex] :
          ( ( A3 = zero_zero_complex )
         => ( B2 = zero_zero_complex ) ) ) ) ).

% dvd_field_iff
thf(fact_2767_dvd__field__iff,axiom,
    ( dvd_dvd_real
    = ( ^ [A3: real,B2: real] :
          ( ( A3 = zero_zero_real )
         => ( B2 = zero_zero_real ) ) ) ) ).

% dvd_field_iff
thf(fact_2768_dvd__field__iff,axiom,
    ( dvd_dvd_rat
    = ( ^ [A3: rat,B2: rat] :
          ( ( A3 = zero_zero_rat )
         => ( B2 = zero_zero_rat ) ) ) ) ).

% dvd_field_iff
thf(fact_2769_dvd__0__left,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ A )
     => ( A = zero_z3403309356797280102nteger ) ) ).

% dvd_0_left
thf(fact_2770_dvd__0__left,axiom,
    ! [A: complex] :
      ( ( dvd_dvd_complex @ zero_zero_complex @ A )
     => ( A = zero_zero_complex ) ) ).

% dvd_0_left
thf(fact_2771_dvd__0__left,axiom,
    ! [A: real] :
      ( ( dvd_dvd_real @ zero_zero_real @ A )
     => ( A = zero_zero_real ) ) ).

% dvd_0_left
thf(fact_2772_dvd__0__left,axiom,
    ! [A: rat] :
      ( ( dvd_dvd_rat @ zero_zero_rat @ A )
     => ( A = zero_zero_rat ) ) ).

% dvd_0_left
thf(fact_2773_dvd__0__left,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ zero_zero_nat @ A )
     => ( A = zero_zero_nat ) ) ).

% dvd_0_left
thf(fact_2774_dvd__0__left,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ zero_zero_int @ A )
     => ( A = zero_zero_int ) ) ).

% dvd_0_left
thf(fact_2775_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_2776_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_2777_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_2778_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_2779_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_2780_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_2781_one__dvd,axiom,
    ! [A: code_integer] : ( dvd_dvd_Code_integer @ one_one_Code_integer @ A ) ).

% one_dvd
thf(fact_2782_one__dvd,axiom,
    ! [A: complex] : ( dvd_dvd_complex @ one_one_complex @ A ) ).

% one_dvd
thf(fact_2783_one__dvd,axiom,
    ! [A: real] : ( dvd_dvd_real @ one_one_real @ A ) ).

% one_dvd
thf(fact_2784_one__dvd,axiom,
    ! [A: rat] : ( dvd_dvd_rat @ one_one_rat @ A ) ).

% one_dvd
thf(fact_2785_one__dvd,axiom,
    ! [A: nat] : ( dvd_dvd_nat @ one_one_nat @ A ) ).

% one_dvd
thf(fact_2786_one__dvd,axiom,
    ! [A: int] : ( dvd_dvd_int @ one_one_int @ A ) ).

% one_dvd
thf(fact_2787_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_2788_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_2789_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_2790_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_2791_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_2792_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_2793_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_2794_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_2795_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_2796_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_2797_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_2798_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_2799_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_2800_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_2801_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_2802_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_2803_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_2804_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_2805_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_2806_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_2807_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_2808_dvd__div__eq__cancel,axiom,
    ! [A: complex,C: complex,B: complex] :
      ( ( ( divide1717551699836669952omplex @ A @ C )
        = ( divide1717551699836669952omplex @ B @ C ) )
     => ( ( dvd_dvd_complex @ C @ A )
       => ( ( dvd_dvd_complex @ C @ B )
         => ( A = B ) ) ) ) ).

% dvd_div_eq_cancel
thf(fact_2809_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_2810_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_2811_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_2812_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_2813_dvd__div__eq__iff,axiom,
    ! [C: complex,A: complex,B: complex] :
      ( ( dvd_dvd_complex @ C @ A )
     => ( ( dvd_dvd_complex @ C @ B )
       => ( ( ( divide1717551699836669952omplex @ A @ C )
            = ( divide1717551699836669952omplex @ B @ C ) )
          = ( A = B ) ) ) ) ).

% dvd_div_eq_iff
thf(fact_2814_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_2815_dvd__power__same,axiom,
    ! [X4: code_integer,Y: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ X4 @ Y )
     => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ N2 ) @ ( power_8256067586552552935nteger @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_2816_dvd__power__same,axiom,
    ! [X4: nat,Y: nat,N2: nat] :
      ( ( dvd_dvd_nat @ X4 @ Y )
     => ( dvd_dvd_nat @ ( power_power_nat @ X4 @ N2 ) @ ( power_power_nat @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_2817_dvd__power__same,axiom,
    ! [X4: real,Y: real,N2: nat] :
      ( ( dvd_dvd_real @ X4 @ Y )
     => ( dvd_dvd_real @ ( power_power_real @ X4 @ N2 ) @ ( power_power_real @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_2818_dvd__power__same,axiom,
    ! [X4: int,Y: int,N2: nat] :
      ( ( dvd_dvd_int @ X4 @ Y )
     => ( dvd_dvd_int @ ( power_power_int @ X4 @ N2 ) @ ( power_power_int @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_2819_dvd__power__same,axiom,
    ! [X4: complex,Y: complex,N2: nat] :
      ( ( dvd_dvd_complex @ X4 @ Y )
     => ( dvd_dvd_complex @ ( power_power_complex @ X4 @ N2 ) @ ( power_power_complex @ Y @ N2 ) ) ) ).

% dvd_power_same
thf(fact_2820_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_2821_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_2822_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_2823_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_2824_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_2825_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_2826_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_2827_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_2828_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_2829_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_2830_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_2831_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_2832_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_2833_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_2834_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_2835_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_2836_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_2837_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_2838_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_2839_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_2840_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_2841_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_2842_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_2843_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_2844_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_2845_mult_Ocomm__neutral,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ A @ one_one_rat )
      = A ) ).

% mult.comm_neutral
thf(fact_2846_mult_Ocomm__neutral,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ one_one_complex )
      = A ) ).

% mult.comm_neutral
thf(fact_2847_mult_Ocomm__neutral,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ one_one_real )
      = A ) ).

% mult.comm_neutral
thf(fact_2848_mult_Ocomm__neutral,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ one_one_nat )
      = A ) ).

% mult.comm_neutral
thf(fact_2849_mult_Ocomm__neutral,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ one_one_int )
      = A ) ).

% mult.comm_neutral
thf(fact_2850_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_2851_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_2852_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_2853_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_2854_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_2855_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_2856_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_2857_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_2858_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_2859_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_2860_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_2861_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_2862_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_2863_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_2864_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_2865_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_2866_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_2867_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_2868_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_2869_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_2870_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_2871_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_2872_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_2873_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_2874_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_2875_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_2876_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_2877_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_2878_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_2879_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_2880_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_2881_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_2882_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_2883_dvd__mod,axiom,
    ! [K: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ K @ M )
     => ( ( dvd_dvd_nat @ K @ N2 )
       => ( dvd_dvd_nat @ K @ ( modulo_modulo_nat @ M @ N2 ) ) ) ) ).

% dvd_mod
thf(fact_2884_dvd__mod,axiom,
    ! [K: int,M: int,N2: int] :
      ( ( dvd_dvd_int @ K @ M )
     => ( ( dvd_dvd_int @ K @ N2 )
       => ( dvd_dvd_int @ K @ ( modulo_modulo_int @ M @ N2 ) ) ) ) ).

% dvd_mod
thf(fact_2885_dvd__mod,axiom,
    ! [K: code_integer,M: code_integer,N2: code_integer] :
      ( ( dvd_dvd_Code_integer @ K @ M )
     => ( ( dvd_dvd_Code_integer @ K @ N2 )
       => ( dvd_dvd_Code_integer @ K @ ( modulo364778990260209775nteger @ M @ N2 ) ) ) ) ).

% dvd_mod
thf(fact_2886_mod__mod__cancel,axiom,
    ! [C: nat,B: nat,A: nat] :
      ( ( dvd_dvd_nat @ C @ B )
     => ( ( modulo_modulo_nat @ ( modulo_modulo_nat @ A @ B ) @ C )
        = ( modulo_modulo_nat @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2887_mod__mod__cancel,axiom,
    ! [C: int,B: int,A: int] :
      ( ( dvd_dvd_int @ C @ B )
     => ( ( modulo_modulo_int @ ( modulo_modulo_int @ A @ B ) @ C )
        = ( modulo_modulo_int @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2888_mod__mod__cancel,axiom,
    ! [C: code_integer,B: code_integer,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ C @ B )
     => ( ( modulo364778990260209775nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C )
        = ( modulo364778990260209775nteger @ A @ C ) ) ) ).

% mod_mod_cancel
thf(fact_2889_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_2890_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_2891_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_2892_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_2893_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_2894_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_2895_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_2896_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_2897_divide__divide__times__eq,axiom,
    ! [X4: real,Y: real,Z: real,W: real] :
      ( ( divide_divide_real @ ( divide_divide_real @ X4 @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X4 @ W ) @ ( times_times_real @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_2898_divide__divide__times__eq,axiom,
    ! [X4: complex,Y: complex,Z: complex,W: complex] :
      ( ( divide1717551699836669952omplex @ ( divide1717551699836669952omplex @ X4 @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X4 @ W ) @ ( times_times_complex @ Y @ Z ) ) ) ).

% divide_divide_times_eq
thf(fact_2899_times__divide__times__eq,axiom,
    ! [X4: real,Y: real,Z: real,W: real] :
      ( ( times_times_real @ ( divide_divide_real @ X4 @ Y ) @ ( divide_divide_real @ Z @ W ) )
      = ( divide_divide_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_2900_times__divide__times__eq,axiom,
    ! [X4: complex,Y: complex,Z: complex,W: complex] :
      ( ( times_times_complex @ ( divide1717551699836669952omplex @ X4 @ Y ) @ ( divide1717551699836669952omplex @ Z @ W ) )
      = ( divide1717551699836669952omplex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ Y @ W ) ) ) ).

% times_divide_times_eq
thf(fact_2901_power__commuting__commutes,axiom,
    ! [X4: complex,Y: complex,N2: nat] :
      ( ( ( times_times_complex @ X4 @ Y )
        = ( times_times_complex @ Y @ X4 ) )
     => ( ( times_times_complex @ ( power_power_complex @ X4 @ N2 ) @ Y )
        = ( times_times_complex @ Y @ ( power_power_complex @ X4 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2902_power__commuting__commutes,axiom,
    ! [X4: real,Y: real,N2: nat] :
      ( ( ( times_times_real @ X4 @ Y )
        = ( times_times_real @ Y @ X4 ) )
     => ( ( times_times_real @ ( power_power_real @ X4 @ N2 ) @ Y )
        = ( times_times_real @ Y @ ( power_power_real @ X4 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2903_power__commuting__commutes,axiom,
    ! [X4: nat,Y: nat,N2: nat] :
      ( ( ( times_times_nat @ X4 @ Y )
        = ( times_times_nat @ Y @ X4 ) )
     => ( ( times_times_nat @ ( power_power_nat @ X4 @ N2 ) @ Y )
        = ( times_times_nat @ Y @ ( power_power_nat @ X4 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2904_power__commuting__commutes,axiom,
    ! [X4: int,Y: int,N2: nat] :
      ( ( ( times_times_int @ X4 @ Y )
        = ( times_times_int @ Y @ X4 ) )
     => ( ( times_times_int @ ( power_power_int @ X4 @ N2 ) @ Y )
        = ( times_times_int @ Y @ ( power_power_int @ X4 @ N2 ) ) ) ) ).

% power_commuting_commutes
thf(fact_2905_power__mult__distrib,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( power_power_complex @ ( times_times_complex @ A @ B ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ ( power_power_complex @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2906_power__mult__distrib,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( times_times_real @ A @ B ) @ N2 )
      = ( times_times_real @ ( power_power_real @ A @ N2 ) @ ( power_power_real @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2907_power__mult__distrib,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( power_power_nat @ ( times_times_nat @ A @ B ) @ N2 )
      = ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2908_power__mult__distrib,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( times_times_int @ A @ B ) @ N2 )
      = ( times_times_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ).

% power_mult_distrib
thf(fact_2909_power__commutes,axiom,
    ! [A: complex,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ A @ N2 ) @ A )
      = ( times_times_complex @ A @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_2910_power__commutes,axiom,
    ! [A: real,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ A @ N2 ) @ A )
      = ( times_times_real @ A @ ( power_power_real @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_2911_power__commutes,axiom,
    ! [A: nat,N2: nat] :
      ( ( times_times_nat @ ( power_power_nat @ A @ N2 ) @ A )
      = ( times_times_nat @ A @ ( power_power_nat @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_2912_power__commutes,axiom,
    ! [A: int,N2: nat] :
      ( ( times_times_int @ ( power_power_int @ A @ N2 ) @ A )
      = ( times_times_int @ A @ ( power_power_int @ A @ N2 ) ) ) ).

% power_commutes
thf(fact_2913_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_2914_power__mult,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_nat @ ( power_power_nat @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_2915_power__mult,axiom,
    ! [A: real,M: nat,N2: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_real @ ( power_power_real @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_2916_power__mult,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_int @ ( power_power_int @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_2917_power__mult,axiom,
    ! [A: complex,M: nat,N2: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ M @ N2 ) )
      = ( power_power_complex @ ( power_power_complex @ A @ M ) @ N2 ) ) ).

% power_mult
thf(fact_2918_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X4: produc3368934014287244435at_num] :
      ~ ! [F2: nat > num > num,A5: nat,B5: nat,Acc: num] :
          ( X4
         != ( produc851828971589881931at_num @ F2 @ ( produc1195630363706982562at_num @ A5 @ ( product_Pair_nat_num @ B5 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_2919_fold__atLeastAtMost__nat_Ocases,axiom,
    ! [X4: produc4471711990508489141at_nat] :
      ~ ! [F2: nat > nat > nat,A5: nat,B5: nat,Acc: nat] :
          ( X4
         != ( produc3209952032786966637at_nat @ F2 @ ( produc487386426758144856at_nat @ A5 @ ( product_Pair_nat_nat @ B5 @ Acc ) ) ) ) ).

% fold_atLeastAtMost_nat.cases
thf(fact_2920_mult__0,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% mult_0
thf(fact_2921_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_2922_mult__le__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ).

% mult_le_mono2
thf(fact_2923_mult__le__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ).

% mult_le_mono1
thf(fact_2924_mult__le__mono,axiom,
    ! [I2: nat,J: nat,K: nat,L: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ K @ L )
       => ( ord_less_eq_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ L ) ) ) ) ).

% mult_le_mono
thf(fact_2925_le__square,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ M ) ) ).

% le_square
thf(fact_2926_le__cube,axiom,
    ! [M: nat] : ( ord_less_eq_nat @ M @ ( times_times_nat @ M @ ( times_times_nat @ M @ M ) ) ) ).

% le_cube
thf(fact_2927_mod__mult__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_2928_mod__mult__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_2929_mod__mult__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_eq
thf(fact_2930_mod__mult__cong,axiom,
    ! [A: nat,C: nat,A4: nat,B: nat,B4: nat] :
      ( ( ( modulo_modulo_nat @ A @ C )
        = ( modulo_modulo_nat @ A4 @ C ) )
     => ( ( ( modulo_modulo_nat @ B @ C )
          = ( modulo_modulo_nat @ B4 @ C ) )
       => ( ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C )
          = ( modulo_modulo_nat @ ( times_times_nat @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_2931_mod__mult__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( times_times_int @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_2932_mod__mult__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_mult_cong
thf(fact_2933_mod__mult__mult2,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ C ) @ ( times_times_nat @ B @ C ) )
      = ( times_times_nat @ ( modulo_modulo_nat @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_2934_mod__mult__mult2,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ C ) @ ( times_times_int @ B @ C ) )
      = ( times_times_int @ ( modulo_modulo_int @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_2935_mod__mult__mult2,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ C ) @ ( times_3573771949741848930nteger @ B @ C ) )
      = ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ B ) @ C ) ) ).

% mod_mult_mult2
thf(fact_2936_mult__mod__right,axiom,
    ! [C: nat,A: nat,B: nat] :
      ( ( times_times_nat @ C @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_nat @ ( times_times_nat @ C @ A ) @ ( times_times_nat @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_2937_mult__mod__right,axiom,
    ! [C: int,A: int,B: int] :
      ( ( times_times_int @ C @ ( modulo_modulo_int @ A @ B ) )
      = ( modulo_modulo_int @ ( times_times_int @ C @ A ) @ ( times_times_int @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_2938_mult__mod__right,axiom,
    ! [C: code_integer,A: code_integer,B: code_integer] :
      ( ( times_3573771949741848930nteger @ C @ ( modulo364778990260209775nteger @ A @ B ) )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ C @ A ) @ ( times_3573771949741848930nteger @ C @ B ) ) ) ).

% mult_mod_right
thf(fact_2939_mod__mult__left__eq,axiom,
    ! [A: nat,C: nat,B: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ ( modulo_modulo_nat @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_2940_mod__mult__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_2941_mod__mult__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_left_eq
thf(fact_2942_mod__mult__right__eq,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( modulo_modulo_nat @ ( times_times_nat @ A @ ( modulo_modulo_nat @ B @ C ) ) @ C )
      = ( modulo_modulo_nat @ ( times_times_nat @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_2943_mod__mult__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( times_times_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( times_times_int @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_2944_mod__mult__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( times_3573771949741848930nteger @ A @ B ) @ C ) ) ).

% mod_mult_right_eq
thf(fact_2945_left__add__mult__distrib,axiom,
    ! [I2: nat,U: nat,J: nat,K: nat] :
      ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ K ) )
      = ( plus_plus_nat @ ( times_times_nat @ ( plus_plus_nat @ I2 @ J ) @ U ) @ K ) ) ).

% left_add_mult_distrib
thf(fact_2946_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_2947_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_2948_nat__mult__1__right,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ N2 @ one_one_nat )
      = N2 ) ).

% nat_mult_1_right
thf(fact_2949_nat__mult__1,axiom,
    ! [N2: nat] :
      ( ( times_times_nat @ one_one_nat @ N2 )
      = N2 ) ).

% nat_mult_1
thf(fact_2950_div__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( divide_divide_nat @ M @ ( times_times_nat @ N2 @ Q3 ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ M @ N2 ) @ Q3 ) ) ).

% div_mult2_eq
thf(fact_2951_div__mod__decomp__int,axiom,
    ! [A2: int,N2: int] :
      ( A2
      = ( plus_plus_int @ ( times_times_int @ ( divide_divide_int @ A2 @ N2 ) @ N2 ) @ ( modulo_modulo_int @ A2 @ N2 ) ) ) ).

% div_mod_decomp_int
thf(fact_2952_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 )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) )
        & ( ( ord_less_int @ K @ zero_zero_int )
         => ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 ) ) ) ) ) ).

% split_zdiv
thf(fact_2953_int__div__neg__eq,axiom,
    ! [A: int,B: int,Q3: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ R3 @ zero_zero_int )
       => ( ( ord_less_int @ B @ R3 )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_neg_eq
thf(fact_2954_int__div__pos__eq,axiom,
    ! [A: int,B: int,Q3: int,R3: int] :
      ( ( A
        = ( plus_plus_int @ ( times_times_int @ B @ Q3 ) @ R3 ) )
     => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
       => ( ( ord_less_int @ R3 @ B )
         => ( ( divide_divide_int @ A @ B )
            = Q3 ) ) ) ) ).

% int_div_pos_eq
thf(fact_2955_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 ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_int @ K @ J3 )
                & ( ord_less_eq_int @ J3 @ zero_zero_int )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_neg_lemma
thf(fact_2956_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 ) )
        = ( ! [I3: int,J3: int] :
              ( ( ( ord_less_eq_int @ zero_zero_int @ J3 )
                & ( ord_less_int @ J3 @ K )
                & ( N2
                  = ( plus_plus_int @ ( times_times_int @ K @ I3 ) @ J3 ) ) )
             => ( P @ I3 @ J3 ) ) ) ) ) ).

% split_pos_lemma
thf(fact_2957_zdiv__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( divide_divide_int @ A @ ( times_times_int @ B @ C ) )
        = ( divide_divide_int @ ( divide_divide_int @ A @ B ) @ C ) ) ) ).

% zdiv_zmult2_eq
thf(fact_2958_zmod__zmult2__eq,axiom,
    ! [C: int,A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ C )
     => ( ( modulo_modulo_int @ A @ ( times_times_int @ B @ C ) )
        = ( plus_plus_int @ ( times_times_int @ B @ ( modulo_modulo_int @ ( divide_divide_int @ A @ B ) @ C ) ) @ ( modulo_modulo_int @ A @ B ) ) ) ) ).

% zmod_zmult2_eq
thf(fact_2959_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_2960_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_2961_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_2962_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_2963_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_2964_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_2965_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_2966_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_2967_is__unitE,axiom,
    ! [A: nat,C: nat] :
      ( ( dvd_dvd_nat @ A @ one_one_nat )
     => ~ ( ( A != zero_zero_nat )
         => ! [B5: nat] :
              ( ( B5 != zero_zero_nat )
             => ( ( dvd_dvd_nat @ B5 @ one_one_nat )
               => ( ( ( divide_divide_nat @ one_one_nat @ A )
                    = B5 )
                 => ( ( ( divide_divide_nat @ one_one_nat @ B5 )
                      = A )
                   => ( ( ( times_times_nat @ A @ B5 )
                        = one_one_nat )
                     => ( ( divide_divide_nat @ C @ A )
                       != ( times_times_nat @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2968_is__unitE,axiom,
    ! [A: int,C: int] :
      ( ( dvd_dvd_int @ A @ one_one_int )
     => ~ ( ( A != zero_zero_int )
         => ! [B5: int] :
              ( ( B5 != zero_zero_int )
             => ( ( dvd_dvd_int @ B5 @ one_one_int )
               => ( ( ( divide_divide_int @ one_one_int @ A )
                    = B5 )
                 => ( ( ( divide_divide_int @ one_one_int @ B5 )
                      = A )
                   => ( ( ( times_times_int @ A @ B5 )
                        = one_one_int )
                     => ( ( divide_divide_int @ C @ A )
                       != ( times_times_int @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2969_is__unitE,axiom,
    ! [A: code_integer,C: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ one_one_Code_integer )
     => ~ ( ( A != zero_z3403309356797280102nteger )
         => ! [B5: code_integer] :
              ( ( B5 != zero_z3403309356797280102nteger )
             => ( ( dvd_dvd_Code_integer @ B5 @ one_one_Code_integer )
               => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ A )
                    = B5 )
                 => ( ( ( divide6298287555418463151nteger @ one_one_Code_integer @ B5 )
                      = A )
                   => ( ( ( times_3573771949741848930nteger @ A @ B5 )
                        = one_one_Code_integer )
                     => ( ( divide6298287555418463151nteger @ C @ A )
                       != ( times_3573771949741848930nteger @ C @ B5 ) ) ) ) ) ) ) ) ) ).

% is_unitE
thf(fact_2970_evenE,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_2971_evenE,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_2972_evenE,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) ) ) ).

% evenE
thf(fact_2973_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_2974_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_2975_of__bool__odd__eq__mod__2,axiom,
    ! [A: nat] :
      ( ( zero_n2687167440665602831ol_nat
        @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_2976_of__bool__odd__eq__mod__2,axiom,
    ! [A: int] :
      ( ( zero_n2684676970156552555ol_int
        @ ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_2977_of__bool__odd__eq__mod__2,axiom,
    ! [A: code_integer] :
      ( ( zero_n356916108424825756nteger
        @ ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% of_bool_odd_eq_mod_2
thf(fact_2978_even__two__times__div__two,axiom,
    ! [A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_2979_even__two__times__div__two,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_2980_even__two__times__div__two,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) )
        = A ) ) ).

% even_two_times_div_two
thf(fact_2981_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_2982_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_2983_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_2984_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_2985_zero__less__eq__of__bool,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( zero_n356916108424825756nteger @ P ) ) ).

% zero_less_eq_of_bool
thf(fact_2986_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_2987_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_2988_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_2989_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_2990_of__bool__less__eq__one,axiom,
    ! [P: $o] : ( ord_le3102999989581377725nteger @ ( zero_n356916108424825756nteger @ P ) @ one_one_Code_integer ) ).

% of_bool_less_eq_one
thf(fact_2991_split__of__bool__asm,axiom,
    ! [P: complex > $o,P2: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_complex ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_complex ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2992_split__of__bool__asm,axiom,
    ! [P: real > $o,P2: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_real ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_real ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2993_split__of__bool__asm,axiom,
    ! [P: rat > $o,P2: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_rat ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_rat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2994_split__of__bool__asm,axiom,
    ! [P: nat > $o,P2: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_nat ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_nat ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2995_split__of__bool__asm,axiom,
    ! [P: int > $o,P2: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_int ) )
            | ( ~ P2
              & ~ ( P @ zero_zero_int ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2996_split__of__bool__asm,axiom,
    ! [P: code_integer > $o,P2: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
      = ( ~ ( ( P2
              & ~ ( P @ one_one_Code_integer ) )
            | ( ~ P2
              & ~ ( P @ zero_z3403309356797280102nteger ) ) ) ) ) ).

% split_of_bool_asm
thf(fact_2997_split__of__bool,axiom,
    ! [P: complex > $o,P2: $o] :
      ( ( P @ ( zero_n1201886186963655149omplex @ P2 ) )
      = ( ( P2
         => ( P @ one_one_complex ) )
        & ( ~ P2
         => ( P @ zero_zero_complex ) ) ) ) ).

% split_of_bool
thf(fact_2998_split__of__bool,axiom,
    ! [P: real > $o,P2: $o] :
      ( ( P @ ( zero_n3304061248610475627l_real @ P2 ) )
      = ( ( P2
         => ( P @ one_one_real ) )
        & ( ~ P2
         => ( P @ zero_zero_real ) ) ) ) ).

% split_of_bool
thf(fact_2999_split__of__bool,axiom,
    ! [P: rat > $o,P2: $o] :
      ( ( P @ ( zero_n2052037380579107095ol_rat @ P2 ) )
      = ( ( P2
         => ( P @ one_one_rat ) )
        & ( ~ P2
         => ( P @ zero_zero_rat ) ) ) ) ).

% split_of_bool
thf(fact_3000_split__of__bool,axiom,
    ! [P: nat > $o,P2: $o] :
      ( ( P @ ( zero_n2687167440665602831ol_nat @ P2 ) )
      = ( ( P2
         => ( P @ one_one_nat ) )
        & ( ~ P2
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_of_bool
thf(fact_3001_split__of__bool,axiom,
    ! [P: int > $o,P2: $o] :
      ( ( P @ ( zero_n2684676970156552555ol_int @ P2 ) )
      = ( ( P2
         => ( P @ one_one_int ) )
        & ( ~ P2
         => ( P @ zero_zero_int ) ) ) ) ).

% split_of_bool
thf(fact_3002_split__of__bool,axiom,
    ! [P: code_integer > $o,P2: $o] :
      ( ( P @ ( zero_n356916108424825756nteger @ P2 ) )
      = ( ( P2
         => ( P @ one_one_Code_integer ) )
        & ( ~ P2
         => ( P @ zero_z3403309356797280102nteger ) ) ) ) ).

% split_of_bool
thf(fact_3003_of__bool__def,axiom,
    ( zero_n1201886186963655149omplex
    = ( ^ [P5: $o] : ( if_complex @ P5 @ one_one_complex @ zero_zero_complex ) ) ) ).

% of_bool_def
thf(fact_3004_of__bool__def,axiom,
    ( zero_n3304061248610475627l_real
    = ( ^ [P5: $o] : ( if_real @ P5 @ one_one_real @ zero_zero_real ) ) ) ).

% of_bool_def
thf(fact_3005_of__bool__def,axiom,
    ( zero_n2052037380579107095ol_rat
    = ( ^ [P5: $o] : ( if_rat @ P5 @ one_one_rat @ zero_zero_rat ) ) ) ).

% of_bool_def
thf(fact_3006_of__bool__def,axiom,
    ( zero_n2687167440665602831ol_nat
    = ( ^ [P5: $o] : ( if_nat @ P5 @ one_one_nat @ zero_zero_nat ) ) ) ).

% of_bool_def
thf(fact_3007_of__bool__def,axiom,
    ( zero_n2684676970156552555ol_int
    = ( ^ [P5: $o] : ( if_int @ P5 @ one_one_int @ zero_zero_int ) ) ) ).

% of_bool_def
thf(fact_3008_of__bool__def,axiom,
    ( zero_n356916108424825756nteger
    = ( ^ [P5: $o] : ( if_Code_integer @ P5 @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) ) ).

% of_bool_def
thf(fact_3009_not__is__unit__0,axiom,
    ~ ( dvd_dvd_Code_integer @ zero_z3403309356797280102nteger @ one_one_Code_integer ) ).

% not_is_unit_0
thf(fact_3010_not__is__unit__0,axiom,
    ~ ( dvd_dvd_nat @ zero_zero_nat @ one_one_nat ) ).

% not_is_unit_0
thf(fact_3011_not__is__unit__0,axiom,
    ~ ( dvd_dvd_int @ zero_zero_int @ one_one_int ) ).

% not_is_unit_0
thf(fact_3012_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_3013_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_3014_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_3015_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_3016_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_3017_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_3018_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_3019_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_3020_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_3021_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_3022_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_3023_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_3024_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_3025_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_3026_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_3027_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_3028_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_3029_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_3030_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_3031_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_3032_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_3033_div__power,axiom,
    ! [B: nat,A: nat,N2: nat] :
      ( ( dvd_dvd_nat @ B @ A )
     => ( ( power_power_nat @ ( divide_divide_nat @ A @ B ) @ N2 )
        = ( divide_divide_nat @ ( power_power_nat @ A @ N2 ) @ ( power_power_nat @ B @ N2 ) ) ) ) ).

% div_power
thf(fact_3034_div__power,axiom,
    ! [B: int,A: int,N2: nat] :
      ( ( dvd_dvd_int @ B @ A )
     => ( ( power_power_int @ ( divide_divide_int @ A @ B ) @ N2 )
        = ( divide_divide_int @ ( power_power_int @ A @ N2 ) @ ( power_power_int @ B @ N2 ) ) ) ) ).

% div_power
thf(fact_3035_div__power,axiom,
    ! [B: code_integer,A: code_integer,N2: nat] :
      ( ( dvd_dvd_Code_integer @ B @ A )
     => ( ( power_8256067586552552935nteger @ ( divide6298287555418463151nteger @ A @ B ) @ N2 )
        = ( divide6298287555418463151nteger @ ( power_8256067586552552935nteger @ A @ N2 ) @ ( power_8256067586552552935nteger @ B @ N2 ) ) ) ) ).

% div_power
thf(fact_3036_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_3037_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_3038_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_3039_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_nat
    = ( ^ [A3: nat,B2: nat] :
          ( ( modulo_modulo_nat @ B2 @ A3 )
          = zero_zero_nat ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3040_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_int
    = ( ^ [A3: int,B2: int] :
          ( ( modulo_modulo_int @ B2 @ A3 )
          = zero_zero_int ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3041_dvd__eq__mod__eq__0,axiom,
    ( dvd_dvd_Code_integer
    = ( ^ [A3: code_integer,B2: code_integer] :
          ( ( modulo364778990260209775nteger @ B2 @ A3 )
          = zero_z3403309356797280102nteger ) ) ) ).

% dvd_eq_mod_eq_0
thf(fact_3042_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_3043_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_3044_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_3045_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_3046_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_3047_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_3048_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_3049_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_3050_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_3051_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_3052_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_3053_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_3054_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_3055_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_3056_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_3057_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_3058_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_3059_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_3060_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_3061_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_3062_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_3063_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_3064_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_3065_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_3066_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_3067_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_3068_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_3069_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_3070_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_3071_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_3072_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_3073_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_3074_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_3075_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_3076_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_3077_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_3078_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_3079_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_3080_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_3081_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_3082_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_3083_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_3084_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_3085_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_3086_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_3087_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_3088_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_3089_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_3090_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_3091_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_3092_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_3093_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_3094_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_3095_zero__le__square,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( times_times_real @ A @ A ) ) ).

% zero_le_square
thf(fact_3096_zero__le__square,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( times_times_rat @ A @ A ) ) ).

% zero_le_square
thf(fact_3097_zero__le__square,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( times_times_int @ A @ A ) ) ).

% zero_le_square
thf(fact_3098_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_3099_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_3100_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_3101_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_3102_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_3103_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_3104_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_3105_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_3106_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_3107_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_3108_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_3109_not__square__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( times_times_real @ A @ A ) @ zero_zero_real ) ).

% not_square_less_zero
thf(fact_3110_not__square__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( times_times_rat @ A @ A ) @ zero_zero_rat ) ).

% not_square_less_zero
thf(fact_3111_not__square__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( times_times_int @ A @ A ) @ zero_zero_int ) ).

% not_square_less_zero
thf(fact_3112_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_3113_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_3114_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_3115_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_3116_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_3117_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_3118_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_3119_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_3120_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_3121_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_3122_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_3123_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_3124_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_3125_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_3126_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_3127_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_3128_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_3129_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_3130_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_3131_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_3132_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_3133_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_3134_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_3135_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_3136_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_3137_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_3138_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_3139_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_3140_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_3141_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_3142_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_3143_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_3144_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_3145_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_3146_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_3147_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_3148_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_3149_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_3150_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_3151_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_3152_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_3153_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_3154_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_3155_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_3156_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_3157_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_3158_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_3159_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_3160_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_3161_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_3162_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_3163_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_3164_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_3165_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_3166_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_3167_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_3168_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_3169_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_3170_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_3171_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_3172_dvd__power__le,axiom,
    ! [X4: code_integer,Y: code_integer,N2: nat,M: nat] :
      ( ( dvd_dvd_Code_integer @ X4 @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ N2 ) @ ( power_8256067586552552935nteger @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3173_dvd__power__le,axiom,
    ! [X4: nat,Y: nat,N2: nat,M: nat] :
      ( ( dvd_dvd_nat @ X4 @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_nat @ ( power_power_nat @ X4 @ N2 ) @ ( power_power_nat @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3174_dvd__power__le,axiom,
    ! [X4: real,Y: real,N2: nat,M: nat] :
      ( ( dvd_dvd_real @ X4 @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_real @ ( power_power_real @ X4 @ N2 ) @ ( power_power_real @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3175_dvd__power__le,axiom,
    ! [X4: int,Y: int,N2: nat,M: nat] :
      ( ( dvd_dvd_int @ X4 @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_int @ ( power_power_int @ X4 @ N2 ) @ ( power_power_int @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3176_dvd__power__le,axiom,
    ! [X4: complex,Y: complex,N2: nat,M: nat] :
      ( ( dvd_dvd_complex @ X4 @ Y )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ( dvd_dvd_complex @ ( power_power_complex @ X4 @ N2 ) @ ( power_power_complex @ Y @ M ) ) ) ) ).

% dvd_power_le
thf(fact_3177_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_3178_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_3179_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_3180_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_3181_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_3182_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_3183_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_3184_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_3185_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_3186_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_3187_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_3188_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_3189_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_3190_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_3191_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_3192_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_3193_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_3194_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_3195_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_3196_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_3197_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_3198_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_3199_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_3200_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_3201_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_3202_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_3203_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_3204_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_3205_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_3206_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_3207_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_3208_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_3209_frac__eq__eq,axiom,
    ! [Y: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ( divide_divide_rat @ X4 @ Y )
            = ( divide_divide_rat @ W @ Z ) )
          = ( ( times_times_rat @ X4 @ Z )
            = ( times_times_rat @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3210_frac__eq__eq,axiom,
    ! [Y: real,Z: real,X4: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ( divide_divide_real @ X4 @ Y )
            = ( divide_divide_real @ W @ Z ) )
          = ( ( times_times_real @ X4 @ Z )
            = ( times_times_real @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3211_frac__eq__eq,axiom,
    ! [Y: complex,Z: complex,X4: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( ( divide1717551699836669952omplex @ X4 @ Y )
            = ( divide1717551699836669952omplex @ W @ Z ) )
          = ( ( times_times_complex @ X4 @ Z )
            = ( times_times_complex @ W @ Y ) ) ) ) ) ).

% frac_eq_eq
thf(fact_3212_mult__numeral__1__right,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ A @ ( numera1916890842035813515d_enat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3213_mult__numeral__1__right,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ A @ ( numera6690914467698888265omplex @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3214_mult__numeral__1__right,axiom,
    ! [A: real] :
      ( ( times_times_real @ A @ ( numeral_numeral_real @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3215_mult__numeral__1__right,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ A @ ( numeral_numeral_nat @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3216_mult__numeral__1__right,axiom,
    ! [A: int] :
      ( ( times_times_int @ A @ ( numeral_numeral_int @ one ) )
      = A ) ).

% mult_numeral_1_right
thf(fact_3217_mult__numeral__1,axiom,
    ! [A: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3218_mult__numeral__1,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3219_mult__numeral__1,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3220_mult__numeral__1,axiom,
    ! [A: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3221_mult__numeral__1,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ one ) @ A )
      = A ) ).

% mult_numeral_1
thf(fact_3222_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_3223_div__mult2__numeral__eq,axiom,
    ! [A: nat,K: num,L: num] :
      ( ( divide_divide_nat @ ( divide_divide_nat @ A @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ L ) )
      = ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_3224_div__mult2__numeral__eq,axiom,
    ! [A: int,K: num,L: num] :
      ( ( divide_divide_int @ ( divide_divide_int @ A @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ L ) )
      = ( divide_divide_int @ A @ ( numeral_numeral_int @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_3225_div__mult2__numeral__eq,axiom,
    ! [A: code_integer,K: num,L: num] :
      ( ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ K ) ) @ ( numera6620942414471956472nteger @ L ) )
      = ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( times_times_num @ K @ L ) ) ) ) ).

% div_mult2_numeral_eq
thf(fact_3226_left__right__inverse__power,axiom,
    ! [X4: rat,Y: rat,N2: nat] :
      ( ( ( times_times_rat @ X4 @ Y )
        = one_one_rat )
     => ( ( times_times_rat @ ( power_power_rat @ X4 @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
        = one_one_rat ) ) ).

% left_right_inverse_power
thf(fact_3227_left__right__inverse__power,axiom,
    ! [X4: complex,Y: complex,N2: nat] :
      ( ( ( times_times_complex @ X4 @ Y )
        = one_one_complex )
     => ( ( times_times_complex @ ( power_power_complex @ X4 @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
        = one_one_complex ) ) ).

% left_right_inverse_power
thf(fact_3228_left__right__inverse__power,axiom,
    ! [X4: real,Y: real,N2: nat] :
      ( ( ( times_times_real @ X4 @ Y )
        = one_one_real )
     => ( ( times_times_real @ ( power_power_real @ X4 @ N2 ) @ ( power_power_real @ Y @ N2 ) )
        = one_one_real ) ) ).

% left_right_inverse_power
thf(fact_3229_left__right__inverse__power,axiom,
    ! [X4: nat,Y: nat,N2: nat] :
      ( ( ( times_times_nat @ X4 @ Y )
        = one_one_nat )
     => ( ( times_times_nat @ ( power_power_nat @ X4 @ N2 ) @ ( power_power_nat @ Y @ N2 ) )
        = one_one_nat ) ) ).

% left_right_inverse_power
thf(fact_3230_left__right__inverse__power,axiom,
    ! [X4: int,Y: int,N2: nat] :
      ( ( ( times_times_int @ X4 @ Y )
        = one_one_int )
     => ( ( times_times_int @ ( power_power_int @ X4 @ N2 ) @ ( power_power_int @ Y @ N2 ) )
        = one_one_int ) ) ).

% left_right_inverse_power
thf(fact_3231_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_3232_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_3233_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_3234_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_3235_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_3236_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_3237_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_3238_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_3239_mod__eqE,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
     => ~ ! [D3: int] :
            ( B
           != ( plus_plus_int @ A @ ( times_times_int @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_3240_mod__eqE,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
     => ~ ! [D3: code_integer] :
            ( B
           != ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ C @ D3 ) ) ) ) ).

% mod_eqE
thf(fact_3241_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_3242_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_3243_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_3244_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_3245_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_3246_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_3247_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_3248_mult__less__mono2,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ K @ I2 ) @ ( times_times_nat @ K @ J ) ) ) ) ).

% mult_less_mono2
thf(fact_3249_mult__less__mono1,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_nat @ ( times_times_nat @ I2 @ K ) @ ( times_times_nat @ J @ K ) ) ) ) ).

% mult_less_mono1
thf(fact_3250_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_3251_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_3252_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_3253_less__mult__imp__div__less,axiom,
    ! [M: nat,I2: nat,N2: nat] :
      ( ( ord_less_nat @ M @ ( times_times_nat @ I2 @ N2 ) )
     => ( ord_less_nat @ ( divide_divide_nat @ M @ N2 ) @ I2 ) ) ).

% less_mult_imp_div_less
thf(fact_3254_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_3255_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_3256_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_3257_nat__mod__eq__iff,axiom,
    ! [X4: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X4 @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
      = ( ? [Q1: nat,Q22: nat] :
            ( ( plus_plus_nat @ X4 @ ( times_times_nat @ N2 @ Q1 ) )
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q22 ) ) ) ) ) ).

% nat_mod_eq_iff
thf(fact_3258_bits__induct,axiom,
    ! [P: nat > $o,A: nat] :
      ( ! [A5: nat] :
          ( ( ( divide_divide_nat @ A5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: nat,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_nat @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_nat @ ( zero_n2687167440665602831ol_nat @ B5 ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3259_bits__induct,axiom,
    ! [P: int > $o,A: int] :
      ( ! [A5: int] :
          ( ( ( divide_divide_int @ A5 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: int,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide_divide_int @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ B5 ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3260_bits__induct,axiom,
    ! [P: code_integer > $o,A: code_integer] :
      ( ! [A5: code_integer] :
          ( ( ( divide6298287555418463151nteger @ A5 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = A5 )
         => ( P @ A5 ) )
     => ( ! [A5: code_integer,B5: $o] :
            ( ( P @ A5 )
           => ( ( ( divide6298287555418463151nteger @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
                = A5 )
             => ( P @ ( plus_p5714425477246183910nteger @ ( zero_n356916108424825756nteger @ B5 ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A5 ) ) ) ) )
       => ( P @ A ) ) ) ).

% bits_induct
thf(fact_3261_oddE,axiom,
    ! [A: code_integer] :
      ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B5 ) @ one_one_Code_integer ) ) ) ).

% oddE
thf(fact_3262_oddE,axiom,
    ! [A: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B5 ) @ one_one_nat ) ) ) ).

% oddE
thf(fact_3263_oddE,axiom,
    ! [A: int] :
      ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B5 ) @ one_one_int ) ) ) ).

% oddE
thf(fact_3264_VEBT__internal_Ovalid_H_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,D3: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ D3 ) )
     => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT,Deg3: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Deg3 ) ) ) ).

% VEBT_internal.valid'.cases
thf(fact_3265_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_3266_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_3267_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_3268_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_3269_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_3270_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_3271_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_3272_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_3273_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_3274_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_3275_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_3276_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_3277_even__numeral,axiom,
    ! [N2: num] : ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ).

% even_numeral
thf(fact_3278_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_3279_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_3280_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_3281_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_3282_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_3283_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_3284_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_3285_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_3286_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_3287_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_3288_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_3289_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_3290_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_3291_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_3292_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_3293_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_3294_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_3295_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_3296_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_3297_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_3298_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_3299_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_3300_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_3301_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_3302_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_3303_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_3304_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_3305_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_3306_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_3307_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_3308_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_3309_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_3310_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_3311_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_3312_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_3313_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_3314_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_3315_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_3316_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_3317_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_3318_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_3319_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_3320_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_3321_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_3322_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_3323_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_3324_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_3325_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_3326_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_3327_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_3328_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_3329_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_3330_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_3331_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_3332_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_3333_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_3334_mult__left__le__one__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ Y @ X4 ) @ X4 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3335_mult__left__le__one__le,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ Y @ X4 ) @ X4 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3336_mult__left__le__one__le,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ Y @ X4 ) @ X4 ) ) ) ) ).

% mult_left_le_one_le
thf(fact_3337_mult__right__le__one__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( times_times_real @ X4 @ Y ) @ X4 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3338_mult__right__le__one__le,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
       => ( ( ord_less_eq_rat @ Y @ one_one_rat )
         => ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Y ) @ X4 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3339_mult__right__le__one__le,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( ord_less_eq_int @ Y @ one_one_int )
         => ( ord_less_eq_int @ ( times_times_int @ X4 @ Y ) @ X4 ) ) ) ) ).

% mult_right_le_one_le
thf(fact_3340_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_3341_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_3342_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_3343_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_3344_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_3345_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_3346_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_3347_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_3348_sum__squares__ge__zero,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3349_sum__squares__ge__zero,axiom,
    ! [X4: rat,Y: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3350_sum__squares__ge__zero,axiom,
    ! [X4: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) ) ).

% sum_squares_ge_zero
thf(fact_3351_sum__squares__le__zero__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real )
      = ( ( X4 = zero_zero_real )
        & ( Y = zero_zero_real ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3352_sum__squares__le__zero__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat )
      = ( ( X4 = zero_zero_rat )
        & ( Y = zero_zero_rat ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3353_sum__squares__le__zero__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int )
      = ( ( X4 = zero_zero_int )
        & ( Y = zero_zero_int ) ) ) ).

% sum_squares_le_zero_iff
thf(fact_3354_not__sum__squares__lt__zero,axiom,
    ! [X4: real,Y: real] :
      ~ ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) @ zero_zero_real ) ).

% not_sum_squares_lt_zero
thf(fact_3355_not__sum__squares__lt__zero,axiom,
    ! [X4: rat,Y: rat] :
      ~ ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) ) @ zero_zero_rat ) ).

% not_sum_squares_lt_zero
thf(fact_3356_not__sum__squares__lt__zero,axiom,
    ! [X4: int,Y: int] :
      ~ ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) @ zero_zero_int ) ).

% not_sum_squares_lt_zero
thf(fact_3357_sum__squares__gt__zero__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) )
      = ( ( X4 != zero_zero_real )
        | ( Y != zero_zero_real ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3358_sum__squares__gt__zero__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) ) )
      = ( ( X4 != zero_zero_rat )
        | ( Y != zero_zero_rat ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3359_sum__squares__gt__zero__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) ) )
      = ( ( X4 != zero_zero_int )
        | ( Y != zero_zero_int ) ) ) ).

% sum_squares_gt_zero_iff
thf(fact_3360_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_3361_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_3362_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_3363_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_3364_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_3365_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_3366_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_3367_mult__imp__less__div__pos,axiom,
    ! [Y: rat,Z: rat,X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ ( times_times_rat @ Z @ Y ) @ X4 )
       => ( ord_less_rat @ Z @ ( divide_divide_rat @ X4 @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3368_mult__imp__less__div__pos,axiom,
    ! [Y: real,Z: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ ( times_times_real @ Z @ Y ) @ X4 )
       => ( ord_less_real @ Z @ ( divide_divide_real @ X4 @ Y ) ) ) ) ).

% mult_imp_less_div_pos
thf(fact_3369_mult__imp__div__pos__less,axiom,
    ! [Y: rat,X4: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_rat @ X4 @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3370_mult__imp__div__pos__less,axiom,
    ! [Y: real,X4: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ X4 @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_real @ ( divide_divide_real @ X4 @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_less
thf(fact_3371_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_3372_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_3373_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_3374_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_3375_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_3376_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_3377_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_3378_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_3379_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_3380_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_3381_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_3382_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_3383_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( numeral_numeral_rat @ W )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3384_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( numeral_numeral_real @ W )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( numeral_numeral_real @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3385_eq__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( numera6690914467698888265omplex @ W )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(1)
thf(fact_3386_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( numeral_numeral_rat @ W ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( numeral_numeral_rat @ W )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3387_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( numeral_numeral_real @ W ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( numeral_numeral_real @ W )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3388_divide__eq__eq__numeral_I1_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( numera6690914467698888265omplex @ W )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(1)
thf(fact_3389_divide__add__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Z ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_3390_divide__add__eq__iff,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Z ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_3391_divide__add__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_add_eq_iff
thf(fact_3392_add__divide__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ X4 @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_3393_add__divide__eq__iff,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ X4 @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X4 @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_3394_add__divide__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ X4 @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X4 @ Z ) @ Y ) @ Z ) ) ) ).

% add_divide_eq_iff
thf(fact_3395_add__num__frac,axiom,
    ! [Y: rat,Z: rat,X4: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ Z @ ( divide_divide_rat @ X4 @ Y ) )
        = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3396_add__num__frac,axiom,
    ! [Y: real,Z: real,X4: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ Z @ ( divide_divide_real @ X4 @ Y ) )
        = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3397_add__num__frac,axiom,
    ! [Y: complex,Z: complex,X4: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ X4 @ Y ) )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_num_frac
thf(fact_3398_add__frac__num,axiom,
    ! [Y: rat,X4: rat,Z: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Y ) @ Z )
        = ( divide_divide_rat @ ( plus_plus_rat @ X4 @ ( times_times_rat @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3399_add__frac__num,axiom,
    ! [Y: real,X4: real,Z: real] :
      ( ( Y != zero_zero_real )
     => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Y ) @ Z )
        = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( times_times_real @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3400_add__frac__num,axiom,
    ! [Y: complex,X4: complex,Z: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Y ) @ Z )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ X4 @ ( times_times_complex @ Z @ Y ) ) @ Y ) ) ) ).

% add_frac_num
thf(fact_3401_add__frac__eq,axiom,
    ! [Y: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ X4 @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_3402_add__frac__eq,axiom,
    ! [Y: real,Z: real,X4: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ X4 @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_3403_add__frac__eq,axiom,
    ! [Y: complex,Z: complex,X4: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ X4 @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% add_frac_eq
thf(fact_3404_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3405_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3406_add__divide__eq__if__simps_I1_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(1)
thf(fact_3407_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3408_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3409_add__divide__eq__if__simps_I2_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = B ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(2)
thf(fact_3410_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_3411_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_3412_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_3413_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_3414_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_3415_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_3416_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_3417_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_3418_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_3419_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_3420_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_3421_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_3422_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_3423_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_3424_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_3425_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_3426_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_3427_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_3428_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_3429_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_3430_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_3431_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_3432_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_3433_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_3434_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_3435_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_3436_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_3437_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_3438_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_3439_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_3440_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_3441_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_3442_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_3443_pos__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ B @ A ) ) ) ) ) ).

% pos_zmod_mult_2
thf(fact_3444_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_3445_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_3446_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_3447_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_3448_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_3449_div__less__iff__less__mult,axiom,
    ! [Q3: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_nat @ ( divide_divide_nat @ M @ Q3 ) @ N2 )
        = ( ord_less_nat @ M @ ( times_times_nat @ N2 @ Q3 ) ) ) ) ).

% div_less_iff_less_mult
thf(fact_3450_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_3451_mod__eq__nat1E,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ N2 @ M )
       => ~ ! [S3: nat] :
              ( M
             != ( plus_plus_nat @ N2 @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).

% mod_eq_nat1E
thf(fact_3452_mod__eq__nat2E,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( ( modulo_modulo_nat @ M @ Q3 )
        = ( modulo_modulo_nat @ N2 @ Q3 ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ~ ! [S3: nat] :
              ( N2
             != ( plus_plus_nat @ M @ ( times_times_nat @ Q3 @ S3 ) ) ) ) ) ).

% mod_eq_nat2E
thf(fact_3453_nat__mod__eq__lemma,axiom,
    ! [X4: nat,N2: nat,Y: nat] :
      ( ( ( modulo_modulo_nat @ X4 @ N2 )
        = ( modulo_modulo_nat @ Y @ N2 ) )
     => ( ( ord_less_eq_nat @ Y @ X4 )
       => ? [Q2: nat] :
            ( X4
            = ( plus_plus_nat @ Y @ ( times_times_nat @ N2 @ Q2 ) ) ) ) ) ).

% nat_mod_eq_lemma
thf(fact_3454_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_3455_mod__mult2__eq,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( modulo_modulo_nat @ M @ ( times_times_nat @ N2 @ Q3 ) )
      = ( plus_plus_nat @ ( times_times_nat @ N2 @ ( modulo_modulo_nat @ ( divide_divide_nat @ M @ N2 ) @ Q3 ) ) @ ( modulo_modulo_nat @ M @ N2 ) ) ) ).

% mod_mult2_eq
thf(fact_3456_even__zero,axiom,
    dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ zero_z3403309356797280102nteger ).

% even_zero
thf(fact_3457_even__zero,axiom,
    dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ zero_zero_nat ).

% even_zero
thf(fact_3458_even__zero,axiom,
    dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ zero_zero_int ).

% even_zero
thf(fact_3459_odd__one,axiom,
    ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ one_one_Code_integer ) ).

% odd_one
thf(fact_3460_odd__one,axiom,
    ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ one_one_nat ) ).

% odd_one
thf(fact_3461_odd__one,axiom,
    ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ one_one_int ) ).

% odd_one
thf(fact_3462_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_3463_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_3464_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_3465_bit__eq__rec,axiom,
    ( ( ^ [Y6: nat,Z4: nat] : Y6 = Z4 )
    = ( ^ [A3: nat,B2: nat] :
          ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_nat @ A3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
            = ( divide_divide_nat @ B2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_3466_bit__eq__rec,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A3: int,B2: int] :
          ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide_divide_int @ A3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
            = ( divide_divide_int @ B2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_3467_bit__eq__rec,axiom,
    ( ( ^ [Y6: code_integer,Z4: code_integer] : Y6 = Z4 )
    = ( ^ [A3: code_integer,B2: code_integer] :
          ( ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A3 )
            = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B2 ) )
          & ( ( divide6298287555418463151nteger @ A3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
            = ( divide6298287555418463151nteger @ B2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_eq_rec
thf(fact_3468_dvd__power__iff,axiom,
    ! [X4: code_integer,M: nat,N2: nat] :
      ( ( X4 != zero_z3403309356797280102nteger )
     => ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ X4 @ M ) @ ( power_8256067586552552935nteger @ X4 @ N2 ) )
        = ( ( dvd_dvd_Code_integer @ X4 @ one_one_Code_integer )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3469_dvd__power__iff,axiom,
    ! [X4: nat,M: nat,N2: nat] :
      ( ( X4 != zero_zero_nat )
     => ( ( dvd_dvd_nat @ ( power_power_nat @ X4 @ M ) @ ( power_power_nat @ X4 @ N2 ) )
        = ( ( dvd_dvd_nat @ X4 @ one_one_nat )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3470_dvd__power__iff,axiom,
    ! [X4: int,M: nat,N2: nat] :
      ( ( X4 != zero_zero_int )
     => ( ( dvd_dvd_int @ ( power_power_int @ X4 @ M ) @ ( power_power_int @ X4 @ N2 ) )
        = ( ( dvd_dvd_int @ X4 @ one_one_int )
          | ( ord_less_eq_nat @ M @ N2 ) ) ) ) ).

% dvd_power_iff
thf(fact_3471_dvd__power,axiom,
    ! [N2: nat,X4: code_integer] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X4 = one_one_Code_integer ) )
     => ( dvd_dvd_Code_integer @ X4 @ ( power_8256067586552552935nteger @ X4 @ N2 ) ) ) ).

% dvd_power
thf(fact_3472_dvd__power,axiom,
    ! [N2: nat,X4: rat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X4 = one_one_rat ) )
     => ( dvd_dvd_rat @ X4 @ ( power_power_rat @ X4 @ N2 ) ) ) ).

% dvd_power
thf(fact_3473_dvd__power,axiom,
    ! [N2: nat,X4: nat] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X4 = one_one_nat ) )
     => ( dvd_dvd_nat @ X4 @ ( power_power_nat @ X4 @ N2 ) ) ) ).

% dvd_power
thf(fact_3474_dvd__power,axiom,
    ! [N2: nat,X4: real] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X4 = one_one_real ) )
     => ( dvd_dvd_real @ X4 @ ( power_power_real @ X4 @ N2 ) ) ) ).

% dvd_power
thf(fact_3475_dvd__power,axiom,
    ! [N2: nat,X4: int] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X4 = one_one_int ) )
     => ( dvd_dvd_int @ X4 @ ( power_power_int @ X4 @ N2 ) ) ) ).

% dvd_power
thf(fact_3476_dvd__power,axiom,
    ! [N2: nat,X4: complex] :
      ( ( ( ord_less_nat @ zero_zero_nat @ N2 )
        | ( X4 = one_one_complex ) )
     => ( dvd_dvd_complex @ X4 @ ( power_power_complex @ X4 @ N2 ) ) ) ).

% dvd_power
thf(fact_3477_field__le__mult__one__interval,axiom,
    ! [X4: real,Y: real] :
      ( ! [Z2: real] :
          ( ( ord_less_real @ zero_zero_real @ Z2 )
         => ( ( ord_less_real @ Z2 @ one_one_real )
           => ( ord_less_eq_real @ ( times_times_real @ Z2 @ X4 ) @ Y ) ) )
     => ( ord_less_eq_real @ X4 @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3478_field__le__mult__one__interval,axiom,
    ! [X4: rat,Y: rat] :
      ( ! [Z2: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ Z2 )
         => ( ( ord_less_rat @ Z2 @ one_one_rat )
           => ( ord_less_eq_rat @ ( times_times_rat @ Z2 @ X4 ) @ Y ) ) )
     => ( ord_less_eq_rat @ X4 @ Y ) ) ).

% field_le_mult_one_interval
thf(fact_3479_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_3480_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_3481_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_3482_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_3483_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_3484_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_3485_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_3486_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_3487_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_3488_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_3489_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_3490_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_3491_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_3492_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_3493_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_3494_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_3495_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_3496_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_3497_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_3498_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_3499_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_3500_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_3501_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_3502_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_3503_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_3504_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_3505_mult__imp__le__div__pos,axiom,
    ! [Y: real,Z: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ Y ) @ X4 )
       => ( ord_less_eq_real @ Z @ ( divide_divide_real @ X4 @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3506_mult__imp__le__div__pos,axiom,
    ! [Y: rat,Z: rat,X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ Y ) @ X4 )
       => ( ord_less_eq_rat @ Z @ ( divide_divide_rat @ X4 @ Y ) ) ) ) ).

% mult_imp_le_div_pos
thf(fact_3507_mult__imp__div__pos__le,axiom,
    ! [Y: real,X4: real,Z: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ X4 @ ( times_times_real @ Z @ Y ) )
       => ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3508_mult__imp__div__pos__le,axiom,
    ! [Y: rat,X4: rat,Z: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ X4 @ ( times_times_rat @ Z @ Y ) )
       => ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y ) @ Z ) ) ) ).

% mult_imp_div_pos_le
thf(fact_3509_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_3510_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_3511_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_3512_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_3513_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_3514_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_3515_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_3516_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_3517_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_3518_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_3519_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_3520_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_3521_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_3522_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_3523_convex__bound__le,axiom,
    ! [X4: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_eq_real @ X4 @ A )
     => ( ( ord_less_eq_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3524_convex__bound__le,axiom,
    ! [X4: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_eq_rat @ X4 @ A )
     => ( ( ord_less_eq_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_eq_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3525_convex__bound__le,axiom,
    ! [X4: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_eq_int @ X4 @ A )
     => ( ( ord_less_eq_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_eq_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_le
thf(fact_3526_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ 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 @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3527_less__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ 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 @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(1)
thf(fact_3528_divide__less__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3529_divide__less__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(1)
thf(fact_3530_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_3531_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_3532_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_3533_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_3534_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_3535_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_3536_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_3537_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_3538_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_3539_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_3540_mult__2__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( numeral_numeral_rat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3541_mult__2__right,axiom,
    ! [Z: extended_enat] :
      ( ( times_7803423173614009249d_enat @ Z @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3542_mult__2__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3543_mult__2__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3544_mult__2__right,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3545_mult__2__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2_right
thf(fact_3546_mult__2,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_rat @ Z @ Z ) ) ).

% mult_2
thf(fact_3547_mult__2,axiom,
    ! [Z: extended_enat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ Z )
      = ( plus_p3455044024723400733d_enat @ Z @ Z ) ) ).

% mult_2
thf(fact_3548_mult__2,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_complex @ Z @ Z ) ) ).

% mult_2
thf(fact_3549_mult__2,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_real @ Z @ Z ) ) ).

% mult_2
thf(fact_3550_mult__2,axiom,
    ! [Z: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_nat @ Z @ Z ) ) ).

% mult_2
thf(fact_3551_mult__2,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Z )
      = ( plus_plus_int @ Z @ Z ) ) ).

% mult_2
thf(fact_3552_power2__eq__square,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_complex @ A @ A ) ) ).

% power2_eq_square
thf(fact_3553_power2__eq__square,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ A @ A ) ) ).

% power2_eq_square
thf(fact_3554_power2__eq__square,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_nat @ A @ A ) ) ).

% power2_eq_square
thf(fact_3555_power2__eq__square,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_int @ A @ A ) ) ).

% power2_eq_square
thf(fact_3556_power4__eq__xxxx,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ ( times_times_complex @ ( times_times_complex @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).

% power4_eq_xxxx
thf(fact_3557_power4__eq__xxxx,axiom,
    ! [X4: real] :
      ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( times_times_real @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).

% power4_eq_xxxx
thf(fact_3558_power4__eq__xxxx,axiom,
    ! [X4: nat] :
      ( ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( times_times_nat @ ( times_times_nat @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).

% power4_eq_xxxx
thf(fact_3559_power4__eq__xxxx,axiom,
    ! [X4: int] :
      ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( times_times_int @ ( times_times_int @ X4 @ X4 ) @ X4 ) @ X4 ) ) ).

% power4_eq_xxxx
thf(fact_3560_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_3561_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_3562_power__even__eq,axiom,
    ! [A: nat,N2: nat] :
      ( ( power_power_nat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_nat @ ( power_power_nat @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_3563_power__even__eq,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_real @ ( power_power_real @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_3564_power__even__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_int @ ( power_power_int @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_3565_power__even__eq,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_complex @ ( power_power_complex @ A @ N2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power_even_eq
thf(fact_3566_power__dvd__imp__le,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ I2 @ M ) @ ( power_power_nat @ I2 @ N2 ) )
     => ( ( ord_less_nat @ one_one_nat @ I2 )
       => ( ord_less_eq_nat @ M @ N2 ) ) ) ).

% power_dvd_imp_le
thf(fact_3567_neg__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) ) ).

% neg_zdiv_mult_2
thf(fact_3568_pos__zdiv__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( divide_divide_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( divide_divide_int @ B @ A ) ) ) ).

% pos_zdiv_mult_2
thf(fact_3569_div__nat__eqI,axiom,
    ! [N2: nat,Q3: nat,M: nat] :
      ( ( ord_less_eq_nat @ ( times_times_nat @ N2 @ Q3 ) @ M )
     => ( ( ord_less_nat @ M @ ( times_times_nat @ N2 @ ( suc @ Q3 ) ) )
       => ( ( divide_divide_nat @ M @ N2 )
          = Q3 ) ) ) ).

% div_nat_eqI
thf(fact_3570_less__eq__div__iff__mult__less__eq,axiom,
    ! [Q3: nat,M: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Q3 )
     => ( ( ord_less_eq_nat @ M @ ( divide_divide_nat @ N2 @ Q3 ) )
        = ( ord_less_eq_nat @ ( times_times_nat @ M @ Q3 ) @ N2 ) ) ) ).

% less_eq_div_iff_mult_less_eq
thf(fact_3571_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_3572_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_3573_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 )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
               => ( P @ I3 ) ) ) ) ) ) ).

% split_div
thf(fact_3574_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 )
         => ! [I3: nat,J3: nat] :
              ( ( ord_less_nat @ J3 @ N2 )
             => ( ( M
                  = ( plus_plus_nat @ ( times_times_nat @ N2 @ I3 ) @ J3 ) )
               => ( P @ J3 ) ) ) ) ) ) ).

% split_mod
thf(fact_3575_VEBT__internal_Onaive__member_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [A5: $o,B5: $o,X5: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ X5 ) )
     => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT,Ux2: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Ux2 ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT,X5: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ X5 ) ) ) ) ).

% VEBT_internal.naive_member.cases
thf(fact_3576_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_3577_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_3578_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_3579_odd__iff__mod__2__eq__one,axiom,
    ! [A: nat] :
      ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_nat ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3580_odd__iff__mod__2__eq__one,axiom,
    ! [A: int] :
      ( ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = one_one_int ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3581_odd__iff__mod__2__eq__one,axiom,
    ! [A: code_integer] :
      ( ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) )
      = ( ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = one_one_Code_integer ) ) ).

% odd_iff_mod_2_eq_one
thf(fact_3582_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_3583_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_3584_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_3585_convex__bound__lt,axiom,
    ! [X4: real,A: real,Y: real,U: real,V: real] :
      ( ( ord_less_real @ X4 @ A )
     => ( ( ord_less_real @ Y @ A )
       => ( ( ord_less_eq_real @ zero_zero_real @ U )
         => ( ( ord_less_eq_real @ zero_zero_real @ V )
           => ( ( ( plus_plus_real @ U @ V )
                = one_one_real )
             => ( ord_less_real @ ( plus_plus_real @ ( times_times_real @ U @ X4 ) @ ( times_times_real @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3586_convex__bound__lt,axiom,
    ! [X4: rat,A: rat,Y: rat,U: rat,V: rat] :
      ( ( ord_less_rat @ X4 @ A )
     => ( ( ord_less_rat @ Y @ A )
       => ( ( ord_less_eq_rat @ zero_zero_rat @ U )
         => ( ( ord_less_eq_rat @ zero_zero_rat @ V )
           => ( ( ( plus_plus_rat @ U @ V )
                = one_one_rat )
             => ( ord_less_rat @ ( plus_plus_rat @ ( times_times_rat @ U @ X4 ) @ ( times_times_rat @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3587_convex__bound__lt,axiom,
    ! [X4: int,A: int,Y: int,U: int,V: int] :
      ( ( ord_less_int @ X4 @ A )
     => ( ( ord_less_int @ Y @ A )
       => ( ( ord_less_eq_int @ zero_zero_int @ U )
         => ( ( ord_less_eq_int @ zero_zero_int @ V )
           => ( ( ( plus_plus_int @ U @ V )
                = one_one_int )
             => ( ord_less_int @ ( plus_plus_int @ ( times_times_int @ U @ X4 ) @ ( times_times_int @ V @ Y ) ) @ A ) ) ) ) ) ) ).

% convex_bound_lt
thf(fact_3588_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ 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 @ W ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( numeral_numeral_real @ W ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3589_le__divide__eq__numeral_I1_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ 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 @ W ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( numeral_numeral_rat @ W ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(1)
thf(fact_3590_divide__le__eq__numeral_I1_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( numeral_numeral_real @ W ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3591_divide__le__eq__numeral_I1_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( numeral_numeral_rat @ W ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(1)
thf(fact_3592_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_3593_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_3594_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_3595_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_3596_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_3597_num_Osize__gen_I1_J,axiom,
    ( ( size_num @ one )
    = zero_zero_nat ) ).

% num.size_gen(1)
thf(fact_3598_four__x__squared,axiom,
    ! [X4: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% four_x_squared
thf(fact_3599_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_3600_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_3601_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_3602_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_3603_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_3604_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_3605_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_3606_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_3607_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_3608_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_3609_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_3610_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_3611_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_3612_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_3613_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_3614_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_3615_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_3616_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_3617_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_3618_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_3619_power2__sum,axiom,
    ! [X4: rat,Y: rat] :
      ( ( power_power_rat @ ( plus_plus_rat @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_sum
thf(fact_3620_power2__sum,axiom,
    ! [X4: extended_enat,Y: extended_enat] :
      ( ( power_8040749407984259932d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_p3455044024723400733d_enat @ ( plus_p3455044024723400733d_enat @ ( power_8040749407984259932d_enat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8040749407984259932d_enat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_7803423173614009249d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_sum
thf(fact_3621_power2__sum,axiom,
    ! [X4: complex,Y: complex] :
      ( ( power_power_complex @ ( plus_plus_complex @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( plus_plus_complex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_sum
thf(fact_3622_power2__sum,axiom,
    ! [X4: real,Y: real] :
      ( ( power_power_real @ ( plus_plus_real @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_sum
thf(fact_3623_power2__sum,axiom,
    ! [X4: nat,Y: nat] :
      ( ( power_power_nat @ ( plus_plus_nat @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_nat @ ( plus_plus_nat @ ( power_power_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_sum
thf(fact_3624_power2__sum,axiom,
    ! [X4: int,Y: int] :
      ( ( power_power_int @ ( plus_plus_int @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_sum
thf(fact_3625_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_3626_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_3627_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_3628_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_3629_L2__set__mult__ineq__lemma,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_real @ A @ C ) ) @ ( times_times_real @ B @ D ) ) @ ( plus_plus_real @ ( times_times_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% L2_set_mult_ineq_lemma
thf(fact_3630_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_3631_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_3632_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_3633_Euclid__induct,axiom,
    ! [P: nat > nat > $o,A: nat,B: nat] :
      ( ! [A5: nat,B5: nat] :
          ( ( P @ A5 @ B5 )
          = ( P @ B5 @ A5 ) )
     => ( ! [A5: nat] : ( P @ A5 @ zero_zero_nat )
       => ( ! [A5: nat,B5: nat] :
              ( ( P @ A5 @ B5 )
             => ( P @ A5 @ ( plus_plus_nat @ A5 @ B5 ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Euclid_induct
thf(fact_3634_sum__squares__bound,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y ) @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_3635_sum__squares__bound,axiom,
    ! [X4: rat,Y: rat] : ( ord_less_eq_rat @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y ) @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sum_squares_bound
thf(fact_3636_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_3637_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_3638_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_3639_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_3640_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_3641_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_3642_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_3643_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_3644_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_3645_VEBT__internal_Omembermima_Ocases,axiom,
    ! [X4: produc9072475918466114483BT_nat] :
      ( ! [Uu2: $o,Uv2: $o,Uw2: nat] :
          ( X4
         != ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Uw2 ) )
     => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT,Uz2: nat] :
            ( X4
           != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Uz2 ) )
       => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT,X5: nat] :
              ( X4
             != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ X5 ) )
         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT,X5: nat] :
                ( X4
               != ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) @ X5 ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT,X5: nat] :
                  ( X4
                 != ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) @ X5 ) ) ) ) ) ) ).

% VEBT_internal.membermima.cases
thf(fact_3646_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_3647_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_3648_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_3649_mod__double__modulus,axiom,
    ! [M: code_integer,X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ M )
     => ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
       => ( ( ( modulo364778990260209775nteger @ X4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( modulo364778990260209775nteger @ X4 @ M ) )
          | ( ( modulo364778990260209775nteger @ X4 @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ M ) )
            = ( plus_p5714425477246183910nteger @ ( modulo364778990260209775nteger @ X4 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3650_mod__double__modulus,axiom,
    ! [M: nat,X4: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ M )
     => ( ( ord_less_eq_nat @ zero_zero_nat @ X4 )
       => ( ( ( modulo_modulo_nat @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_nat @ X4 @ M ) )
          | ( ( modulo_modulo_nat @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_nat @ ( modulo_modulo_nat @ X4 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3651_mod__double__modulus,axiom,
    ! [M: int,X4: int] :
      ( ( ord_less_int @ zero_zero_int @ M )
     => ( ( ord_less_eq_int @ zero_zero_int @ X4 )
       => ( ( ( modulo_modulo_int @ X4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( modulo_modulo_int @ X4 @ M ) )
          | ( ( modulo_modulo_int @ X4 @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ M ) )
            = ( plus_plus_int @ ( modulo_modulo_int @ X4 @ M ) @ M ) ) ) ) ) ).

% mod_double_modulus
thf(fact_3652_option_Osize__gen_I1_J,axiom,
    ! [X4: product_prod_nat_nat > nat] :
      ( ( size_o8335143837870341156at_nat @ X4 @ none_P5556105721700978146at_nat )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_3653_option_Osize__gen_I1_J,axiom,
    ! [X4: num > nat] :
      ( ( size_option_num @ X4 @ none_num )
      = ( suc @ zero_zero_nat ) ) ).

% option.size_gen(1)
thf(fact_3654_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_3655_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_3656_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_3657_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_3658_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_3659_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_3660_even__even__mod__4__iff,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( modulo_modulo_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ).

% even_even_mod_4_iff
thf(fact_3661_div2__even__ext__nat,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( divide_divide_nat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
     => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 )
          = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Y ) )
       => ( X4 = Y ) ) ) ).

% div2_even_ext_nat
thf(fact_3662_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_3663_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_3664_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_3665_incr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int] :
            ( ( P @ X5 )
           => ( P @ ( plus_plus_int @ X5 @ 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_3666_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_3667_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_3668_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_3669_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_3670_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_3671_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_3672_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_3673_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_3674_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_3675_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_3676_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_3677_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_3678_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_3679_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_3680_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_3681_unset__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% unset_bit_nonnegative_int_iff
thf(fact_3682_set__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se7879613467334960850it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% set_bit_nonnegative_int_iff
thf(fact_3683_flip__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se2159334234014336723it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% flip_bit_nonnegative_int_iff
thf(fact_3684_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_3685_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_3686_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_3687_unset__bit__less__eq,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ ( bit_se4203085406695923979it_int @ N2 @ K ) @ K ) ).

% unset_bit_less_eq
thf(fact_3688_set__bit__greater__eq,axiom,
    ! [K: int,N2: nat] : ( ord_less_eq_int @ K @ ( bit_se7879613467334960850it_int @ N2 @ K ) ) ).

% set_bit_greater_eq
thf(fact_3689_minf_I7_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z2 )
     => ~ ( ord_less_real @ T2 @ X2 ) ) ).

% minf(7)
thf(fact_3690_minf_I7_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z2 )
     => ~ ( ord_less_rat @ T2 @ X2 ) ) ).

% minf(7)
thf(fact_3691_minf_I7_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z2 )
     => ~ ( ord_less_num @ T2 @ X2 ) ) ).

% minf(7)
thf(fact_3692_minf_I7_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z2 )
     => ~ ( ord_less_nat @ T2 @ X2 ) ) ).

% minf(7)
thf(fact_3693_minf_I7_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z2 )
     => ~ ( ord_less_int @ T2 @ X2 ) ) ).

% minf(7)
thf(fact_3694_minf_I5_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z2 )
     => ( ord_less_real @ X2 @ T2 ) ) ).

% minf(5)
thf(fact_3695_minf_I5_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z2 )
     => ( ord_less_rat @ X2 @ T2 ) ) ).

% minf(5)
thf(fact_3696_minf_I5_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z2 )
     => ( ord_less_num @ X2 @ T2 ) ) ).

% minf(5)
thf(fact_3697_minf_I5_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z2 )
     => ( ord_less_nat @ X2 @ T2 ) ) ).

% minf(5)
thf(fact_3698_minf_I5_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z2 )
     => ( ord_less_int @ X2 @ T2 ) ) ).

% minf(5)
thf(fact_3699_minf_I4_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(4)
thf(fact_3700_minf_I4_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(4)
thf(fact_3701_minf_I4_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(4)
thf(fact_3702_minf_I4_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(4)
thf(fact_3703_minf_I4_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(4)
thf(fact_3704_minf_I3_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(3)
thf(fact_3705_minf_I3_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(3)
thf(fact_3706_minf_I3_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(3)
thf(fact_3707_minf_I3_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(3)
thf(fact_3708_minf_I3_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z2 )
     => ( X2 != T2 ) ) ).

% minf(3)
thf(fact_3709_minf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3710_minf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3711_minf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: num] :
          ! [X2: num] :
            ( ( ord_less_num @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3712_minf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3713_minf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(2)
thf(fact_3714_minf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X5: real] :
          ( ( ord_less_real @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: real] :
          ! [X2: real] :
            ( ( ord_less_real @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3715_minf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3716_minf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X5: num] :
          ( ( ord_less_num @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: num] :
          ! [X2: num] :
            ( ( ord_less_num @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3717_minf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3718_minf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X5: int] :
          ( ( ord_less_int @ X5 @ Z3 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z3 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: int] :
          ! [X2: int] :
            ( ( ord_less_int @ X2 @ Z2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% minf(1)
thf(fact_3719_pinf_I7_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ( ord_less_real @ T2 @ X2 ) ) ).

% pinf(7)
thf(fact_3720_pinf_I7_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z2 @ X2 )
     => ( ord_less_rat @ T2 @ X2 ) ) ).

% pinf(7)
thf(fact_3721_pinf_I7_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z2 @ X2 )
     => ( ord_less_num @ T2 @ X2 ) ) ).

% pinf(7)
thf(fact_3722_pinf_I7_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z2 @ X2 )
     => ( ord_less_nat @ T2 @ X2 ) ) ).

% pinf(7)
thf(fact_3723_pinf_I7_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z2 @ X2 )
     => ( ord_less_int @ T2 @ X2 ) ) ).

% pinf(7)
thf(fact_3724_pinf_I5_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ~ ( ord_less_real @ X2 @ T2 ) ) ).

% pinf(5)
thf(fact_3725_pinf_I5_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z2 @ X2 )
     => ~ ( ord_less_rat @ X2 @ T2 ) ) ).

% pinf(5)
thf(fact_3726_pinf_I5_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z2 @ X2 )
     => ~ ( ord_less_num @ X2 @ T2 ) ) ).

% pinf(5)
thf(fact_3727_pinf_I5_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z2 @ X2 )
     => ~ ( ord_less_nat @ X2 @ T2 ) ) ).

% pinf(5)
thf(fact_3728_pinf_I5_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z2 @ X2 )
     => ~ ( ord_less_int @ X2 @ T2 ) ) ).

% pinf(5)
thf(fact_3729_pinf_I4_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(4)
thf(fact_3730_pinf_I4_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(4)
thf(fact_3731_pinf_I4_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(4)
thf(fact_3732_pinf_I4_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(4)
thf(fact_3733_pinf_I4_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(4)
thf(fact_3734_pinf_I3_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(3)
thf(fact_3735_pinf_I3_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(3)
thf(fact_3736_pinf_I3_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(3)
thf(fact_3737_pinf_I3_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(3)
thf(fact_3738_pinf_I3_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z2 @ X2 )
     => ( X2 != T2 ) ) ).

% pinf(3)
thf(fact_3739_pinf_I2_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3740_pinf_I2_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3741_pinf_I2_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: num] :
          ! [X2: num] :
            ( ( ord_less_num @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3742_pinf_I2_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3743_pinf_I2_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                | ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                | ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(2)
thf(fact_3744_pinf_I1_J,axiom,
    ! [P: real > $o,P6: real > $o,Q: real > $o,Q6: real > $o] :
      ( ? [Z3: real] :
        ! [X5: real] :
          ( ( ord_less_real @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: real] :
          ! [X5: real] :
            ( ( ord_less_real @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: real] :
          ! [X2: real] :
            ( ( ord_less_real @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3745_pinf_I1_J,axiom,
    ! [P: rat > $o,P6: rat > $o,Q: rat > $o,Q6: rat > $o] :
      ( ? [Z3: rat] :
        ! [X5: rat] :
          ( ( ord_less_rat @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: rat] :
          ! [X5: rat] :
            ( ( ord_less_rat @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: rat] :
          ! [X2: rat] :
            ( ( ord_less_rat @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3746_pinf_I1_J,axiom,
    ! [P: num > $o,P6: num > $o,Q: num > $o,Q6: num > $o] :
      ( ? [Z3: num] :
        ! [X5: num] :
          ( ( ord_less_num @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: num] :
          ! [X5: num] :
            ( ( ord_less_num @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: num] :
          ! [X2: num] :
            ( ( ord_less_num @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3747_pinf_I1_J,axiom,
    ! [P: nat > $o,P6: nat > $o,Q: nat > $o,Q6: nat > $o] :
      ( ? [Z3: nat] :
        ! [X5: nat] :
          ( ( ord_less_nat @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: nat] :
          ! [X5: nat] :
            ( ( ord_less_nat @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: nat] :
          ! [X2: nat] :
            ( ( ord_less_nat @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3748_pinf_I1_J,axiom,
    ! [P: int > $o,P6: int > $o,Q: int > $o,Q6: int > $o] :
      ( ? [Z3: int] :
        ! [X5: int] :
          ( ( ord_less_int @ Z3 @ X5 )
         => ( ( P @ X5 )
            = ( P6 @ X5 ) ) )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z3 @ X5 )
           => ( ( Q @ X5 )
              = ( Q6 @ X5 ) ) )
       => ? [Z2: int] :
          ! [X2: int] :
            ( ( ord_less_int @ Z2 @ X2 )
           => ( ( ( P @ X2 )
                & ( Q @ X2 ) )
              = ( ( P6 @ X2 )
                & ( Q6 @ X2 ) ) ) ) ) ) ).

% pinf(1)
thf(fact_3749_minf_I8_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z2 )
     => ~ ( ord_less_eq_real @ T2 @ X2 ) ) ).

% minf(8)
thf(fact_3750_minf_I8_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z2 )
     => ~ ( ord_less_eq_rat @ T2 @ X2 ) ) ).

% minf(8)
thf(fact_3751_minf_I8_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z2 )
     => ~ ( ord_less_eq_num @ T2 @ X2 ) ) ).

% minf(8)
thf(fact_3752_minf_I8_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z2 )
     => ~ ( ord_less_eq_nat @ T2 @ X2 ) ) ).

% minf(8)
thf(fact_3753_minf_I8_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z2 )
     => ~ ( ord_less_eq_int @ T2 @ X2 ) ) ).

% minf(8)
thf(fact_3754_minf_I6_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z2 )
     => ( ord_less_eq_real @ X2 @ T2 ) ) ).

% minf(6)
thf(fact_3755_minf_I6_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z2 )
     => ( ord_less_eq_rat @ X2 @ T2 ) ) ).

% minf(6)
thf(fact_3756_minf_I6_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ X2 @ Z2 )
     => ( ord_less_eq_num @ X2 @ T2 ) ) ).

% minf(6)
thf(fact_3757_minf_I6_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z2 )
     => ( ord_less_eq_nat @ X2 @ T2 ) ) ).

% minf(6)
thf(fact_3758_minf_I6_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z2 )
     => ( ord_less_eq_int @ X2 @ T2 ) ) ).

% minf(6)
thf(fact_3759_pinf_I8_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ( ord_less_eq_real @ T2 @ X2 ) ) ).

% pinf(8)
thf(fact_3760_pinf_I8_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z2 @ X2 )
     => ( ord_less_eq_rat @ T2 @ X2 ) ) ).

% pinf(8)
thf(fact_3761_pinf_I8_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z2 @ X2 )
     => ( ord_less_eq_num @ T2 @ X2 ) ) ).

% pinf(8)
thf(fact_3762_pinf_I8_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z2 @ X2 )
     => ( ord_less_eq_nat @ T2 @ X2 ) ) ).

% pinf(8)
thf(fact_3763_pinf_I8_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z2 @ X2 )
     => ( ord_less_eq_int @ T2 @ X2 ) ) ).

% pinf(8)
thf(fact_3764_pinf_I6_J,axiom,
    ! [T2: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ~ ( ord_less_eq_real @ X2 @ T2 ) ) ).

% pinf(6)
thf(fact_3765_pinf_I6_J,axiom,
    ! [T2: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z2 @ X2 )
     => ~ ( ord_less_eq_rat @ X2 @ T2 ) ) ).

% pinf(6)
thf(fact_3766_pinf_I6_J,axiom,
    ! [T2: num] :
    ? [Z2: num] :
    ! [X2: num] :
      ( ( ord_less_num @ Z2 @ X2 )
     => ~ ( ord_less_eq_num @ X2 @ T2 ) ) ).

% pinf(6)
thf(fact_3767_pinf_I6_J,axiom,
    ! [T2: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z2 @ X2 )
     => ~ ( ord_less_eq_nat @ X2 @ T2 ) ) ).

% pinf(6)
thf(fact_3768_pinf_I6_J,axiom,
    ! [T2: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z2 @ X2 )
     => ~ ( ord_less_eq_int @ X2 @ T2 ) ) ).

% pinf(6)
thf(fact_3769_list__decode_Ocases,axiom,
    ! [X4: nat] :
      ( ( X4 != zero_zero_nat )
     => ~ ! [N3: nat] :
            ( X4
           != ( suc @ N3 ) ) ) ).

% list_decode.cases
thf(fact_3770_imp__le__cong,axiom,
    ! [X4: int,X7: int,P: $o,P6: $o] :
      ( ( X4 = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
           => P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
           => P6 ) ) ) ) ).

% imp_le_cong
thf(fact_3771_conj__le__cong,axiom,
    ! [X4: int,X7: int,P: $o,P6: $o] :
      ( ( X4 = X7 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X7 )
         => ( P = P6 ) )
       => ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
            & P )
          = ( ( ord_less_eq_int @ zero_zero_int @ X7 )
            & P6 ) ) ) ) ).

% conj_le_cong
thf(fact_3772_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_3773_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_3774_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_3775_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_3776_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_3777_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_3778_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_3779_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_3780_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_3781_minf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X2 @ Z2 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3782_minf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z2 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3783_minf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z2 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3784_minf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z2 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3785_minf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z2 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).

% minf(10)
thf(fact_3786_minf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ X2 @ Z2 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3787_minf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ X2 @ Z2 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3788_minf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ X2 @ Z2 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3789_minf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ X2 @ Z2 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3790_minf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ X2 @ Z2 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).

% minf(9)
thf(fact_3791_pinf_I10_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ X2 )
     => ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3792_pinf_I10_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3793_pinf_I10_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z2 @ X2 )
     => ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3794_pinf_I10_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z2 @ X2 )
     => ( ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3795_pinf_I10_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z2 @ X2 )
     => ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) )
        = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ) ).

% pinf(10)
thf(fact_3796_pinf_I9_J,axiom,
    ! [D: code_integer,S: code_integer] :
    ? [Z2: code_integer] :
    ! [X2: code_integer] :
      ( ( ord_le6747313008572928689nteger @ Z2 @ X2 )
     => ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) )
        = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3797_pinf_I9_J,axiom,
    ! [D: real,S: real] :
    ? [Z2: real] :
    ! [X2: real] :
      ( ( ord_less_real @ Z2 @ X2 )
     => ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) )
        = ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3798_pinf_I9_J,axiom,
    ! [D: rat,S: rat] :
    ? [Z2: rat] :
    ! [X2: rat] :
      ( ( ord_less_rat @ Z2 @ X2 )
     => ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) )
        = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3799_pinf_I9_J,axiom,
    ! [D: nat,S: nat] :
    ? [Z2: nat] :
    ! [X2: nat] :
      ( ( ord_less_nat @ Z2 @ X2 )
     => ( ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) )
        = ( dvd_dvd_nat @ D @ ( plus_plus_nat @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3800_pinf_I9_J,axiom,
    ! [D: int,S: int] :
    ? [Z2: int] :
    ! [X2: int] :
      ( ( ord_less_int @ Z2 @ X2 )
     => ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) )
        = ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ S ) ) ) ) ).

% pinf(9)
thf(fact_3801_length__mul__elem,axiom,
    ! [Xs: list_list_VEBT_VEBT,N2: nat] :
      ( ! [X5: list_VEBT_VEBT] :
          ( ( member2936631157270082147T_VEBT @ X5 @ ( set_list_VEBT_VEBT2 @ Xs ) )
         => ( ( size_s6755466524823107622T_VEBT @ X5 )
            = N2 ) )
     => ( ( size_s6755466524823107622T_VEBT @ ( concat_VEBT_VEBT @ Xs ) )
        = ( times_times_nat @ ( size_s8217280938318005548T_VEBT @ Xs ) @ N2 ) ) ) ).

% length_mul_elem
thf(fact_3802_length__mul__elem,axiom,
    ! [Xs: list_list_o,N2: nat] :
      ( ! [X5: list_o] :
          ( ( member_list_o @ X5 @ ( set_list_o2 @ Xs ) )
         => ( ( size_size_list_o @ X5 )
            = N2 ) )
     => ( ( size_size_list_o @ ( concat_o @ Xs ) )
        = ( times_times_nat @ ( size_s2710708370519433104list_o @ Xs ) @ N2 ) ) ) ).

% length_mul_elem
thf(fact_3803_length__mul__elem,axiom,
    ! [Xs: list_list_nat,N2: nat] :
      ( ! [X5: list_nat] :
          ( ( member_list_nat @ X5 @ ( set_list_nat2 @ Xs ) )
         => ( ( size_size_list_nat @ X5 )
            = N2 ) )
     => ( ( size_size_list_nat @ ( concat_nat @ Xs ) )
        = ( times_times_nat @ ( size_s3023201423986296836st_nat @ Xs ) @ N2 ) ) ) ).

% length_mul_elem
thf(fact_3804_length__mul__elem,axiom,
    ! [Xs: list_list_int,N2: nat] :
      ( ! [X5: list_int] :
          ( ( member_list_int @ X5 @ ( set_list_int2 @ Xs ) )
         => ( ( size_size_list_int @ X5 )
            = N2 ) )
     => ( ( size_size_list_int @ ( concat_int @ Xs ) )
        = ( times_times_nat @ ( size_s533118279054570080st_int @ Xs ) @ N2 ) ) ) ).

% length_mul_elem
thf(fact_3805_mult__le__cancel__iff2,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ Z @ X4 ) @ ( times_times_real @ Z @ Y ) )
        = ( ord_less_eq_real @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3806_mult__le__cancel__iff2,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ Z @ X4 ) @ ( times_times_rat @ Z @ Y ) )
        = ( ord_less_eq_rat @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3807_mult__le__cancel__iff2,axiom,
    ! [Z: int,X4: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ Z @ X4 ) @ ( times_times_int @ Z @ Y ) )
        = ( ord_less_eq_int @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff2
thf(fact_3808_mult__le__cancel__iff1,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_eq_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_eq_real @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3809_mult__le__cancel__iff1,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_eq_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_eq_rat @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3810_mult__le__cancel__iff1,axiom,
    ! [Z: int,X4: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_eq_int @ X4 @ Y ) ) ) ).

% mult_le_cancel_iff1
thf(fact_3811_product__nth,axiom,
    ! [N2: nat,Xs: list_num,Ys: list_num] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_size_list_num @ Xs ) @ ( size_size_list_num @ Ys ) ) )
     => ( ( nth_Pr6456567536196504476um_num @ ( product_num_num @ Xs @ Ys ) @ N2 )
        = ( product_Pair_num_num @ ( nth_num @ Xs @ ( divide_divide_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) @ ( nth_num @ Ys @ ( modulo_modulo_nat @ N2 @ ( size_size_list_num @ Ys ) ) ) ) ) ) ).

% product_nth
thf(fact_3812_product__nth,axiom,
    ! [N2: nat,Xs: list_Code_integer,Ys: list_o] :
      ( ( ord_less_nat @ N2 @ ( times_times_nat @ ( size_s3445333598471063425nteger @ Xs ) @ ( size_size_list_o @ Ys ) ) )
     => ( ( nth_Pr8522763379788166057eger_o @ ( produc3607205314601156340eger_o @ Xs @ Ys ) @ N2 )
        = ( produc6677183202524767010eger_o @ ( nth_Code_integer @ 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_3813_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_3814_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_3815_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_3816_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_3817_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_3818_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_3819_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_3820_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_3821_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_3822_pos__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R3 ) )
       => ( 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 @ Q3 @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) ) ) ) ) ) ).

% pos_eucl_rel_int_mult_2
thf(fact_3823_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_3824_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_3825_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_3826_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_3827_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_3828_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_3829_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_3830_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_3831_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_3832_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_3833_triangle__Suc,axiom,
    ! [N2: nat] :
      ( ( nat_triangle @ ( suc @ N2 ) )
      = ( plus_plus_nat @ ( nat_triangle @ N2 ) @ ( suc @ N2 ) ) ) ).

% triangle_Suc
thf(fact_3834_unique__remainder,axiom,
    ! [A: int,B: int,Q3: int,R3: int,Q4: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R3 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R4 ) )
       => ( R3 = R4 ) ) ) ).

% unique_remainder
thf(fact_3835_unique__quotient,axiom,
    ! [A: int,B: int,Q3: int,R3: int,Q4: int,R4: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R3 ) )
     => ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q4 @ R4 ) )
       => ( Q3 = Q4 ) ) ) ).

% unique_quotient
thf(fact_3836_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_3837_div__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R3 ) )
     => ( ( divide_divide_int @ K @ L )
        = Q3 ) ) ).

% div_int_unique
thf(fact_3838_mod__int__unique,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R3 ) )
     => ( ( modulo_modulo_int @ K @ L )
        = R3 ) ) ).

% mod_int_unique
thf(fact_3839_eucl__rel__int__dividesI,axiom,
    ! [L: int,K: int,Q3: int] :
      ( ( L != zero_zero_int )
     => ( ( K
          = ( times_times_int @ Q3 @ L ) )
       => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ zero_zero_int ) ) ) ) ).

% eucl_rel_int_dividesI
thf(fact_3840_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_3841_eucl__rel__int__iff,axiom,
    ! [K: int,L: int,Q3: int,R3: int] :
      ( ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R3 ) )
      = ( ( K
          = ( plus_plus_int @ ( times_times_int @ L @ Q3 ) @ R3 ) )
        & ( ( ord_less_int @ zero_zero_int @ L )
         => ( ( ord_less_eq_int @ zero_zero_int @ R3 )
            & ( ord_less_int @ R3 @ L ) ) )
        & ( ~ ( ord_less_int @ zero_zero_int @ L )
         => ( ( ( ord_less_int @ L @ zero_zero_int )
             => ( ( ord_less_int @ L @ R3 )
                & ( ord_less_eq_int @ R3 @ zero_zero_int ) ) )
            & ( ~ ( ord_less_int @ L @ zero_zero_int )
             => ( Q3 = zero_zero_int ) ) ) ) ) ) ).

% eucl_rel_int_iff
thf(fact_3842_mult__less__iff1,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( ( ord_less_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ Y @ Z ) )
        = ( ord_less_real @ X4 @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3843_mult__less__iff1,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Z )
     => ( ( ord_less_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ Y @ Z ) )
        = ( ord_less_rat @ X4 @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3844_mult__less__iff1,axiom,
    ! [Z: int,X4: int,Y: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_int @ ( times_times_int @ X4 @ Z ) @ ( times_times_int @ Y @ Z ) )
        = ( ord_less_int @ X4 @ Y ) ) ) ).

% mult_less_iff1
thf(fact_3845_Divides_Oadjust__div__eq,axiom,
    ! [Q3: int,R3: int] :
      ( ( adjust_div @ ( product_Pair_int_int @ Q3 @ R3 ) )
      = ( plus_plus_int @ Q3 @ ( zero_n2684676970156552555ol_int @ ( R3 != zero_zero_int ) ) ) ) ).

% Divides.adjust_div_eq
thf(fact_3846_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_3847_neg__eucl__rel__int__mult__2,axiom,
    ! [B: int,A: int,Q3: int,R3: int] :
      ( ( ord_less_eq_int @ B @ zero_zero_int )
     => ( ( eucl_rel_int @ ( plus_plus_int @ A @ one_one_int ) @ B @ ( product_Pair_int_int @ Q3 @ R3 ) )
       => ( 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 @ Q3 @ ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ R3 ) @ one_one_int ) ) ) ) ) ).

% neg_eucl_rel_int_mult_2
thf(fact_3848_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_3849_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_3850_set__decode__Suc,axiom,
    ! [N2: nat,X4: nat] :
      ( ( member_nat @ ( suc @ N2 ) @ ( nat_set_decode @ X4 ) )
      = ( member_nat @ N2 @ ( nat_set_decode @ ( divide_divide_nat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% set_decode_Suc
thf(fact_3851_set__decode__0,axiom,
    ! [X4: nat] :
      ( ( member_nat @ zero_zero_nat @ ( nat_set_decode @ X4 ) )
      = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ X4 ) ) ) ).

% set_decode_0
thf(fact_3852_add__scale__eq__noteq,axiom,
    ! [R3: rat,A: rat,B: rat,C: rat,D: rat] :
      ( ( R3 != zero_zero_rat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_rat @ A @ ( times_times_rat @ R3 @ C ) )
         != ( plus_plus_rat @ B @ ( times_times_rat @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3853_add__scale__eq__noteq,axiom,
    ! [R3: complex,A: complex,B: complex,C: complex,D: complex] :
      ( ( R3 != zero_zero_complex )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_complex @ A @ ( times_times_complex @ R3 @ C ) )
         != ( plus_plus_complex @ B @ ( times_times_complex @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3854_add__scale__eq__noteq,axiom,
    ! [R3: real,A: real,B: real,C: real,D: real] :
      ( ( R3 != zero_zero_real )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_real @ A @ ( times_times_real @ R3 @ C ) )
         != ( plus_plus_real @ B @ ( times_times_real @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3855_add__scale__eq__noteq,axiom,
    ! [R3: nat,A: nat,B: nat,C: nat,D: nat] :
      ( ( R3 != zero_zero_nat )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_nat @ A @ ( times_times_nat @ R3 @ C ) )
         != ( plus_plus_nat @ B @ ( times_times_nat @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3856_add__scale__eq__noteq,axiom,
    ! [R3: int,A: int,B: int,C: int,D: int] :
      ( ( R3 != zero_zero_int )
     => ( ( ( A = B )
          & ( C != D ) )
       => ( ( plus_plus_int @ A @ ( times_times_int @ R3 @ C ) )
         != ( plus_plus_int @ B @ ( times_times_int @ R3 @ D ) ) ) ) ) ).

% add_scale_eq_noteq
thf(fact_3857_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_3858_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_3859_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_3860_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_3861_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_3862_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_3863_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_3864_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_3865_diff__zero,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_zero
thf(fact_3866_diff__zero,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_zero
thf(fact_3867_diff__zero,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_zero
thf(fact_3868_diff__zero,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ A @ zero_zero_nat )
      = A ) ).

% diff_zero
thf(fact_3869_diff__zero,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_zero
thf(fact_3870_zero__diff,axiom,
    ! [A: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_diff
thf(fact_3871_diff__0__right,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ zero_zero_complex )
      = A ) ).

% diff_0_right
thf(fact_3872_diff__0__right,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ zero_zero_real )
      = A ) ).

% diff_0_right
thf(fact_3873_diff__0__right,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ zero_zero_rat )
      = A ) ).

% diff_0_right
thf(fact_3874_diff__0__right,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ zero_zero_int )
      = A ) ).

% diff_0_right
thf(fact_3875_diff__self,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ A @ A )
      = zero_zero_complex ) ).

% diff_self
thf(fact_3876_diff__self,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ A @ A )
      = zero_zero_real ) ).

% diff_self
thf(fact_3877_diff__self,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ A @ A )
      = zero_zero_rat ) ).

% diff_self
thf(fact_3878_diff__self,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ A @ A )
      = zero_zero_int ) ).

% diff_self
thf(fact_3879_add__diff__cancel,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( plus_plus_real @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3880_add__diff__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( plus_plus_rat @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3881_add__diff__cancel,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( plus_plus_int @ A @ B ) @ B )
      = A ) ).

% add_diff_cancel
thf(fact_3882_diff__add__cancel,axiom,
    ! [A: real,B: real] :
      ( ( plus_plus_real @ ( minus_minus_real @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3883_diff__add__cancel,axiom,
    ! [A: rat,B: rat] :
      ( ( plus_plus_rat @ ( minus_minus_rat @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3884_diff__add__cancel,axiom,
    ! [A: int,B: int] :
      ( ( plus_plus_int @ ( minus_minus_int @ A @ B ) @ B )
      = A ) ).

% diff_add_cancel
thf(fact_3885_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_3886_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_3887_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_3888_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_3889_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_3890_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_3891_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_3892_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_3893_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_3894_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_3895_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_3896_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_3897_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_3898_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_3899_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_3900_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_3901_minus__mod__self2,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ B )
      = ( modulo_modulo_int @ A @ B ) ) ).

% minus_mod_self2
thf(fact_3902_minus__mod__self2,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ B )
      = ( modulo364778990260209775nteger @ A @ B ) ) ).

% minus_mod_self2
thf(fact_3903_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_3904_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_3905_diff__self__eq__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ M )
      = zero_zero_nat ) ).

% diff_self_eq_0
thf(fact_3906_diff__0__eq__0,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% diff_0_eq_0
thf(fact_3907_diff__diff__cancel,axiom,
    ! [I2: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ N2 )
     => ( ( minus_minus_nat @ N2 @ ( minus_minus_nat @ N2 @ I2 ) )
        = I2 ) ) ).

% diff_diff_cancel
thf(fact_3908_diff__diff__left,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ I2 @ ( plus_plus_nat @ J @ K ) ) ) ).

% diff_diff_left
thf(fact_3909_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_3910_concat__bit__0,axiom,
    ! [K: int,L: int] :
      ( ( bit_concat_bit @ zero_zero_nat @ K @ L )
      = L ) ).

% concat_bit_0
thf(fact_3911_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_3912_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_3913_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_3914_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_3915_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_3916_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_3917_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_3918_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_3919_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_3920_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_3921_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_3922_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_3923_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_3924_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_3925_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ one_one_complex )
    = zero_zero_complex ) ).

% diff_numeral_special(9)
thf(fact_3926_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_real @ one_one_real @ one_one_real )
    = zero_zero_real ) ).

% diff_numeral_special(9)
thf(fact_3927_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ one_one_rat )
    = zero_zero_rat ) ).

% diff_numeral_special(9)
thf(fact_3928_diff__numeral__special_I9_J,axiom,
    ( ( minus_minus_int @ one_one_int @ one_one_int )
    = zero_zero_int ) ).

% diff_numeral_special(9)
thf(fact_3929_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_3930_right__diff__distrib__numeral,axiom,
    ! [V: num,B: rat,C: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( minus_minus_rat @ B @ C ) )
      = ( minus_minus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ B ) @ ( times_times_rat @ ( numeral_numeral_rat @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3931_right__diff__distrib__numeral,axiom,
    ! [V: num,B: complex,C: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( minus_minus_complex @ B @ C ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ B ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3932_right__diff__distrib__numeral,axiom,
    ! [V: num,B: real,C: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( minus_minus_real @ B @ C ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ V ) @ B ) @ ( times_times_real @ ( numeral_numeral_real @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3933_right__diff__distrib__numeral,axiom,
    ! [V: num,B: int,C: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( minus_minus_int @ B @ C ) )
      = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ V ) @ B ) @ ( times_times_int @ ( numeral_numeral_int @ V ) @ C ) ) ) ).

% right_diff_distrib_numeral
thf(fact_3934_left__diff__distrib__numeral,axiom,
    ! [A: rat,B: rat,V: num] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ B ) @ ( numeral_numeral_rat @ V ) )
      = ( minus_minus_rat @ ( times_times_rat @ A @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ B @ ( numeral_numeral_rat @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3935_left__diff__distrib__numeral,axiom,
    ! [A: complex,B: complex,V: num] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ B ) @ ( numera6690914467698888265omplex @ V ) )
      = ( minus_minus_complex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ B @ ( numera6690914467698888265omplex @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3936_left__diff__distrib__numeral,axiom,
    ! [A: real,B: real,V: num] :
      ( ( times_times_real @ ( minus_minus_real @ A @ B ) @ ( numeral_numeral_real @ V ) )
      = ( minus_minus_real @ ( times_times_real @ A @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ B @ ( numeral_numeral_real @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3937_left__diff__distrib__numeral,axiom,
    ! [A: int,B: int,V: num] :
      ( ( times_times_int @ ( minus_minus_int @ A @ B ) @ ( numeral_numeral_int @ V ) )
      = ( minus_minus_int @ ( times_times_int @ A @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ B @ ( numeral_numeral_int @ V ) ) ) ) ).

% left_diff_distrib_numeral
thf(fact_3938_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_3939_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_3940_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_3941_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_3942_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_3943_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_3944_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_3945_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_3946_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_3947_of__bool__not__iff,axiom,
    ! [P: $o] :
      ( ( zero_n356916108424825756nteger @ ~ P )
      = ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( zero_n356916108424825756nteger @ P ) ) ) ).

% of_bool_not_iff
thf(fact_3948_Nat_Odiff__diff__right,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).

% Nat.diff_diff_right
thf(fact_3949_Nat_Oadd__diff__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
        = ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K ) ) ) ).

% Nat.add_diff_assoc2
thf(fact_3950_Nat_Oadd__diff__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K ) ) ) ).

% Nat.add_diff_assoc
thf(fact_3951_diff__Suc__1,axiom,
    ! [N2: nat] :
      ( ( minus_minus_nat @ ( suc @ N2 ) @ one_one_nat )
      = N2 ) ).

% diff_Suc_1
thf(fact_3952_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_3953_signed__take__bit__numeral__of__1,axiom,
    ! [K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ K ) @ one_one_int )
      = one_one_int ) ).

% signed_take_bit_numeral_of_1
thf(fact_3954_concat__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_concat_bit @ N2 @ K @ L ) )
      = ( ord_less_eq_int @ zero_zero_int @ L ) ) ).

% concat_bit_nonnegative_iff
thf(fact_3955_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_3956_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_3957_diff__Suc__diff__eq2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( suc @ ( minus_minus_nat @ J @ K ) ) @ I2 )
        = ( minus_minus_nat @ ( suc @ J ) @ ( plus_plus_nat @ K @ I2 ) ) ) ) ).

% diff_Suc_diff_eq2
thf(fact_3958_diff__Suc__diff__eq1,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ I2 @ ( suc @ ( minus_minus_nat @ J @ K ) ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ I2 @ K ) @ ( suc @ J ) ) ) ) ).

% diff_Suc_diff_eq1
thf(fact_3959_zle__diff1__eq,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ W @ ( minus_minus_int @ Z @ one_one_int ) )
      = ( ord_less_int @ W @ Z ) ) ).

% zle_diff1_eq
thf(fact_3960_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_3961_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_3962_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_3963_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_3964_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_3965_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_3966_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_3967_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_3968_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_3969_odd__two__times__div__two__nat,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% odd_two_times_div_two_nat
thf(fact_3970_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_3971_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_3972_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_3973_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_3974_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_3975_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_3976_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_3977_signed__take__bit__diff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( minus_minus_int @ K @ L ) ) ) ).

% signed_take_bit_diff
thf(fact_3978_diff__commute,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( minus_minus_nat @ ( minus_minus_nat @ I2 @ J ) @ K )
      = ( minus_minus_nat @ ( minus_minus_nat @ I2 @ K ) @ J ) ) ).

% diff_commute
thf(fact_3979_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_3980_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_3981_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_3982_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_3983_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_3984_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_3985_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_3986_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_3987_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_3988_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_3989_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_3990_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_3991_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: complex,Z4: complex] : Y6 = Z4 )
    = ( ^ [A3: complex,B2: complex] :
          ( ( minus_minus_complex @ A3 @ B2 )
          = zero_zero_complex ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3992_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: real,Z4: real] : Y6 = Z4 )
    = ( ^ [A3: real,B2: real] :
          ( ( minus_minus_real @ A3 @ B2 )
          = zero_zero_real ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3993_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: rat,Z4: rat] : Y6 = Z4 )
    = ( ^ [A3: rat,B2: rat] :
          ( ( minus_minus_rat @ A3 @ B2 )
          = zero_zero_rat ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3994_eq__iff__diff__eq__0,axiom,
    ( ( ^ [Y6: int,Z4: int] : Y6 = Z4 )
    = ( ^ [A3: int,B2: int] :
          ( ( minus_minus_int @ A3 @ B2 )
          = zero_zero_int ) ) ) ).

% eq_iff_diff_eq_0
thf(fact_3995_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_3996_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_3997_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_3998_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_3999_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_4000_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_4001_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_4002_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_4003_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_4004_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_4005_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_4006_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_4007_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_4008_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_4009_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_4010_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_4011_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_4012_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_4013_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_4014_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_4015_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_4016_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_4017_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_4018_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_4019_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_4020_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_4021_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_4022_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_4023_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_4024_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_4025_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_4026_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_4027_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_4028_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_4029_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_4030_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_4031_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_4032_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_4033_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_4034_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_4035_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_4036_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_4037_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_4038_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_4039_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_4040_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_4041_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_4042_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_4043_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_4044_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_4045_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_4046_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_4047_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_4048_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_4049_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_4050_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_4051_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_4052_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_4053_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_4054_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_4055_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_4056_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_4057_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_4058_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_4059_diff__divide__distrib,axiom,
    ! [A: complex,B: complex,C: complex] :
      ( ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ B ) @ C )
      = ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ C ) @ ( divide1717551699836669952omplex @ B @ C ) ) ) ).

% diff_divide_distrib
thf(fact_4060_dvd__diff,axiom,
    ! [X4: code_integer,Y: code_integer,Z: code_integer] :
      ( ( dvd_dvd_Code_integer @ X4 @ Y )
     => ( ( dvd_dvd_Code_integer @ X4 @ Z )
       => ( dvd_dvd_Code_integer @ X4 @ ( minus_8373710615458151222nteger @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4061_dvd__diff,axiom,
    ! [X4: real,Y: real,Z: real] :
      ( ( dvd_dvd_real @ X4 @ Y )
     => ( ( dvd_dvd_real @ X4 @ Z )
       => ( dvd_dvd_real @ X4 @ ( minus_minus_real @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4062_dvd__diff,axiom,
    ! [X4: rat,Y: rat,Z: rat] :
      ( ( dvd_dvd_rat @ X4 @ Y )
     => ( ( dvd_dvd_rat @ X4 @ Z )
       => ( dvd_dvd_rat @ X4 @ ( minus_minus_rat @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4063_dvd__diff,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( dvd_dvd_int @ X4 @ Y )
     => ( ( dvd_dvd_int @ X4 @ Z )
       => ( dvd_dvd_int @ X4 @ ( minus_minus_int @ Y @ Z ) ) ) ) ).

% dvd_diff
thf(fact_4064_dvd__diff__commute,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ C @ B ) )
      = ( dvd_dvd_Code_integer @ A @ ( minus_8373710615458151222nteger @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_4065_dvd__diff__commute,axiom,
    ! [A: int,C: int,B: int] :
      ( ( dvd_dvd_int @ A @ ( minus_minus_int @ C @ B ) )
      = ( dvd_dvd_int @ A @ ( minus_minus_int @ B @ C ) ) ) ).

% dvd_diff_commute
thf(fact_4066_zero__induct__lemma,axiom,
    ! [P: nat > $o,K: nat,I2: nat] :
      ( ( P @ K )
     => ( ! [N3: nat] :
            ( ( P @ ( suc @ N3 ) )
           => ( P @ N3 ) )
       => ( P @ ( minus_minus_nat @ K @ I2 ) ) ) ) ).

% zero_induct_lemma
thf(fact_4067_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_4068_minus__nat_Odiff__0,axiom,
    ! [M: nat] :
      ( ( minus_minus_nat @ M @ zero_zero_nat )
      = M ) ).

% minus_nat.diff_0
thf(fact_4069_mod__diff__right__eq,axiom,
    ! [A: int,B: int,C: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ A @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_4070_mod__diff__right__eq,axiom,
    ! [A: code_integer,B: code_integer,C: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_right_eq
thf(fact_4071_mod__diff__left__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ B ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_4072_mod__diff__left__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ B ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_left_eq
thf(fact_4073_mod__diff__cong,axiom,
    ! [A: int,C: int,A4: int,B: int,B4: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ A4 @ C ) )
     => ( ( ( modulo_modulo_int @ B @ C )
          = ( modulo_modulo_int @ B4 @ C ) )
       => ( ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C )
          = ( modulo_modulo_int @ ( minus_minus_int @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_4074_mod__diff__cong,axiom,
    ! [A: code_integer,C: code_integer,A4: code_integer,B: code_integer,B4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ A4 @ C ) )
     => ( ( ( modulo364778990260209775nteger @ B @ C )
          = ( modulo364778990260209775nteger @ B4 @ C ) )
       => ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C )
          = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A4 @ B4 ) @ C ) ) ) ) ).

% mod_diff_cong
thf(fact_4075_mod__diff__eq,axiom,
    ! [A: int,C: int,B: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ ( modulo_modulo_int @ A @ C ) @ ( modulo_modulo_int @ B @ C ) ) @ C )
      = ( modulo_modulo_int @ ( minus_minus_int @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_4076_mod__diff__eq,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ ( modulo364778990260209775nteger @ A @ C ) @ ( modulo364778990260209775nteger @ B @ C ) ) @ C )
      = ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ A @ B ) @ C ) ) ).

% mod_diff_eq
thf(fact_4077_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_4078_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_4079_signed__take__bit__mult,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( times_times_int @ K @ L ) ) ) ).

% signed_take_bit_mult
thf(fact_4080_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_4081_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_4082_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_4083_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_4084_diff__le__self,axiom,
    ! [M: nat,N2: nat] : ( ord_less_eq_nat @ ( minus_minus_nat @ M @ N2 ) @ M ) ).

% diff_le_self
thf(fact_4085_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_4086_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_4087_signed__take__bit__add,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ ( bit_ri631733984087533419it_int @ N2 @ L ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( plus_plus_int @ K @ L ) ) ) ).

% signed_take_bit_add
thf(fact_4088_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_4089_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_4090_diff__add__inverse,axiom,
    ! [N2: nat,M: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ N2 @ M ) @ N2 )
      = M ) ).

% diff_add_inverse
thf(fact_4091_diff__add__inverse2,axiom,
    ! [M: nat,N2: nat] :
      ( ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ N2 )
      = M ) ).

% diff_add_inverse2
thf(fact_4092_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_4093_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_4094_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_4095_concat__bit__assoc,axiom,
    ! [N2: nat,K: int,M: nat,L: int,R3: int] :
      ( ( bit_concat_bit @ N2 @ K @ ( bit_concat_bit @ M @ L @ R3 ) )
      = ( bit_concat_bit @ ( plus_plus_nat @ M @ N2 ) @ ( bit_concat_bit @ N2 @ K @ L ) @ R3 ) ) ).

% concat_bit_assoc
thf(fact_4096_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_4097_le__iff__diff__le__0,axiom,
    ( ord_less_eq_real
    = ( ^ [A3: real,B2: real] : ( ord_less_eq_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).

% le_iff_diff_le_0
thf(fact_4098_le__iff__diff__le__0,axiom,
    ( ord_less_eq_rat
    = ( ^ [A3: rat,B2: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).

% le_iff_diff_le_0
thf(fact_4099_le__iff__diff__le__0,axiom,
    ( ord_less_eq_int
    = ( ^ [A3: int,B2: int] : ( ord_less_eq_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).

% le_iff_diff_le_0
thf(fact_4100_less__iff__diff__less__0,axiom,
    ( ord_less_real
    = ( ^ [A3: real,B2: real] : ( ord_less_real @ ( minus_minus_real @ A3 @ B2 ) @ zero_zero_real ) ) ) ).

% less_iff_diff_less_0
thf(fact_4101_less__iff__diff__less__0,axiom,
    ( ord_less_rat
    = ( ^ [A3: rat,B2: rat] : ( ord_less_rat @ ( minus_minus_rat @ A3 @ B2 ) @ zero_zero_rat ) ) ) ).

% less_iff_diff_less_0
thf(fact_4102_less__iff__diff__less__0,axiom,
    ( ord_less_int
    = ( ^ [A3: int,B2: int] : ( ord_less_int @ ( minus_minus_int @ A3 @ B2 ) @ zero_zero_int ) ) ) ).

% less_iff_diff_less_0
thf(fact_4103_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_4104_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_4105_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_4106_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_4107_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_4108_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_4109_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_4110_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_4111_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_4112_add__le__add__imp__diff__le,axiom,
    ! [I2: real,K: real,N2: real,J: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N2 )
     => ( ( ord_less_eq_real @ N2 @ ( plus_plus_real @ J @ K ) )
       => ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ 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_4113_add__le__add__imp__diff__le,axiom,
    ! [I2: rat,K: rat,N2: rat,J: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
     => ( ( ord_less_eq_rat @ N2 @ ( plus_plus_rat @ J @ K ) )
       => ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ 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_4114_add__le__add__imp__diff__le,axiom,
    ! [I2: nat,K: nat,N2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ ( plus_plus_nat @ J @ K ) )
       => ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ 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_4115_add__le__add__imp__diff__le,axiom,
    ! [I2: int,K: int,N2: int,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
     => ( ( ord_less_eq_int @ N2 @ ( plus_plus_int @ J @ K ) )
       => ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ 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_4116_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_4117_add__le__imp__le__diff,axiom,
    ! [I2: real,K: real,N2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ I2 @ K ) @ N2 )
     => ( ord_less_eq_real @ I2 @ ( minus_minus_real @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4118_add__le__imp__le__diff,axiom,
    ! [I2: rat,K: rat,N2: rat] :
      ( ( ord_less_eq_rat @ ( plus_plus_rat @ I2 @ K ) @ N2 )
     => ( ord_less_eq_rat @ I2 @ ( minus_minus_rat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4119_add__le__imp__le__diff,axiom,
    ! [I2: nat,K: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ N2 )
     => ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4120_add__le__imp__le__diff,axiom,
    ! [I2: int,K: int,N2: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ K ) @ N2 )
     => ( ord_less_eq_int @ I2 @ ( minus_minus_int @ N2 @ K ) ) ) ).

% add_le_imp_le_diff
thf(fact_4121_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_4122_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_4123_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_4124_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_4125_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_4126_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_4127_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_4128_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_4129_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_4130_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_4131_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_4132_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_4133_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_4134_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_4135_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_4136_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_4137_square__diff__square__factored,axiom,
    ! [X4: rat,Y: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ ( times_times_rat @ Y @ Y ) )
      = ( times_times_rat @ ( plus_plus_rat @ X4 @ Y ) @ ( minus_minus_rat @ X4 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_4138_square__diff__square__factored,axiom,
    ! [X4: complex,Y: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X4 @ X4 ) @ ( times_times_complex @ Y @ Y ) )
      = ( times_times_complex @ ( plus_plus_complex @ X4 @ Y ) @ ( minus_minus_complex @ X4 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_4139_square__diff__square__factored,axiom,
    ! [X4: real,Y: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) )
      = ( times_times_real @ ( plus_plus_real @ X4 @ Y ) @ ( minus_minus_real @ X4 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_4140_square__diff__square__factored,axiom,
    ! [X4: int,Y: int] :
      ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ ( times_times_int @ Y @ Y ) )
      = ( times_times_int @ ( plus_plus_int @ X4 @ Y ) @ ( minus_minus_int @ X4 @ Y ) ) ) ).

% square_diff_square_factored
thf(fact_4141_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_4142_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_4143_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_4144_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_4145_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_4146_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_4147_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_4148_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_4149_mult__diff__mult,axiom,
    ! [X4: rat,Y: rat,A: rat,B: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X4 @ Y ) @ ( times_times_rat @ A @ B ) )
      = ( plus_plus_rat @ ( times_times_rat @ X4 @ ( minus_minus_rat @ Y @ B ) ) @ ( times_times_rat @ ( minus_minus_rat @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_4150_mult__diff__mult,axiom,
    ! [X4: complex,Y: complex,A: complex,B: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X4 @ Y ) @ ( times_times_complex @ A @ B ) )
      = ( plus_plus_complex @ ( times_times_complex @ X4 @ ( minus_minus_complex @ Y @ B ) ) @ ( times_times_complex @ ( minus_minus_complex @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_4151_mult__diff__mult,axiom,
    ! [X4: real,Y: real,A: real,B: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ Y ) @ ( times_times_real @ A @ B ) )
      = ( plus_plus_real @ ( times_times_real @ X4 @ ( minus_minus_real @ Y @ B ) ) @ ( times_times_real @ ( minus_minus_real @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_4152_mult__diff__mult,axiom,
    ! [X4: int,Y: int,A: int,B: int] :
      ( ( minus_minus_int @ ( times_times_int @ X4 @ Y ) @ ( times_times_int @ A @ B ) )
      = ( plus_plus_int @ ( times_times_int @ X4 @ ( minus_minus_int @ Y @ B ) ) @ ( times_times_int @ ( minus_minus_int @ X4 @ A ) @ B ) ) ) ).

% mult_diff_mult
thf(fact_4153_mod__eq__dvd__iff,axiom,
    ! [A: int,C: int,B: int] :
      ( ( ( modulo_modulo_int @ A @ C )
        = ( modulo_modulo_int @ B @ C ) )
      = ( dvd_dvd_int @ C @ ( minus_minus_int @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_4154_mod__eq__dvd__iff,axiom,
    ! [A: code_integer,C: code_integer,B: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ C )
        = ( modulo364778990260209775nteger @ B @ C ) )
      = ( dvd_dvd_Code_integer @ C @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% mod_eq_dvd_iff
thf(fact_4155_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_4156_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_4157_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_4158_diff__less__Suc,axiom,
    ! [M: nat,N2: nat] : ( ord_less_nat @ ( minus_minus_nat @ M @ N2 ) @ ( suc @ M ) ) ).

% diff_less_Suc
thf(fact_4159_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_4160_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_4161_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_4162_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_4163_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_4164_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_4165_less__diff__conv,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
      = ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ).

% less_diff_conv
thf(fact_4166_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_4167_Nat_Ole__imp__diff__is__add,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ( minus_minus_nat @ J @ I2 )
          = K )
        = ( J
          = ( plus_plus_nat @ K @ I2 ) ) ) ) ).

% Nat.le_imp_diff_is_add
thf(fact_4168_Nat_Odiff__add__assoc2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ J @ I2 ) @ K )
        = ( plus_plus_nat @ ( minus_minus_nat @ J @ K ) @ I2 ) ) ) ).

% Nat.diff_add_assoc2
thf(fact_4169_Nat_Odiff__add__assoc,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ ( minus_minus_nat @ J @ K ) ) ) ) ).

% Nat.diff_add_assoc
thf(fact_4170_Nat_Ole__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_eq_nat @ I2 @ ( minus_minus_nat @ J @ K ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ K ) @ J ) ) ) ).

% Nat.le_diff_conv2
thf(fact_4171_le__diff__conv,axiom,
    ! [J: nat,K: nat,I2: nat] :
      ( ( ord_less_eq_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
      = ( ord_less_eq_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ).

% le_diff_conv
thf(fact_4172_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_4173_int__le__induct,axiom,
    ! [I2: int,K: int,P: int > $o] :
      ( ( ord_less_eq_int @ I2 @ K )
     => ( ( P @ K )
       => ( ! [I4: int] :
              ( ( ord_less_eq_int @ I4 @ K )
             => ( ( P @ I4 )
               => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_le_induct
thf(fact_4174_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_4175_int__less__induct,axiom,
    ! [I2: int,K: int,P: int > $o] :
      ( ( ord_less_int @ I2 @ K )
     => ( ( P @ ( minus_minus_int @ K @ one_one_int ) )
       => ( ! [I4: int] :
              ( ( ord_less_int @ I4 @ K )
             => ( ( P @ I4 )
               => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_less_induct
thf(fact_4176_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_4177_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_4178_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_4179_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_4180_mod__if,axiom,
    ( modulo_modulo_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( ord_less_nat @ M6 @ N ) @ M6 @ ( modulo_modulo_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ).

% mod_if
thf(fact_4181_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_4182_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_4183_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_4184_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_4185_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_4186_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_4187_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_4188_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_4189_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_4190_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_4191_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_4192_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_4193_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_4194_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_4195_divide__diff__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ X4 @ Z ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ X4 @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_4196_divide__diff__eq__iff,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ X4 @ Z ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ X4 @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_4197_divide__diff__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X4 @ Z ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ X4 @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% divide_diff_eq_iff
thf(fact_4198_diff__divide__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ X4 @ ( divide_divide_rat @ Y @ Z ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_4199_diff__divide__eq__iff,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ X4 @ ( divide_divide_real @ Y @ Z ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_4200_diff__divide__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ X4 @ ( divide1717551699836669952omplex @ Y @ Z ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X4 @ Z ) @ Y ) @ Z ) ) ) ).

% diff_divide_eq_iff
thf(fact_4201_diff__frac__eq,axiom,
    ! [Y: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ X4 @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4202_diff__frac__eq,axiom,
    ! [Y: real,Z: real,X4: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ X4 @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4203_diff__frac__eq,axiom,
    ! [Y: complex,Z: complex,X4: complex,W: complex] :
      ( ( Y != zero_zero_complex )
     => ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ X4 @ Y ) @ ( divide1717551699836669952omplex @ W @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ X4 @ Z ) @ ( times_times_complex @ W @ Y ) ) @ ( times_times_complex @ Y @ Z ) ) ) ) ) ).

% diff_frac_eq
thf(fact_4204_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ A @ ( divide_divide_rat @ B @ Z ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4205_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ A @ ( divide_divide_real @ B @ Z ) )
          = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4206_add__divide__eq__if__simps_I4_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = A ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ A @ ( divide1717551699836669952omplex @ B @ Z ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( times_times_complex @ A @ Z ) @ B ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(4)
thf(fact_4207_square__diff__one__factored,axiom,
    ! [X4: rat] :
      ( ( minus_minus_rat @ ( times_times_rat @ X4 @ X4 ) @ one_one_rat )
      = ( times_times_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) @ ( minus_minus_rat @ X4 @ one_one_rat ) ) ) ).

% square_diff_one_factored
thf(fact_4208_square__diff__one__factored,axiom,
    ! [X4: complex] :
      ( ( minus_minus_complex @ ( times_times_complex @ X4 @ X4 ) @ one_one_complex )
      = ( times_times_complex @ ( plus_plus_complex @ X4 @ one_one_complex ) @ ( minus_minus_complex @ X4 @ one_one_complex ) ) ) ).

% square_diff_one_factored
thf(fact_4209_square__diff__one__factored,axiom,
    ! [X4: real] :
      ( ( minus_minus_real @ ( times_times_real @ X4 @ X4 ) @ one_one_real )
      = ( times_times_real @ ( plus_plus_real @ X4 @ one_one_real ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).

% square_diff_one_factored
thf(fact_4210_square__diff__one__factored,axiom,
    ! [X4: int] :
      ( ( minus_minus_int @ ( times_times_int @ X4 @ X4 ) @ one_one_int )
      = ( times_times_int @ ( plus_plus_int @ X4 @ one_one_int ) @ ( minus_minus_int @ X4 @ one_one_int ) ) ) ).

% square_diff_one_factored
thf(fact_4211_inf__period_I4_J,axiom,
    ! [D: code_integer,D4: code_integer,T2: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X2: code_integer,K4: code_integer] :
          ( ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T2 ) ) )
          = ( ~ ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).

% inf_period(4)
thf(fact_4212_inf__period_I4_J,axiom,
    ! [D: rat,D4: rat,T2: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X2: rat,K4: rat] :
          ( ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T2 ) ) )
          = ( ~ ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).

% inf_period(4)
thf(fact_4213_inf__period_I4_J,axiom,
    ! [D: complex,D4: complex,T2: complex] :
      ( ( dvd_dvd_complex @ D @ D4 )
     => ! [X2: complex,K4: complex] :
          ( ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T2 ) ) )
          = ( ~ ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).

% inf_period(4)
thf(fact_4214_inf__period_I4_J,axiom,
    ! [D: real,D4: real,T2: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X2: real,K4: real] :
          ( ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T2 ) ) )
          = ( ~ ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).

% inf_period(4)
thf(fact_4215_inf__period_I4_J,axiom,
    ! [D: int,D4: int,T2: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X2: int,K4: int] :
          ( ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T2 ) ) )
          = ( ~ ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D4 ) ) @ T2 ) ) ) ) ) ).

% inf_period(4)
thf(fact_4216_inf__period_I3_J,axiom,
    ! [D: code_integer,D4: code_integer,T2: code_integer] :
      ( ( dvd_dvd_Code_integer @ D @ D4 )
     => ! [X2: code_integer,K4: code_integer] :
          ( ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ X2 @ T2 ) )
          = ( dvd_dvd_Code_integer @ D @ ( plus_p5714425477246183910nteger @ ( minus_8373710615458151222nteger @ X2 @ ( times_3573771949741848930nteger @ K4 @ D4 ) ) @ T2 ) ) ) ) ).

% inf_period(3)
thf(fact_4217_inf__period_I3_J,axiom,
    ! [D: rat,D4: rat,T2: rat] :
      ( ( dvd_dvd_rat @ D @ D4 )
     => ! [X2: rat,K4: rat] :
          ( ( dvd_dvd_rat @ D @ ( plus_plus_rat @ X2 @ T2 ) )
          = ( dvd_dvd_rat @ D @ ( plus_plus_rat @ ( minus_minus_rat @ X2 @ ( times_times_rat @ K4 @ D4 ) ) @ T2 ) ) ) ) ).

% inf_period(3)
thf(fact_4218_inf__period_I3_J,axiom,
    ! [D: complex,D4: complex,T2: complex] :
      ( ( dvd_dvd_complex @ D @ D4 )
     => ! [X2: complex,K4: complex] :
          ( ( dvd_dvd_complex @ D @ ( plus_plus_complex @ X2 @ T2 ) )
          = ( dvd_dvd_complex @ D @ ( plus_plus_complex @ ( minus_minus_complex @ X2 @ ( times_times_complex @ K4 @ D4 ) ) @ T2 ) ) ) ) ).

% inf_period(3)
thf(fact_4219_inf__period_I3_J,axiom,
    ! [D: real,D4: real,T2: real] :
      ( ( dvd_dvd_real @ D @ D4 )
     => ! [X2: real,K4: real] :
          ( ( dvd_dvd_real @ D @ ( plus_plus_real @ X2 @ T2 ) )
          = ( dvd_dvd_real @ D @ ( plus_plus_real @ ( minus_minus_real @ X2 @ ( times_times_real @ K4 @ D4 ) ) @ T2 ) ) ) ) ).

% inf_period(3)
thf(fact_4220_inf__period_I3_J,axiom,
    ! [D: int,D4: int,T2: int] :
      ( ( dvd_dvd_int @ D @ D4 )
     => ! [X2: int,K4: int] :
          ( ( dvd_dvd_int @ D @ ( plus_plus_int @ X2 @ T2 ) )
          = ( dvd_dvd_int @ D @ ( plus_plus_int @ ( minus_minus_int @ X2 @ ( times_times_int @ K4 @ D4 ) ) @ T2 ) ) ) ) ).

% inf_period(3)
thf(fact_4221_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_4222_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_4223_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_4224_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_4225_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_4226_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_4227_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_4228_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_4229_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_4230_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_4231_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_4232_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_4233_diff__Suc__less,axiom,
    ! [N2: nat,I2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ord_less_nat @ ( minus_minus_nat @ N2 @ ( suc @ I2 ) ) @ N2 ) ) ).

% diff_Suc_less
thf(fact_4234_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_4235_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_4236_less__diff__conv2,axiom,
    ! [K: nat,J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ K @ J )
     => ( ( ord_less_nat @ ( minus_minus_nat @ J @ K ) @ I2 )
        = ( ord_less_nat @ J @ ( plus_plus_nat @ I2 @ K ) ) ) ) ).

% less_diff_conv2
thf(fact_4237_nat__diff__add__eq2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_diff_add_eq2
thf(fact_4238_nat__diff__add__eq1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( minus_minus_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_diff_add_eq1
thf(fact_4239_nat__le__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_le_add_iff2
thf(fact_4240_nat__le__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_eq_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_le_add_iff1
thf(fact_4241_nat__eq__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( M
          = ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_eq_add_iff2
thf(fact_4242_nat__eq__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M )
          = ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M )
          = N2 ) ) ) ).

% nat_eq_add_iff1
thf(fact_4243_plusinfinity,axiom,
    ! [D: int,P6: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K2: int] :
            ( ( P6 @ X5 )
            = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z3: int] :
            ! [X5: int] :
              ( ( ord_less_int @ Z3 @ X5 )
             => ( ( P @ X5 )
                = ( P6 @ X5 ) ) )
         => ( ? [X_1: int] : ( P6 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% plusinfinity
thf(fact_4244_minusinfinity,axiom,
    ! [D: int,P1: int > $o,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K2: int] :
            ( ( P1 @ X5 )
            = ( P1 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ? [Z3: int] :
            ! [X5: int] :
              ( ( ord_less_int @ X5 @ Z3 )
             => ( ( P @ X5 )
                = ( P1 @ X5 ) ) )
         => ( ? [X_1: int] : ( P1 @ X_1 )
           => ? [X_12: int] : ( P @ X_12 ) ) ) ) ) ).

% minusinfinity
thf(fact_4245_int__induct,axiom,
    ! [P: int > $o,K: int,I2: int] :
      ( ( P @ K )
     => ( ! [I4: int] :
            ( ( ord_less_eq_int @ K @ I4 )
           => ( ( P @ I4 )
             => ( P @ ( plus_plus_int @ I4 @ one_one_int ) ) ) )
       => ( ! [I4: int] :
              ( ( ord_less_eq_int @ I4 @ K )
             => ( ( P @ I4 )
               => ( P @ ( minus_minus_int @ I4 @ one_one_int ) ) ) )
         => ( P @ I2 ) ) ) ) ).

% int_induct
thf(fact_4246_mod__eq__dvd__iff__nat,axiom,
    ! [N2: nat,M: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ N2 @ M )
     => ( ( ( modulo_modulo_nat @ M @ Q3 )
          = ( modulo_modulo_nat @ N2 @ Q3 ) )
        = ( dvd_dvd_nat @ Q3 @ ( minus_minus_nat @ M @ N2 ) ) ) ) ).

% mod_eq_dvd_iff_nat
thf(fact_4247_modulo__nat__def,axiom,
    ( modulo_modulo_nat
    = ( ^ [M6: nat,N: nat] : ( minus_minus_nat @ M6 @ ( times_times_nat @ ( divide_divide_nat @ M6 @ N ) @ N ) ) ) ) ).

% modulo_nat_def
thf(fact_4248_frac__le__eq,axiom,
    ! [Y: real,Z: real,X4: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ X4 @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_eq_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_le_eq
thf(fact_4249_frac__le__eq,axiom,
    ! [Y: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( divide_divide_rat @ X4 @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_eq_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_le_eq
thf(fact_4250_frac__less__eq,axiom,
    ! [Y: rat,Z: rat,X4: rat,W: rat] :
      ( ( Y != zero_zero_rat )
     => ( ( Z != zero_zero_rat )
       => ( ( ord_less_rat @ ( divide_divide_rat @ X4 @ Y ) @ ( divide_divide_rat @ W @ Z ) )
          = ( ord_less_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( times_times_rat @ X4 @ Z ) @ ( times_times_rat @ W @ Y ) ) @ ( times_times_rat @ Y @ Z ) ) @ zero_zero_rat ) ) ) ) ).

% frac_less_eq
thf(fact_4251_frac__less__eq,axiom,
    ! [Y: real,Z: real,X4: real,W: real] :
      ( ( Y != zero_zero_real )
     => ( ( Z != zero_zero_real )
       => ( ( ord_less_real @ ( divide_divide_real @ X4 @ Y ) @ ( divide_divide_real @ W @ Z ) )
          = ( ord_less_real @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ X4 @ Z ) @ ( times_times_real @ W @ Y ) ) @ ( times_times_real @ Y @ Z ) ) @ zero_zero_real ) ) ) ) ).

% frac_less_eq
thf(fact_4252_power2__commute,axiom,
    ! [X4: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ ( minus_minus_complex @ Y @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_4253_power2__commute,axiom,
    ! [X4: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ ( minus_minus_real @ Y @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_4254_power2__commute,axiom,
    ! [X4: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ ( minus_minus_rat @ Y @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_4255_power2__commute,axiom,
    ! [X4: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ ( minus_minus_int @ Y @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_commute
thf(fact_4256_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_4257_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_4258_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_4259_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_4260_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_4261_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_4262_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_4263_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_4264_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_4265_div__if,axiom,
    ( divide_divide_nat
    = ( ^ [M6: nat,N: nat] :
          ( if_nat
          @ ( ( ord_less_nat @ M6 @ N )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( suc @ ( divide_divide_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ) ).

% div_if
thf(fact_4266_add__eq__if,axiom,
    ( plus_plus_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( suc @ ( plus_plus_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N ) ) ) ) ) ).

% add_eq_if
thf(fact_4267_nat__less__add__iff1,axiom,
    ! [J: nat,I2: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ I2 @ J ) @ U ) @ M ) @ N2 ) ) ) ).

% nat_less_add_iff1
thf(fact_4268_nat__less__add__iff2,axiom,
    ! [I2: nat,J: nat,U: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( ord_less_nat @ ( plus_plus_nat @ ( times_times_nat @ I2 @ U ) @ M ) @ ( plus_plus_nat @ ( times_times_nat @ J @ U ) @ N2 ) )
        = ( ord_less_nat @ M @ ( plus_plus_nat @ ( times_times_nat @ ( minus_minus_nat @ J @ I2 ) @ U ) @ N2 ) ) ) ) ).

% nat_less_add_iff2
thf(fact_4269_mult__eq__if,axiom,
    ( times_times_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ zero_zero_nat @ ( plus_plus_nat @ N @ ( times_times_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) @ N ) ) ) ) ) ).

% mult_eq_if
thf(fact_4270_decr__mult__lemma,axiom,
    ! [D: int,P: int > $o,K: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int] :
            ( ( P @ X5 )
           => ( P @ ( minus_minus_int @ X5 @ 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_4271_dvd__minus__add,axiom,
    ! [Q3: nat,N2: nat,R3: nat,M: nat] :
      ( ( ord_less_eq_nat @ Q3 @ N2 )
     => ( ( ord_less_eq_nat @ Q3 @ ( times_times_nat @ R3 @ M ) )
       => ( ( dvd_dvd_nat @ M @ ( minus_minus_nat @ N2 @ Q3 ) )
          = ( dvd_dvd_nat @ M @ ( plus_plus_nat @ N2 @ ( minus_minus_nat @ ( times_times_nat @ R3 @ M ) @ Q3 ) ) ) ) ) ) ).

% dvd_minus_add
thf(fact_4272_mod__nat__eqI,axiom,
    ! [R3: nat,N2: nat,M: nat] :
      ( ( ord_less_nat @ R3 @ N2 )
     => ( ( ord_less_eq_nat @ R3 @ M )
       => ( ( dvd_dvd_nat @ N2 @ ( minus_minus_nat @ M @ R3 ) )
         => ( ( modulo_modulo_nat @ M @ N2 )
            = R3 ) ) ) ) ).

% mod_nat_eqI
thf(fact_4273_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_4274_scaling__mono,axiom,
    ! [U: real,V: real,R3: real,S: real] :
      ( ( ord_less_eq_real @ U @ V )
     => ( ( ord_less_eq_real @ zero_zero_real @ R3 )
       => ( ( ord_less_eq_real @ R3 @ S )
         => ( ord_less_eq_real @ ( plus_plus_real @ U @ ( divide_divide_real @ ( times_times_real @ R3 @ ( minus_minus_real @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4275_scaling__mono,axiom,
    ! [U: rat,V: rat,R3: rat,S: rat] :
      ( ( ord_less_eq_rat @ U @ V )
     => ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
       => ( ( ord_less_eq_rat @ R3 @ S )
         => ( ord_less_eq_rat @ ( plus_plus_rat @ U @ ( divide_divide_rat @ ( times_times_rat @ R3 @ ( minus_minus_rat @ V @ U ) ) @ S ) ) @ V ) ) ) ) ).

% scaling_mono
thf(fact_4276_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_4277_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_4278_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_4279_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_4280_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_4281_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_4282_power__eq__if,axiom,
    ( power_power_rat
    = ( ^ [P5: rat,M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ P5 @ ( power_power_rat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4283_power__eq__if,axiom,
    ( power_power_complex
    = ( ^ [P5: complex,M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ P5 @ ( power_power_complex @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4284_power__eq__if,axiom,
    ( power_power_real
    = ( ^ [P5: real,M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ P5 @ ( power_power_real @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4285_power__eq__if,axiom,
    ( power_power_nat
    = ( ^ [P5: nat,M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ P5 @ ( power_power_nat @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4286_power__eq__if,axiom,
    ( power_power_int
    = ( ^ [P5: int,M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ P5 @ ( power_power_int @ P5 @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% power_eq_if
thf(fact_4287_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_4288_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_4289_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_4290_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_4291_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_4292_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_4293_even__diff__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( minus_minus_int @ K @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_diff_iff
thf(fact_4294_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri6519982836138164636nteger @ M @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_4295_even__signed__take__bit__iff,axiom,
    ! [M: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri631733984087533419it_int @ M @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_signed_take_bit_iff
thf(fact_4296_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_4297_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_4298_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_4299_add__0__iff,axiom,
    ! [B: complex,A: complex] :
      ( ( B
        = ( plus_plus_complex @ B @ A ) )
      = ( A = zero_zero_complex ) ) ).

% add_0_iff
thf(fact_4300_add__0__iff,axiom,
    ! [B: real,A: real] :
      ( ( B
        = ( plus_plus_real @ B @ A ) )
      = ( A = zero_zero_real ) ) ).

% add_0_iff
thf(fact_4301_add__0__iff,axiom,
    ! [B: rat,A: rat] :
      ( ( B
        = ( plus_plus_rat @ B @ A ) )
      = ( A = zero_zero_rat ) ) ).

% add_0_iff
thf(fact_4302_add__0__iff,axiom,
    ! [B: nat,A: nat] :
      ( ( B
        = ( plus_plus_nat @ B @ A ) )
      = ( A = zero_zero_nat ) ) ).

% add_0_iff
thf(fact_4303_add__0__iff,axiom,
    ! [B: int,A: int] :
      ( ( B
        = ( plus_plus_int @ B @ A ) )
      = ( A = zero_zero_int ) ) ).

% add_0_iff
thf(fact_4304_crossproduct__eq,axiom,
    ! [W: rat,Y: rat,X4: rat,Z: rat] :
      ( ( ( plus_plus_rat @ ( times_times_rat @ W @ Y ) @ ( times_times_rat @ X4 @ Z ) )
        = ( plus_plus_rat @ ( times_times_rat @ W @ Z ) @ ( times_times_rat @ X4 @ Y ) ) )
      = ( ( W = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_4305_crossproduct__eq,axiom,
    ! [W: complex,Y: complex,X4: complex,Z: complex] :
      ( ( ( plus_plus_complex @ ( times_times_complex @ W @ Y ) @ ( times_times_complex @ X4 @ Z ) )
        = ( plus_plus_complex @ ( times_times_complex @ W @ Z ) @ ( times_times_complex @ X4 @ Y ) ) )
      = ( ( W = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_4306_crossproduct__eq,axiom,
    ! [W: real,Y: real,X4: real,Z: real] :
      ( ( ( plus_plus_real @ ( times_times_real @ W @ Y ) @ ( times_times_real @ X4 @ Z ) )
        = ( plus_plus_real @ ( times_times_real @ W @ Z ) @ ( times_times_real @ X4 @ Y ) ) )
      = ( ( W = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_4307_crossproduct__eq,axiom,
    ! [W: nat,Y: nat,X4: nat,Z: nat] :
      ( ( ( plus_plus_nat @ ( times_times_nat @ W @ Y ) @ ( times_times_nat @ X4 @ Z ) )
        = ( plus_plus_nat @ ( times_times_nat @ W @ Z ) @ ( times_times_nat @ X4 @ Y ) ) )
      = ( ( W = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_4308_crossproduct__eq,axiom,
    ! [W: int,Y: int,X4: int,Z: int] :
      ( ( ( plus_plus_int @ ( times_times_int @ W @ Y ) @ ( times_times_int @ X4 @ Z ) )
        = ( plus_plus_int @ ( times_times_int @ W @ Z ) @ ( times_times_int @ X4 @ Y ) ) )
      = ( ( W = X4 )
        | ( Y = Z ) ) ) ).

% crossproduct_eq
thf(fact_4309_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_4310_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_4311_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_4312_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_4313_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_4314_power2__diff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( power_power_rat @ ( minus_minus_rat @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_rat @ ( plus_plus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_diff
thf(fact_4315_power2__diff,axiom,
    ! [X4: complex,Y: complex] :
      ( ( power_power_complex @ ( minus_minus_complex @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( plus_plus_complex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_diff
thf(fact_4316_power2__diff,axiom,
    ! [X4: real,Y: real] :
      ( ( power_power_real @ ( minus_minus_real @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_diff
thf(fact_4317_power2__diff,axiom,
    ! [X4: int,Y: int] :
      ( ( power_power_int @ ( minus_minus_int @ X4 @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_int @ ( plus_plus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 ) @ Y ) ) ) ).

% power2_diff
thf(fact_4318_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_4319_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_4320_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_4321_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_4322_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_4323_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_4324_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_4325_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_4326_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_4327_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_4328_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_4329_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_4330_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_4331_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_4332_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_4333_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_4334_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_4335_neg__zmod__mult__2,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_int @ A @ zero_zero_int )
     => ( ( modulo_modulo_int @ ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) )
        = ( minus_minus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( plus_plus_int @ B @ one_one_int ) @ A ) ) @ one_one_int ) ) ) ).

% neg_zmod_mult_2
thf(fact_4336_divmod__step__eq,axiom,
    ! [L: num,R3: nat,Q3: nat] :
      ( ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R3 ) )
          = ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ one_one_nat ) @ ( minus_minus_nat @ R3 @ ( numeral_numeral_nat @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_nat @ ( numeral_numeral_nat @ L ) @ R3 )
       => ( ( unique5026877609467782581ep_nat @ L @ ( product_Pair_nat_nat @ Q3 @ R3 ) )
          = ( product_Pair_nat_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Q3 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_4337_divmod__step__eq,axiom,
    ! [L: num,R3: int,Q3: int] :
      ( ( ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R3 ) )
          = ( product_Pair_int_int @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ one_one_int ) @ ( minus_minus_int @ R3 @ ( numeral_numeral_int @ L ) ) ) ) )
      & ( ~ ( ord_less_eq_int @ ( numeral_numeral_int @ L ) @ R3 )
       => ( ( unique5024387138958732305ep_int @ L @ ( product_Pair_int_int @ Q3 @ R3 ) )
          = ( product_Pair_int_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Q3 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_4338_divmod__step__eq,axiom,
    ! [L: num,R3: code_integer,Q3: code_integer] :
      ( ( ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ R3 @ ( numera6620942414471956472nteger @ L ) ) ) ) )
      & ( ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ L ) @ R3 )
       => ( ( unique4921790084139445826nteger @ L @ ( produc1086072967326762835nteger @ Q3 @ R3 ) )
          = ( produc1086072967326762835nteger @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Q3 ) @ R3 ) ) ) ) ).

% divmod_step_eq
thf(fact_4339_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_4340_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_4341_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_4342_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_4343_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_4344_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_4345_Bernoulli__inequality__even,axiom,
    ! [N2: nat,X4: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N2 ) ) ) ).

% Bernoulli_inequality_even
thf(fact_4346_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_4347_add_Oinverse__inverse,axiom,
    ! [A: real] :
      ( ( uminus_uminus_real @ ( uminus_uminus_real @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4348_add_Oinverse__inverse,axiom,
    ! [A: int] :
      ( ( uminus_uminus_int @ ( uminus_uminus_int @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4349_add_Oinverse__inverse,axiom,
    ! [A: complex] :
      ( ( uminus1482373934393186551omplex @ ( uminus1482373934393186551omplex @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4350_add_Oinverse__inverse,axiom,
    ! [A: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( uminus1351360451143612070nteger @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4351_add_Oinverse__inverse,axiom,
    ! [A: rat] :
      ( ( uminus_uminus_rat @ ( uminus_uminus_rat @ A ) )
      = A ) ).

% add.inverse_inverse
thf(fact_4352_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_4353_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_4354_neg__equal__iff__equal,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = ( uminus1482373934393186551omplex @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4355_neg__equal__iff__equal,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = ( uminus1351360451143612070nteger @ B ) )
      = ( A = B ) ) ).

% neg_equal_iff_equal
thf(fact_4356_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_4357_Compl__anti__mono,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ B3 ) @ ( uminus1532241313380277803et_int @ A2 ) ) ) ).

% Compl_anti_mono
thf(fact_4358_Compl__subset__Compl__iff,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ A2 ) @ ( uminus1532241313380277803et_int @ B3 ) )
      = ( ord_less_eq_set_int @ B3 @ A2 ) ) ).

% Compl_subset_Compl_iff
thf(fact_4359_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ M )
        = ( semiri5074537144036343181t_real @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_4360_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = ( semiri1314217659103216013at_int @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_4361_of__nat__eq__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ M )
        = ( semiri1316708129612266289at_nat @ N2 ) )
      = ( M = N2 ) ) ).

% of_nat_eq_iff
thf(fact_4362_semiring__norm_I90_J,axiom,
    ! [M: num,N2: num] :
      ( ( ( bit1 @ M )
        = ( bit1 @ N2 ) )
      = ( M = N2 ) ) ).

% semiring_norm(90)
thf(fact_4363_bit_Oxor__left__self,axiom,
    ! [X4: int,Y: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_se6526347334894502574or_int @ X4 @ Y ) )
      = Y ) ).

% bit.xor_left_self
thf(fact_4364_idiff__0,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ zero_z5237406670263579293d_enat @ N2 )
      = zero_z5237406670263579293d_enat ) ).

% idiff_0
thf(fact_4365_idiff__0__right,axiom,
    ! [N2: extended_enat] :
      ( ( minus_3235023915231533773d_enat @ N2 @ zero_z5237406670263579293d_enat )
      = N2 ) ).

% idiff_0_right
thf(fact_4366_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_4367_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_4368_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_4369_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_4370_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_real @ zero_zero_real )
    = zero_zero_real ) ).

% add.inverse_neutral
thf(fact_4371_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_int @ zero_zero_int )
    = zero_zero_int ) ).

% add.inverse_neutral
thf(fact_4372_add_Oinverse__neutral,axiom,
    ( ( uminus1482373934393186551omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% add.inverse_neutral
thf(fact_4373_add_Oinverse__neutral,axiom,
    ( ( uminus1351360451143612070nteger @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% add.inverse_neutral
thf(fact_4374_add_Oinverse__neutral,axiom,
    ( ( uminus_uminus_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% add.inverse_neutral
thf(fact_4375_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_4376_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_4377_neg__0__equal__iff__equal,axiom,
    ! [A: complex] :
      ( ( zero_zero_complex
        = ( uminus1482373934393186551omplex @ A ) )
      = ( zero_zero_complex = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4378_neg__0__equal__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( uminus1351360451143612070nteger @ A ) )
      = ( zero_z3403309356797280102nteger = A ) ) ).

% neg_0_equal_iff_equal
thf(fact_4379_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_4380_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_4381_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_4382_neg__equal__0__iff__equal,axiom,
    ! [A: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% neg_equal_0_iff_equal
thf(fact_4383_neg__equal__0__iff__equal,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_0_iff_equal
thf(fact_4384_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_4385_equal__neg__zero,axiom,
    ! [A: real] :
      ( ( A
        = ( uminus_uminus_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% equal_neg_zero
thf(fact_4386_equal__neg__zero,axiom,
    ! [A: int] :
      ( ( A
        = ( uminus_uminus_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% equal_neg_zero
thf(fact_4387_equal__neg__zero,axiom,
    ! [A: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% equal_neg_zero
thf(fact_4388_equal__neg__zero,axiom,
    ! [A: rat] :
      ( ( A
        = ( uminus_uminus_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% equal_neg_zero
thf(fact_4389_neg__equal__zero,axiom,
    ! [A: real] :
      ( ( ( uminus_uminus_real @ A )
        = A )
      = ( A = zero_zero_real ) ) ).

% neg_equal_zero
thf(fact_4390_neg__equal__zero,axiom,
    ! [A: int] :
      ( ( ( uminus_uminus_int @ A )
        = A )
      = ( A = zero_zero_int ) ) ).

% neg_equal_zero
thf(fact_4391_neg__equal__zero,axiom,
    ! [A: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = A )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% neg_equal_zero
thf(fact_4392_neg__equal__zero,axiom,
    ! [A: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = A )
      = ( A = zero_zero_rat ) ) ).

% neg_equal_zero
thf(fact_4393_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_4394_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_4395_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_4396_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_4397_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4398_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4399_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4400_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4401_neg__numeral__eq__iff,axiom,
    ! [M: num,N2: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( M = N2 ) ) ).

% neg_numeral_eq_iff
thf(fact_4402_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_4403_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_4404_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_4405_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_4406_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_4407_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_4408_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_4409_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_4410_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_4411_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_4412_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_4413_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_4414_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_4415_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_4416_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_4417_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_4418_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_4419_minus__add__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_add_distrib
thf(fact_4420_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_4421_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_4422_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_4423_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_4424_minus__add__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( plus_plus_complex @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4425_minus__add__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ ( plus_p5714425477246183910nteger @ A @ B ) )
      = B ) ).

% minus_add_cancel
thf(fact_4426_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_4427_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_4428_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_4429_add__minus__cancel,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ A @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4430_add__minus__cancel,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) )
      = B ) ).

% add_minus_cancel
thf(fact_4431_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_4432_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_4433_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_4434_minus__diff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) )
      = ( minus_minus_complex @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4435_minus__diff__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) )
      = ( minus_8373710615458151222nteger @ B @ A ) ) ).

% minus_diff_eq
thf(fact_4436_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_4437_div__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ A @ B ) ) ).

% div_minus_minus
thf(fact_4438_div__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ A @ B ) ) ).

% div_minus_minus
thf(fact_4439_dvd__minus__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( dvd_dvd_real @ X4 @ ( uminus_uminus_real @ Y ) )
      = ( dvd_dvd_real @ X4 @ Y ) ) ).

% dvd_minus_iff
thf(fact_4440_dvd__minus__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( dvd_dvd_int @ X4 @ ( uminus_uminus_int @ Y ) )
      = ( dvd_dvd_int @ X4 @ Y ) ) ).

% dvd_minus_iff
thf(fact_4441_dvd__minus__iff,axiom,
    ! [X4: complex,Y: complex] :
      ( ( dvd_dvd_complex @ X4 @ ( uminus1482373934393186551omplex @ Y ) )
      = ( dvd_dvd_complex @ X4 @ Y ) ) ).

% dvd_minus_iff
thf(fact_4442_dvd__minus__iff,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ X4 @ ( uminus1351360451143612070nteger @ Y ) )
      = ( dvd_dvd_Code_integer @ X4 @ Y ) ) ).

% dvd_minus_iff
thf(fact_4443_dvd__minus__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( dvd_dvd_rat @ X4 @ ( uminus_uminus_rat @ Y ) )
      = ( dvd_dvd_rat @ X4 @ Y ) ) ).

% dvd_minus_iff
thf(fact_4444_minus__dvd__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( dvd_dvd_real @ ( uminus_uminus_real @ X4 ) @ Y )
      = ( dvd_dvd_real @ X4 @ Y ) ) ).

% minus_dvd_iff
thf(fact_4445_minus__dvd__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( dvd_dvd_int @ ( uminus_uminus_int @ X4 ) @ Y )
      = ( dvd_dvd_int @ X4 @ Y ) ) ).

% minus_dvd_iff
thf(fact_4446_minus__dvd__iff,axiom,
    ! [X4: complex,Y: complex] :
      ( ( dvd_dvd_complex @ ( uminus1482373934393186551omplex @ X4 ) @ Y )
      = ( dvd_dvd_complex @ X4 @ Y ) ) ).

% minus_dvd_iff
thf(fact_4447_minus__dvd__iff,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( uminus1351360451143612070nteger @ X4 ) @ Y )
      = ( dvd_dvd_Code_integer @ X4 @ Y ) ) ).

% minus_dvd_iff
thf(fact_4448_minus__dvd__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( dvd_dvd_rat @ ( uminus_uminus_rat @ X4 ) @ Y )
      = ( dvd_dvd_rat @ X4 @ Y ) ) ).

% minus_dvd_iff
thf(fact_4449_semiring__norm_I88_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit0 @ M )
     != ( bit1 @ N2 ) ) ).

% semiring_norm(88)
thf(fact_4450_semiring__norm_I89_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit1 @ M )
     != ( bit0 @ N2 ) ) ).

% semiring_norm(89)
thf(fact_4451_semiring__norm_I84_J,axiom,
    ! [N2: num] :
      ( one
     != ( bit1 @ N2 ) ) ).

% semiring_norm(84)
thf(fact_4452_semiring__norm_I86_J,axiom,
    ! [M: num] :
      ( ( bit1 @ M )
     != one ) ).

% semiring_norm(86)
thf(fact_4453_mod__minus__minus,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_4454_mod__minus__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) ) ).

% mod_minus_minus
thf(fact_4455_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% take_bit_of_0
thf(fact_4456_take__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% take_bit_of_0
thf(fact_4457_real__add__minus__iff,axiom,
    ! [X4: real,A: real] :
      ( ( ( plus_plus_real @ X4 @ ( uminus_uminus_real @ A ) )
        = zero_zero_real )
      = ( X4 = A ) ) ).

% real_add_minus_iff
thf(fact_4458_bit_Oxor__self,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ X4 )
      = zero_zero_int ) ).

% bit.xor_self
thf(fact_4459_xor__self__eq,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ A )
      = zero_zero_nat ) ).

% xor_self_eq
thf(fact_4460_xor__self__eq,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ A )
      = zero_zero_int ) ).

% xor_self_eq
thf(fact_4461_xor_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ zero_zero_nat @ A )
      = A ) ).

% xor.left_neutral
thf(fact_4462_xor_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ zero_zero_int @ A )
      = A ) ).

% xor.left_neutral
thf(fact_4463_xor_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( bit_se6528837805403552850or_nat @ A @ zero_zero_nat )
      = A ) ).

% xor.right_neutral
thf(fact_4464_xor_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se6526347334894502574or_int @ A @ zero_zero_int )
      = A ) ).

% xor.right_neutral
thf(fact_4465_take__bit__xor,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se6526347334894502574or_int @ A @ B ) )
      = ( bit_se6526347334894502574or_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) ) ).

% take_bit_xor
thf(fact_4466_take__bit__xor,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se6528837805403552850or_nat @ A @ B ) )
      = ( bit_se6528837805403552850or_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) ) ).

% take_bit_xor
thf(fact_4467_concat__bit__of__zero__2,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_concat_bit @ N2 @ K @ zero_zero_int )
      = ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).

% concat_bit_of_zero_2
thf(fact_4468_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_4469_semiring__norm_I73_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(73)
thf(fact_4470_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_4471_neg__less__eq__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_eq_nonneg
thf(fact_4472_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_4473_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_4474_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_4475_less__eq__neg__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_eq_neg_nonpos
thf(fact_4476_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_4477_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_4478_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_4479_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_4480_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_4481_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_4482_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_4483_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_4484_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_4485_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_4486_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_4487_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_4488_less__neg__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% less_neg_neg
thf(fact_4489_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_4490_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_4491_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_4492_neg__less__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% neg_less_pos
thf(fact_4493_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_4494_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_4495_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_4496_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_4497_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_4498_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_4499_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_4500_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_4501_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_4502_ab__left__minus,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ A ) @ A )
      = zero_zero_real ) ).

% ab_left_minus
thf(fact_4503_ab__left__minus,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ A ) @ A )
      = zero_zero_int ) ).

% ab_left_minus
thf(fact_4504_ab__left__minus,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ A )
      = zero_zero_complex ) ).

% ab_left_minus
thf(fact_4505_ab__left__minus,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ A ) @ A )
      = zero_z3403309356797280102nteger ) ).

% ab_left_minus
thf(fact_4506_ab__left__minus,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ A )
      = zero_zero_rat ) ).

% ab_left_minus
thf(fact_4507_add_Oright__inverse,axiom,
    ! [A: real] :
      ( ( plus_plus_real @ A @ ( uminus_uminus_real @ A ) )
      = zero_zero_real ) ).

% add.right_inverse
thf(fact_4508_add_Oright__inverse,axiom,
    ! [A: int] :
      ( ( plus_plus_int @ A @ ( uminus_uminus_int @ A ) )
      = zero_zero_int ) ).

% add.right_inverse
thf(fact_4509_add_Oright__inverse,axiom,
    ! [A: complex] :
      ( ( plus_plus_complex @ A @ ( uminus1482373934393186551omplex @ A ) )
      = zero_zero_complex ) ).

% add.right_inverse
thf(fact_4510_add_Oright__inverse,axiom,
    ! [A: code_integer] :
      ( ( plus_p5714425477246183910nteger @ A @ ( uminus1351360451143612070nteger @ A ) )
      = zero_z3403309356797280102nteger ) ).

% add.right_inverse
thf(fact_4511_add_Oright__inverse,axiom,
    ! [A: rat] :
      ( ( plus_plus_rat @ A @ ( uminus_uminus_rat @ A ) )
      = zero_zero_rat ) ).

% add.right_inverse
thf(fact_4512_diff__0,axiom,
    ! [A: real] :
      ( ( minus_minus_real @ zero_zero_real @ A )
      = ( uminus_uminus_real @ A ) ) ).

% diff_0
thf(fact_4513_diff__0,axiom,
    ! [A: int] :
      ( ( minus_minus_int @ zero_zero_int @ A )
      = ( uminus_uminus_int @ A ) ) ).

% diff_0
thf(fact_4514_diff__0,axiom,
    ! [A: complex] :
      ( ( minus_minus_complex @ zero_zero_complex @ A )
      = ( uminus1482373934393186551omplex @ A ) ) ).

% diff_0
thf(fact_4515_diff__0,axiom,
    ! [A: code_integer] :
      ( ( minus_8373710615458151222nteger @ zero_z3403309356797280102nteger @ A )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% diff_0
thf(fact_4516_diff__0,axiom,
    ! [A: rat] :
      ( ( minus_minus_rat @ zero_zero_rat @ A )
      = ( uminus_uminus_rat @ A ) ) ).

% diff_0
thf(fact_4517_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_4518_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_4519_add__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( plus_plus_complex @ ( numera6690914467698888265omplex @ M ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ) ).

% add_neg_numeral_simps(3)
thf(fact_4520_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_4521_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_4522_mult__minus1,axiom,
    ! [Z: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ one_one_real ) @ Z )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1
thf(fact_4523_mult__minus1,axiom,
    ! [Z: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ one_one_int ) @ Z )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1
thf(fact_4524_mult__minus1,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ Z )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1
thf(fact_4525_mult__minus1,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ Z )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1
thf(fact_4526_mult__minus1,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ one_one_rat ) @ Z )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1
thf(fact_4527_mult__minus1__right,axiom,
    ! [Z: real] :
      ( ( times_times_real @ Z @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ Z ) ) ).

% mult_minus1_right
thf(fact_4528_mult__minus1__right,axiom,
    ! [Z: int] :
      ( ( times_times_int @ Z @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ Z ) ) ).

% mult_minus1_right
thf(fact_4529_mult__minus1__right,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ Z ) ) ).

% mult_minus1_right
thf(fact_4530_mult__minus1__right,axiom,
    ! [Z: code_integer] :
      ( ( times_3573771949741848930nteger @ Z @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ Z ) ) ).

% mult_minus1_right
thf(fact_4531_mult__minus1__right,axiom,
    ! [Z: rat] :
      ( ( times_times_rat @ Z @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ Z ) ) ).

% mult_minus1_right
thf(fact_4532_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_4533_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_4534_uminus__add__conv__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ B )
      = ( minus_minus_complex @ B @ A ) ) ).

% uminus_add_conv_diff
thf(fact_4535_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_4536_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_4537_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_4538_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_4539_diff__minus__eq__add,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ A @ ( uminus1482373934393186551omplex @ B ) )
      = ( plus_plus_complex @ A @ B ) ) ).

% diff_minus_eq_add
thf(fact_4540_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_4541_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_4542_divide__minus1,axiom,
    ! [X4: real] :
      ( ( divide_divide_real @ X4 @ ( uminus_uminus_real @ one_one_real ) )
      = ( uminus_uminus_real @ X4 ) ) ).

% divide_minus1
thf(fact_4543_divide__minus1,axiom,
    ! [X4: complex] :
      ( ( divide1717551699836669952omplex @ X4 @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( uminus1482373934393186551omplex @ X4 ) ) ).

% divide_minus1
thf(fact_4544_divide__minus1,axiom,
    ! [X4: rat] :
      ( ( divide_divide_rat @ X4 @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( uminus_uminus_rat @ X4 ) ) ).

% divide_minus1
thf(fact_4545_div__minus1__right,axiom,
    ! [A: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ A ) ) ).

% div_minus1_right
thf(fact_4546_div__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ A ) ) ).

% div_minus1_right
thf(fact_4547_of__nat__0,axiom,
    ( ( semiri8010041392384452111omplex @ zero_zero_nat )
    = zero_zero_complex ) ).

% of_nat_0
thf(fact_4548_of__nat__0,axiom,
    ( ( semiri681578069525770553at_rat @ zero_zero_nat )
    = zero_zero_rat ) ).

% of_nat_0
thf(fact_4549_of__nat__0,axiom,
    ( ( semiri5074537144036343181t_real @ zero_zero_nat )
    = zero_zero_real ) ).

% of_nat_0
thf(fact_4550_of__nat__0,axiom,
    ( ( semiri1314217659103216013at_int @ zero_zero_nat )
    = zero_zero_int ) ).

% of_nat_0
thf(fact_4551_of__nat__0,axiom,
    ( ( semiri1316708129612266289at_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% of_nat_0
thf(fact_4552_of__nat__0__eq__iff,axiom,
    ! [N2: nat] :
      ( ( zero_zero_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( zero_zero_nat = N2 ) ) ).

% of_nat_0_eq_iff
thf(fact_4553_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_4554_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_4555_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_4556_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_4557_of__nat__eq__0__iff,axiom,
    ! [M: nat] :
      ( ( ( semiri8010041392384452111omplex @ M )
        = zero_zero_complex )
      = ( M = zero_zero_nat ) ) ).

% of_nat_eq_0_iff
thf(fact_4558_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_4559_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_4560_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_4561_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_4562_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_4563_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_4564_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_4565_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_4566_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_4567_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_4568_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_4569_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_4570_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri4216267220026989637d_enat @ ( numeral_numeral_nat @ N2 ) )
      = ( numera1916890842035813515d_enat @ N2 ) ) ).

% of_nat_numeral
thf(fact_4571_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri8010041392384452111omplex @ ( numeral_numeral_nat @ N2 ) )
      = ( numera6690914467698888265omplex @ N2 ) ) ).

% of_nat_numeral
thf(fact_4572_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri5074537144036343181t_real @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% of_nat_numeral
thf(fact_4573_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% of_nat_numeral
thf(fact_4574_of__nat__numeral,axiom,
    ! [N2: num] :
      ( ( semiri1316708129612266289at_nat @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% of_nat_numeral
thf(fact_4575_minus__mod__self1,axiom,
    ! [B: int,A: int] :
      ( ( modulo_modulo_int @ ( minus_minus_int @ B @ A ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_4576_minus__mod__self1,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( modulo364778990260209775nteger @ ( minus_8373710615458151222nteger @ B @ A ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% minus_mod_self1
thf(fact_4577_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_4578_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_4579_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_4580_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_4581_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_4582_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_4583_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_4584_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_4585_take__bit__0,axiom,
    ! [A: int] :
      ( ( bit_se2923211474154528505it_int @ zero_zero_nat @ A )
      = zero_zero_int ) ).

% take_bit_0
thf(fact_4586_take__bit__0,axiom,
    ! [A: nat] :
      ( ( bit_se2925701944663578781it_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% take_bit_0
thf(fact_4587_of__nat__eq__1__iff,axiom,
    ! [N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ N2 )
        = one_one_complex )
      = ( N2 = one_one_nat ) ) ).

% of_nat_eq_1_iff
thf(fact_4588_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_4589_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_4590_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_4591_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_4592_of__nat__1__eq__iff,axiom,
    ! [N2: nat] :
      ( ( one_one_complex
        = ( semiri8010041392384452111omplex @ N2 ) )
      = ( N2 = one_one_nat ) ) ).

% of_nat_1_eq_iff
thf(fact_4593_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_4594_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_4595_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_4596_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_4597_of__nat__1,axiom,
    ( ( semiri8010041392384452111omplex @ one_one_nat )
    = one_one_complex ) ).

% of_nat_1
thf(fact_4598_of__nat__1,axiom,
    ( ( semiri681578069525770553at_rat @ one_one_nat )
    = one_one_rat ) ).

% of_nat_1
thf(fact_4599_of__nat__1,axiom,
    ( ( semiri5074537144036343181t_real @ one_one_nat )
    = one_one_real ) ).

% of_nat_1
thf(fact_4600_of__nat__1,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% of_nat_1
thf(fact_4601_of__nat__1,axiom,
    ( ( semiri1316708129612266289at_nat @ one_one_nat )
    = one_one_nat ) ).

% of_nat_1
thf(fact_4602_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ ( suc @ N2 ) @ one_one_int )
      = one_one_int ) ).

% take_bit_Suc_1
thf(fact_4603_take__bit__Suc__1,axiom,
    ! [N2: nat] :
      ( ( bit_se2925701944663578781it_nat @ ( suc @ N2 ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_Suc_1
thf(fact_4604_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ one_one_int )
      = one_one_int ) ).

% take_bit_numeral_1
thf(fact_4605_take__bit__numeral__1,axiom,
    ! [L: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ one_one_nat )
      = one_one_nat ) ).

% take_bit_numeral_1
thf(fact_4606_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ( semiri8010041392384452111omplex @ X4 )
        = ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W ) )
      = ( X4
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4607_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ( semiri5074537144036343181t_real @ X4 )
        = ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( X4
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4608_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ( semiri1314217659103216013at_int @ X4 )
        = ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( X4
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4609_of__nat__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ( semiri1316708129612266289at_nat @ X4 )
        = ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( X4
        = ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_eq_of_nat_cancel_iff
thf(fact_4610_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ( power_power_complex @ ( semiri8010041392384452111omplex @ B ) @ W )
        = ( semiri8010041392384452111omplex @ X4 ) )
      = ( ( power_power_nat @ B @ W )
        = X4 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4611_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W )
        = ( semiri5074537144036343181t_real @ X4 ) )
      = ( ( power_power_nat @ B @ W )
        = X4 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4612_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W )
        = ( semiri1314217659103216013at_int @ X4 ) )
      = ( ( power_power_nat @ B @ W )
        = X4 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4613_of__nat__eq__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W )
        = ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ( power_power_nat @ B @ W )
        = X4 ) ) ).

% of_nat_eq_of_nat_power_cancel_iff
thf(fact_4614_of__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_complex @ ( semiri8010041392384452111omplex @ M ) @ N2 ) ) ).

% of_nat_power
thf(fact_4615_of__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_real @ ( semiri5074537144036343181t_real @ M ) @ N2 ) ) ).

% of_nat_power
thf(fact_4616_of__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_int @ ( semiri1314217659103216013at_int @ M ) @ N2 ) ) ).

% of_nat_power
thf(fact_4617_of__nat__power,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( power_power_nat @ M @ N2 ) )
      = ( power_power_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 ) ) ).

% of_nat_power
thf(fact_4618_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri6519982836138164636nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4619_signed__take__bit__of__minus__1,axiom,
    ! [N2: nat] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% signed_take_bit_of_minus_1
thf(fact_4620_semiring__norm_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(9)
thf(fact_4621_semiring__norm_I7_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ M @ N2 ) ) ) ).

% semiring_norm(7)
thf(fact_4622_semiring__norm_I15_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( bit0 @ ( times_times_num @ ( bit1 @ M ) @ N2 ) ) ) ).

% semiring_norm(15)
thf(fact_4623_semiring__norm_I14_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( times_times_num @ M @ ( bit1 @ N2 ) ) ) ) ).

% semiring_norm(14)
thf(fact_4624_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_4625_semiring__norm_I72_J,axiom,
    ! [M: num,N2: num] :
      ( ( ord_less_eq_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( ord_less_eq_num @ M @ N2 ) ) ).

% semiring_norm(72)
thf(fact_4626_semiring__norm_I77_J,axiom,
    ! [N2: num] : ( ord_less_num @ one @ ( bit1 @ N2 ) ) ).

% semiring_norm(77)
thf(fact_4627_semiring__norm_I70_J,axiom,
    ! [M: num] :
      ~ ( ord_less_eq_num @ ( bit1 @ M ) @ one ) ).

% semiring_norm(70)
thf(fact_4628_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_numeral_dbl_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4629_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_numeral_dbl_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4630_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu7009210354673126013omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4631_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu8804712462038260780nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4632_dbl__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_numeral_dbl_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_simps(1)
thf(fact_4633_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri5074537144036343181t_real @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% of_nat_of_bool
thf(fact_4634_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1316708129612266289at_nat @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2687167440665602831ol_nat @ P ) ) ).

% of_nat_of_bool
thf(fact_4635_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri1314217659103216013at_int @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% of_nat_of_bool
thf(fact_4636_of__nat__of__bool,axiom,
    ! [P: $o] :
      ( ( semiri4939895301339042750nteger @ ( zero_n2687167440665602831ol_nat @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% of_nat_of_bool
thf(fact_4637_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_4638_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_4639_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_4640_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_4641_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_4642_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_4643_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_4644_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_4645_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_4646_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_4647_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_4648_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_4649_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_4650_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_4651_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_4652_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4653_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4654_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4655_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4656_numeral__eq__neg__one__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( N2 = one ) ) ).

% numeral_eq_neg_one_iff
thf(fact_4657_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_real @ one_one_real )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4658_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_int @ one_one_int )
        = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4659_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1482373934393186551omplex @ one_one_complex )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4660_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus1351360451143612070nteger @ one_one_Code_integer )
        = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4661_neg__one__eq__numeral__iff,axiom,
    ! [N2: num] :
      ( ( ( uminus_uminus_rat @ one_one_rat )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( N2 = one ) ) ).

% neg_one_eq_numeral_iff
thf(fact_4662_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_4663_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_4664_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_4665_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_4666_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: real] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4667_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: int] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4668_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: complex] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4669_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4670_left__minus__one__mult__self,axiom,
    ! [N2: nat,A: rat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ A ) )
      = A ) ).

% left_minus_one_mult_self
thf(fact_4671_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) )
      = one_one_real ) ).

% minus_one_mult_self
thf(fact_4672_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) )
      = one_one_int ) ).

% minus_one_mult_self
thf(fact_4673_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) )
      = one_one_complex ) ).

% minus_one_mult_self
thf(fact_4674_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) )
      = one_one_Code_integer ) ).

% minus_one_mult_self
thf(fact_4675_minus__one__mult__self,axiom,
    ! [N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) )
      = one_one_rat ) ).

% minus_one_mult_self
thf(fact_4676_mod__minus1__right,axiom,
    ! [A: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = zero_zero_int ) ).

% mod_minus1_right
thf(fact_4677_mod__minus1__right,axiom,
    ! [A: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = zero_z3403309356797280102nteger ) ).

% mod_minus1_right
thf(fact_4678_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ M ) )
      = ( plus_plus_complex @ one_one_complex @ ( semiri8010041392384452111omplex @ M ) ) ) ).

% of_nat_Suc
thf(fact_4679_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ M ) )
      = ( plus_plus_rat @ one_one_rat @ ( semiri681578069525770553at_rat @ M ) ) ) ).

% of_nat_Suc
thf(fact_4680_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ M ) )
      = ( plus_plus_real @ one_one_real @ ( semiri5074537144036343181t_real @ M ) ) ) ).

% of_nat_Suc
thf(fact_4681_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ M ) )
      = ( plus_plus_int @ one_one_int @ ( semiri1314217659103216013at_int @ M ) ) ) ).

% of_nat_Suc
thf(fact_4682_of__nat__Suc,axiom,
    ! [M: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ M ) )
      = ( plus_plus_nat @ one_one_nat @ ( semiri1316708129612266289at_nat @ M ) ) ) ).

% of_nat_Suc
thf(fact_4683_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_4684_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_4685_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4686_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( plus_plus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4687_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4688_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4689_semiring__norm_I168_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(168)
thf(fact_4690_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4691_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4692_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4693_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4694_diff__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ) ).

% diff_numeral_simps(3)
thf(fact_4695_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4696_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4697_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4698_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4699_diff__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ M @ N2 ) ) ) ).

% diff_numeral_simps(2)
thf(fact_4700_zdiv__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) ).

% zdiv_numeral_Bit1
thf(fact_4701_semiring__norm_I3_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit0 @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% semiring_norm(3)
thf(fact_4702_semiring__norm_I4_J,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ N2 @ one ) ) ) ).

% semiring_norm(4)
thf(fact_4703_semiring__norm_I5_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit0 @ M ) @ one )
      = ( bit1 @ M ) ) ).

% semiring_norm(5)
thf(fact_4704_semiring__norm_I8_J,axiom,
    ! [M: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ one )
      = ( bit0 @ ( plus_plus_num @ M @ one ) ) ) ).

% semiring_norm(8)
thf(fact_4705_semiring__norm_I10_J,axiom,
    ! [M: num,N2: num] :
      ( ( plus_plus_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit0 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ one ) ) ) ).

% semiring_norm(10)
thf(fact_4706_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4707_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4708_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4709_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4710_semiring__norm_I172_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) @ Y ) ) ).

% semiring_norm(172)
thf(fact_4711_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ V ) @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4712_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ V ) @ ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4713_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ V ) @ ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4714_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ V ) @ ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4715_semiring__norm_I171_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ V ) @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(171)
thf(fact_4716_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ Y ) )
      = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4717_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( times_times_int @ ( numeral_numeral_int @ W ) @ Y ) )
      = ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4718_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ Y ) )
      = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4719_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ V ) ) @ ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ Y ) )
      = ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4720_semiring__norm_I170_J,axiom,
    ! [V: num,W: num,Y: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ ( times_times_rat @ ( numeral_numeral_rat @ W ) @ Y ) )
      = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ V @ W ) ) ) @ Y ) ) ).

% semiring_norm(170)
thf(fact_4721_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4722_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4723_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4724_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4725_mult__neg__numeral__simps_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(3)
thf(fact_4726_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4727_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4728_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4729_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4730_mult__neg__numeral__simps_I2_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ) ).

% mult_neg_numeral_simps(2)
thf(fact_4731_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4732_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4733_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4734_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4735_mult__neg__numeral__simps_I1_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( times_times_num @ M @ N2 ) ) ) ).

% mult_neg_numeral_simps(1)
thf(fact_4736_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_4737_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_4738_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_4739_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_4740_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_4741_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_4742_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_4743_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_4744_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_4745_semiring__norm_I16_J,axiom,
    ! [M: num,N2: num] :
      ( ( times_times_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( bit1 @ ( plus_plus_num @ ( plus_plus_num @ M @ N2 ) @ ( bit0 @ ( times_times_num @ M @ N2 ) ) ) ) ) ).

% semiring_norm(16)
thf(fact_4746_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_4747_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_4748_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_4749_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_4750_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_4751_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_4752_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_eq_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_eq_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_4753_le__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_eq_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_eq_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% le_divide_eq_numeral1(2)
thf(fact_4754_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_eq_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_4755_divide__le__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_eq_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_le_eq_numeral1(2)
thf(fact_4756_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( A
        = ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4757_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: complex,B: complex,W: num] :
      ( ( A
        = ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
            = B ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4758_eq__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( A
        = ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
            = B ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral1(2)
thf(fact_4759_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
           != zero_zero_real )
         => ( B
            = ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) )
        & ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real )
         => ( A = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4760_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: complex,W: num,A: complex] :
      ( ( ( divide1717551699836669952omplex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
        = A )
      = ( ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
           != zero_zero_complex )
         => ( B
            = ( times_times_complex @ A @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) )
        & ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex )
         => ( A = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4761_divide__eq__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
        = A )
      = ( ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
           != zero_zero_rat )
         => ( B
            = ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) )
        & ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat )
         => ( A = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral1(2)
thf(fact_4762_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_4763_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_4764_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_4765_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_4766_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ord_less_real @ A @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) )
      = ( ord_less_real @ B @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_4767_less__divide__eq__numeral1_I2_J,axiom,
    ! [A: rat,B: rat,W: num] :
      ( ( ord_less_rat @ A @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) )
      = ( ord_less_rat @ B @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ).

% less_divide_eq_numeral1(2)
thf(fact_4768_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: real,W: num,A: real] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ A )
      = ( ord_less_real @ ( times_times_real @ A @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_4769_divide__less__eq__numeral1_I2_J,axiom,
    ! [B: rat,W: num,A: rat] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ A )
      = ( ord_less_rat @ ( times_times_rat @ A @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) @ B ) ) ).

% divide_less_eq_numeral1(2)
thf(fact_4770_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_4771_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_4772_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_4773_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_4774_power2__minus,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4775_power2__minus,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4776_power2__minus,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4777_power2__minus,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4778_power2__minus,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_minus
thf(fact_4779_xor__numerals_I3_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4780_xor__numerals_I3_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% xor_numerals(3)
thf(fact_4781_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4782_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4783_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4784_of__nat__power__less__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_less_of_nat_cancel_iff
thf(fact_4785_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4786_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4787_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4788_of__nat__less__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).

% of_nat_less_of_nat_power_cancel_iff
thf(fact_4789_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4790_xor__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit1 @ Y ) ) ) ).

% xor_numerals(1)
thf(fact_4791_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4792_xor__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6526347334894502574or_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_int @ ( bit0 @ Y ) ) ) ).

% xor_numerals(2)
thf(fact_4793_xor__numerals_I5_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% xor_numerals(5)
thf(fact_4794_xor__numerals_I5_J,axiom,
    ! [X4: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit1 @ X4 ) ) ) ).

% xor_numerals(5)
thf(fact_4795_xor__numerals_I8_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
      = ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).

% xor_numerals(8)
thf(fact_4796_xor__numerals_I8_J,axiom,
    ! [X4: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
      = ( numeral_numeral_int @ ( bit0 @ X4 ) ) ) ).

% xor_numerals(8)
thf(fact_4797_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4798_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4799_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4800_of__nat__power__le__of__nat__cancel__iff,axiom,
    ! [X4: nat,B: nat,W: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ B @ W ) ) ) ).

% of_nat_power_le_of_nat_cancel_iff
thf(fact_4801_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ B ) @ W ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4802_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ B ) @ W ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4803_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ B ) @ W ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4804_of__nat__le__of__nat__power__cancel__iff,axiom,
    ! [B: nat,W: nat,X4: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ B ) @ W ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ B @ W ) @ X4 ) ) ).

% of_nat_le_of_nat_power_cancel_iff
thf(fact_4805_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X4: num,N2: nat] :
      ( ( ( semiri4216267220026989637d_enat @ Y )
        = ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4806_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X4: num,N2: nat] :
      ( ( ( semiri8010041392384452111omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4807_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X4: num,N2: nat] :
      ( ( ( semiri5074537144036343181t_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4808_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X4: num,N2: nat] :
      ( ( ( semiri1314217659103216013at_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4809_real__of__nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: nat,X4: num,N2: nat] :
      ( ( ( semiri1316708129612266289at_nat @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) ) ) ).

% real_of_nat_eq_numeral_power_cancel_iff
thf(fact_4810_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: nat] :
      ( ( ( power_8040749407984259932d_enat @ ( numera1916890842035813515d_enat @ X4 ) @ N2 )
        = ( semiri4216267220026989637d_enat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4811_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: nat] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N2 )
        = ( semiri8010041392384452111omplex @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4812_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: nat] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 )
        = ( semiri5074537144036343181t_real @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4813_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: nat] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 )
        = ( semiri1314217659103216013at_int @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4814_numeral__power__eq__of__nat__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: nat] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 )
        = ( semiri1316708129612266289at_nat @ Y ) )
      = ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_nat_cancel_iff
thf(fact_4815_real__of__nat__less__numeral__iff,axiom,
    ! [N2: nat,W: num] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( numeral_numeral_real @ W ) )
      = ( ord_less_nat @ N2 @ ( numeral_numeral_nat @ W ) ) ) ).

% real_of_nat_less_numeral_iff
thf(fact_4816_numeral__less__real__of__nat__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( ord_less_real @ ( numeral_numeral_real @ W ) @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( ord_less_nat @ ( numeral_numeral_nat @ W ) @ N2 ) ) ).

% numeral_less_real_of_nat_iff
thf(fact_4817_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_4818_take__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ one_one_Code_integer )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% take_bit_of_1
thf(fact_4819_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_4820_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_4821_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_4822_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_4823_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_4824_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_4825_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_4826_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4827_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4828_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ one_one_complex )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4829_diff__numeral__special_I10_J,axiom,
    ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4830_diff__numeral__special_I10_J,axiom,
    ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ one_one_rat )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% diff_numeral_special(10)
thf(fact_4831_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4832_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4833_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4834_diff__numeral__special_I11_J,axiom,
    ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4835_diff__numeral__special_I11_J,axiom,
    ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ).

% diff_numeral_special(11)
thf(fact_4836_minus__1__div__2__eq,axiom,
    ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_2_eq
thf(fact_4837_minus__1__div__2__eq,axiom,
    ( ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% minus_1_div_2_eq
thf(fact_4838_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% bits_minus_1_mod_2_eq
thf(fact_4839_bits__minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% bits_minus_1_mod_2_eq
thf(fact_4840_minus__1__mod__2__eq,axiom,
    ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = one_one_int ) ).

% minus_1_mod_2_eq
thf(fact_4841_minus__1__mod__2__eq,axiom,
    ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
    = one_one_Code_integer ) ).

% minus_1_mod_2_eq
thf(fact_4842_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( power_power_rat @ ( semiri681578069525770553at_rat @ X4 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_4843_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ ( power_power_real @ ( semiri5074537144036343181t_real @ X4 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_4844_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ ( power_power_int @ ( semiri1314217659103216013at_int @ X4 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_4845_of__nat__zero__less__power__iff,axiom,
    ! [X4: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ ( power_power_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ N2 ) )
      = ( ( ord_less_nat @ zero_zero_nat @ X4 )
        | ( N2 = zero_zero_nat ) ) ) ).

% of_nat_zero_less_power_iff
thf(fact_4846_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4847_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4848_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4849_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4850_Power_Oring__1__class_Opower__minus__even,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% Power.ring_1_class.power_minus_even
thf(fact_4851_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
        = ( power_power_real @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4852_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
        = ( power_power_int @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4853_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: complex] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
        = ( power_power_complex @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4854_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
        = ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4855_Parity_Oring__1__class_Opower__minus__even,axiom,
    ! [N2: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% Parity.ring_1_class.power_minus_even
thf(fact_4856_power__minus__odd,axiom,
    ! [N2: nat,A: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
        = ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4857_power__minus__odd,axiom,
    ! [N2: nat,A: int] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
        = ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4858_power__minus__odd,axiom,
    ! [N2: nat,A: complex] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
        = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4859_power__minus__odd,axiom,
    ! [N2: nat,A: code_integer] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
        = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4860_power__minus__odd,axiom,
    ! [N2: nat,A: rat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
        = ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ).

% power_minus_odd
thf(fact_4861_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_4862_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_4863_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_4864_xor__numerals_I7_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4865_xor__numerals_I7_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% xor_numerals(7)
thf(fact_4866_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_4867_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_4868_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_complex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) )
      = ( numera6690914467698888265omplex @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_4869_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_4870_diff__numeral__special_I3_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ ( plus_plus_num @ one @ N2 ) ) ) ).

% diff_numeral_special(3)
thf(fact_4871_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ M ) ) @ one_one_real )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_4872_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_4873_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) ) @ one_one_complex )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_4874_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_4875_diff__numeral__special_I4_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) ) @ one_one_rat )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( plus_plus_num @ M @ one ) ) ) ) ).

% diff_numeral_special(4)
thf(fact_4876_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_4877_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_4878_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_4879_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_4880_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_4881_xor__nat__numerals_I4_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit0 @ X4 ) ) ) ).

% xor_nat_numerals(4)
thf(fact_4882_xor__nat__numerals_I3_J,axiom,
    ! [X4: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% xor_nat_numerals(3)
thf(fact_4883_xor__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit0 @ Y ) ) ) ).

% xor_nat_numerals(2)
thf(fact_4884_xor__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% xor_nat_numerals(1)
thf(fact_4885_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_4886_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_4887_dbl__simps_I4_J,axiom,
    ( ( neg_nu7009210354673126013omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_4888_dbl__simps_I4_J,axiom,
    ( ( neg_nu8804712462038260780nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_4889_dbl__simps_I4_J,axiom,
    ( ( neg_numeral_dbl_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% dbl_simps(4)
thf(fact_4890_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_real ) ).

% power_minus1_even
thf(fact_4891_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_int ) ).

% power_minus1_even
thf(fact_4892_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_complex ) ).

% power_minus1_even
thf(fact_4893_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_Code_integer ) ).

% power_minus1_even
thf(fact_4894_power__minus1__even,axiom,
    ! [N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = one_one_rat ) ).

% power_minus1_even
thf(fact_4895_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
        = one_one_real ) ) ).

% neg_one_even_power
thf(fact_4896_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = one_one_int ) ) ).

% neg_one_even_power
thf(fact_4897_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
        = one_one_complex ) ) ).

% neg_one_even_power
thf(fact_4898_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = one_one_Code_integer ) ) ).

% neg_one_even_power
thf(fact_4899_neg__one__even__power,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = one_one_rat ) ) ).

% neg_one_even_power
thf(fact_4900_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% neg_one_odd_power
thf(fact_4901_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% neg_one_odd_power
thf(fact_4902_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% neg_one_odd_power
thf(fact_4903_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% neg_one_odd_power
thf(fact_4904_neg__one__odd__power,axiom,
    ! [N2: nat] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% neg_one_odd_power
thf(fact_4905_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_4906_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_4907_even__of__nat,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% even_of_nat
thf(fact_4908_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_4909_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_4910_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_4911_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I2: num,N2: nat] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4912_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I2: num,N2: nat] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4913_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I2: num,N2: nat] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4914_of__nat__less__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I2: num,N2: nat] :
      ( ( ord_less_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) )
      = ( ord_less_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_less_numeral_power_cancel_iff
thf(fact_4915_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X4: nat] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X4 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4916_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X4: nat] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X4 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4917_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X4: nat] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X4 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4918_numeral__power__less__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X4: nat] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X4 ) ) ).

% numeral_power_less_of_nat_cancel_iff
thf(fact_4919_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I2: num,N2: nat] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4920_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I2: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ X4 ) @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4921_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I2: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4922_of__nat__le__numeral__power__cancel__iff,axiom,
    ! [X4: nat,I2: num,N2: nat] :
      ( ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) )
      = ( ord_less_eq_nat @ X4 @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) ) ) ).

% of_nat_le_numeral_power_cancel_iff
thf(fact_4923_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X4: nat] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ I2 ) @ N2 ) @ ( semiri5074537144036343181t_real @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X4 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4924_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X4: nat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ I2 ) @ N2 ) @ ( semiri681578069525770553at_rat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X4 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4925_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X4: nat] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ ( semiri1316708129612266289at_nat @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X4 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4926_numeral__power__le__of__nat__cancel__iff,axiom,
    ! [I2: num,N2: nat,X4: nat] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ I2 ) @ N2 ) @ ( semiri1314217659103216013at_int @ X4 ) )
      = ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ I2 ) @ N2 ) @ X4 ) ) ).

% numeral_power_le_of_nat_cancel_iff
thf(fact_4927_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_4928_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_4929_xor__numerals_I4_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_4930_xor__numerals_I4_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% xor_numerals(4)
thf(fact_4931_xor__numerals_I6_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_4932_xor__numerals_I6_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% xor_numerals(6)
thf(fact_4933_take__bit__of__exp,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se1745604003318907178nteger @ M @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_nat @ N2 @ M ) ) @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% take_bit_of_exp
thf(fact_4934_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_4935_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_4936_take__bit__of__2,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% take_bit_of_2
thf(fact_4937_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_4938_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_4939_zmod__numeral__Bit1,axiom,
    ! [V: num,W: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ V ) ) @ ( numeral_numeral_int @ ( bit0 @ W ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ V ) @ ( numeral_numeral_int @ W ) ) ) @ one_one_int ) ) ).

% zmod_numeral_Bit1
thf(fact_4940_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_4941_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_4942_signed__take__bit__minus,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) )
      = ( bit_ri631733984087533419it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).

% signed_take_bit_minus
thf(fact_4943_minus__real__def,axiom,
    ( minus_minus_real
    = ( ^ [X: real,Y5: real] : ( plus_plus_real @ X @ ( uminus_uminus_real @ Y5 ) ) ) ) ).

% minus_real_def
thf(fact_4944_xor_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( bit_se6528837805403552850or_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ C )
      = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).

% xor.assoc
thf(fact_4945_xor_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( bit_se6526347334894502574or_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ C )
      = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).

% xor.assoc
thf(fact_4946_xor_Ocommute,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [A3: nat,B2: nat] : ( bit_se6528837805403552850or_nat @ B2 @ A3 ) ) ) ).

% xor.commute
thf(fact_4947_xor_Ocommute,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [A3: int,B2: int] : ( bit_se6526347334894502574or_int @ B2 @ A3 ) ) ) ).

% xor.commute
thf(fact_4948_of__nat__xor__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se6528837805403552850or_nat @ M @ N2 ) )
      = ( bit_se6528837805403552850or_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_xor_eq
thf(fact_4949_of__nat__xor__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se6528837805403552850or_nat @ M @ N2 ) )
      = ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_xor_eq
thf(fact_4950_take__bit__of__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( semiri1314217659103216013at_int @ M ) )
      = ( semiri1314217659103216013at_int @ ( bit_se2925701944663578781it_nat @ N2 @ M ) ) ) ).

% take_bit_of_nat
thf(fact_4951_take__bit__of__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( semiri1316708129612266289at_nat @ M ) )
      = ( semiri1316708129612266289at_nat @ ( bit_se2925701944663578781it_nat @ N2 @ M ) ) ) ).

% take_bit_of_nat
thf(fact_4952_xor_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( bit_se6528837805403552850or_nat @ B @ ( bit_se6528837805403552850or_nat @ A @ C ) )
      = ( bit_se6528837805403552850or_nat @ A @ ( bit_se6528837805403552850or_nat @ B @ C ) ) ) ).

% xor.left_commute
thf(fact_4953_xor_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( bit_se6526347334894502574or_int @ B @ ( bit_se6526347334894502574or_int @ A @ C ) )
      = ( bit_se6526347334894502574or_int @ A @ ( bit_se6526347334894502574or_int @ B @ C ) ) ) ).

% xor.left_commute
thf(fact_4954_equation__minus__iff,axiom,
    ! [A: real,B: real] :
      ( ( A
        = ( uminus_uminus_real @ B ) )
      = ( B
        = ( uminus_uminus_real @ A ) ) ) ).

% equation_minus_iff
thf(fact_4955_equation__minus__iff,axiom,
    ! [A: int,B: int] :
      ( ( A
        = ( uminus_uminus_int @ B ) )
      = ( B
        = ( uminus_uminus_int @ A ) ) ) ).

% equation_minus_iff
thf(fact_4956_equation__minus__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( A
        = ( uminus1482373934393186551omplex @ B ) )
      = ( B
        = ( uminus1482373934393186551omplex @ A ) ) ) ).

% equation_minus_iff
thf(fact_4957_equation__minus__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( A
        = ( uminus1351360451143612070nteger @ B ) )
      = ( B
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% equation_minus_iff
thf(fact_4958_equation__minus__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( A
        = ( uminus_uminus_rat @ B ) )
      = ( B
        = ( uminus_uminus_rat @ A ) ) ) ).

% equation_minus_iff
thf(fact_4959_minus__equation__iff,axiom,
    ! [A: real,B: real] :
      ( ( ( uminus_uminus_real @ A )
        = B )
      = ( ( uminus_uminus_real @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4960_minus__equation__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( uminus_uminus_int @ A )
        = B )
      = ( ( uminus_uminus_int @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4961_minus__equation__iff,axiom,
    ! [A: complex,B: complex] :
      ( ( ( uminus1482373934393186551omplex @ A )
        = B )
      = ( ( uminus1482373934393186551omplex @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4962_minus__equation__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( uminus1351360451143612070nteger @ A )
        = B )
      = ( ( uminus1351360451143612070nteger @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4963_minus__equation__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ( uminus_uminus_rat @ A )
        = B )
      = ( ( uminus_uminus_rat @ B )
        = A ) ) ).

% minus_equation_iff
thf(fact_4964_power__minus__Bit1,axiom,
    ! [X4: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4965_power__minus__Bit1,axiom,
    ! [X4: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4966_power__minus__Bit1,axiom,
    ! [X4: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4967_power__minus__Bit1,axiom,
    ! [X4: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4968_power__minus__Bit1,axiom,
    ! [X4: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ K ) ) )
      = ( uminus_uminus_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit1 @ K ) ) ) ) ) ).

% power_minus_Bit1
thf(fact_4969_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_4970_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_4971_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_4972_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_4973_take__bit__tightened__less__eq__nat,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ord_less_eq_nat @ ( bit_se2925701944663578781it_nat @ M @ Q3 ) @ ( bit_se2925701944663578781it_nat @ N2 @ Q3 ) ) ) ).

% take_bit_tightened_less_eq_nat
thf(fact_4974_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_4975_mult__of__nat__commute,axiom,
    ! [X4: nat,Y: complex] :
      ( ( times_times_complex @ ( semiri8010041392384452111omplex @ X4 ) @ Y )
      = ( times_times_complex @ Y @ ( semiri8010041392384452111omplex @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_4976_mult__of__nat__commute,axiom,
    ! [X4: nat,Y: real] :
      ( ( times_times_real @ ( semiri5074537144036343181t_real @ X4 ) @ Y )
      = ( times_times_real @ Y @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_4977_mult__of__nat__commute,axiom,
    ! [X4: nat,Y: int] :
      ( ( times_times_int @ ( semiri1314217659103216013at_int @ X4 ) @ Y )
      = ( times_times_int @ Y @ ( semiri1314217659103216013at_int @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_4978_mult__of__nat__commute,axiom,
    ! [X4: nat,Y: nat] :
      ( ( times_times_nat @ ( semiri1316708129612266289at_nat @ X4 ) @ Y )
      = ( times_times_nat @ Y @ ( semiri1316708129612266289at_nat @ X4 ) ) ) ).

% mult_of_nat_commute
thf(fact_4979_take__bit__mult,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( times_times_int @ K @ L ) ) ) ).

% take_bit_mult
thf(fact_4980_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_4981_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_4982_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_4983_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_4984_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_4985_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_4986_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_4987_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_4988_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_4989_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_4990_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_4991_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_4992_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_4993_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_4994_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_4995_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_4996_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_4997_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_4998_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_4999_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_5000_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_5001_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_5002_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_5003_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_5004_verit__eq__simplify_I14_J,axiom,
    ! [X22: num,X32: num] :
      ( ( bit0 @ X22 )
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(14)
thf(fact_5005_verit__eq__simplify_I12_J,axiom,
    ! [X32: num] :
      ( one
     != ( bit1 @ X32 ) ) ).

% verit_eq_simplify(12)
thf(fact_5006_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_real @ M )
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5007_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_int @ M )
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5008_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numera6690914467698888265omplex @ M )
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5009_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numera6620942414471956472nteger @ M )
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5010_numeral__neq__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_rat @ M )
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% numeral_neq_neg_numeral
thf(fact_5011_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_real @ ( numeral_numeral_real @ M ) )
     != ( numeral_numeral_real @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5012_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ M ) )
     != ( numeral_numeral_int @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5013_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ M ) )
     != ( numera6690914467698888265omplex @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5014_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) )
     != ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5015_neg__numeral__neq__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ M ) )
     != ( numeral_numeral_rat @ N2 ) ) ).

% neg_numeral_neq_numeral
thf(fact_5016_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_5017_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_5018_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_5019_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_5020_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_5021_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_5022_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_5023_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_5024_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_5025_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_5026_one__neq__neg__one,axiom,
    ( one_one_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% one_neq_neg_one
thf(fact_5027_one__neq__neg__one,axiom,
    ( one_one_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% one_neq_neg_one
thf(fact_5028_one__neq__neg__one,axiom,
    ( one_one_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% one_neq_neg_one
thf(fact_5029_one__neq__neg__one,axiom,
    ( one_one_Code_integer
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% one_neq_neg_one
thf(fact_5030_one__neq__neg__one,axiom,
    ( one_one_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% one_neq_neg_one
thf(fact_5031_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_5032_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_5033_add_Oinverse__distrib__swap,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% add.inverse_distrib_swap
thf(fact_5034_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_5035_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_5036_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_5037_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_5038_group__cancel_Oneg1,axiom,
    ! [A2: complex,K: complex,A: complex] :
      ( ( A2
        = ( plus_plus_complex @ K @ A ) )
     => ( ( uminus1482373934393186551omplex @ A2 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( uminus1482373934393186551omplex @ A ) ) ) ) ).

% group_cancel.neg1
thf(fact_5039_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_5040_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_5041_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_5042_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_5043_is__num__normalize_I8_J,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( plus_plus_complex @ A @ B ) )
      = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ B ) @ ( uminus1482373934393186551omplex @ A ) ) ) ).

% is_num_normalize(8)
thf(fact_5044_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_5045_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_5046_take__bit__diff,axiom,
    ! [N2: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ ( bit_se2923211474154528505it_int @ N2 @ L ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ L ) ) ) ).

% take_bit_diff
thf(fact_5047_minus__diff__minus,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( uminus_uminus_real @ B ) )
      = ( uminus_uminus_real @ ( minus_minus_real @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5048_minus__diff__minus,axiom,
    ! [A: int,B: int] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ A ) @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( minus_minus_int @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5049_minus__diff__minus,axiom,
    ! [A: complex,B: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( uminus1482373934393186551omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5050_minus__diff__minus,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A ) @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5051_minus__diff__minus,axiom,
    ! [A: rat,B: rat] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( uminus_uminus_rat @ B ) )
      = ( uminus_uminus_rat @ ( minus_minus_rat @ A @ B ) ) ) ).

% minus_diff_minus
thf(fact_5052_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_5053_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_5054_minus__diff__commute,axiom,
    ! [B: complex,A: complex] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ B ) @ A )
      = ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_diff_commute
thf(fact_5055_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_5056_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_5057_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_5058_minus__divide__left,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B ) ) ).

% minus_divide_left
thf(fact_5059_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_5060_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_5061_minus__divide__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ ( uminus1482373934393186551omplex @ B ) )
      = ( divide1717551699836669952omplex @ A @ B ) ) ).

% minus_divide_divide
thf(fact_5062_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_5063_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_5064_minus__divide__right,axiom,
    ! [A: complex,B: complex] :
      ( ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) ) ) ).

% minus_divide_right
thf(fact_5065_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_5066_div__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% div_minus_right
thf(fact_5067_div__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( divide6298287555418463151nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( divide6298287555418463151nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% div_minus_right
thf(fact_5068_Diff__mono,axiom,
    ! [A2: set_nat,C4: set_nat,D4: set_nat,B3: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ C4 )
     => ( ( ord_less_eq_set_nat @ D4 @ B3 )
       => ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ ( minus_minus_set_nat @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_5069_Diff__mono,axiom,
    ! [A2: set_int,C4: set_int,D4: set_int,B3: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ C4 )
     => ( ( ord_less_eq_set_int @ D4 @ B3 )
       => ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ ( minus_minus_set_int @ C4 @ D4 ) ) ) ) ).

% Diff_mono
thf(fact_5070_Diff__subset,axiom,
    ! [A2: set_nat,B3: set_nat] : ( ord_less_eq_set_nat @ ( minus_minus_set_nat @ A2 @ B3 ) @ A2 ) ).

% Diff_subset
thf(fact_5071_Diff__subset,axiom,
    ! [A2: set_int,B3: set_int] : ( ord_less_eq_set_int @ ( minus_minus_set_int @ A2 @ B3 ) @ A2 ) ).

% Diff_subset
thf(fact_5072_double__diff,axiom,
    ! [A2: set_nat,B3: set_nat,C4: set_nat] :
      ( ( ord_less_eq_set_nat @ A2 @ B3 )
     => ( ( ord_less_eq_set_nat @ B3 @ C4 )
       => ( ( minus_minus_set_nat @ B3 @ ( minus_minus_set_nat @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_5073_double__diff,axiom,
    ! [A2: set_int,B3: set_int,C4: set_int] :
      ( ( ord_less_eq_set_int @ A2 @ B3 )
     => ( ( ord_less_eq_set_int @ B3 @ C4 )
       => ( ( minus_minus_set_int @ B3 @ ( minus_minus_set_int @ C4 @ A2 ) )
          = A2 ) ) ) ).

% double_diff
thf(fact_5074_mod__minus__eq,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( modulo_modulo_int @ A @ B ) ) @ B )
      = ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_5075_mod__minus__eq,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ A @ B ) ) @ B )
      = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ).

% mod_minus_eq
thf(fact_5076_mod__minus__cong,axiom,
    ! [A: int,B: int,A4: int] :
      ( ( ( modulo_modulo_int @ A @ B )
        = ( modulo_modulo_int @ A4 @ B ) )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
        = ( modulo_modulo_int @ ( uminus_uminus_int @ A4 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5077_mod__minus__cong,axiom,
    ! [A: code_integer,B: code_integer,A4: code_integer] :
      ( ( ( modulo364778990260209775nteger @ A @ B )
        = ( modulo364778990260209775nteger @ A4 @ B ) )
     => ( ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B )
        = ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A4 ) @ B ) ) ) ).

% mod_minus_cong
thf(fact_5078_mod__minus__right,axiom,
    ! [A: int,B: int] :
      ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
      = ( uminus_uminus_int @ ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_5079_mod__minus__right,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( modulo364778990260209775nteger @ A @ ( uminus1351360451143612070nteger @ B ) )
      = ( uminus1351360451143612070nteger @ ( modulo364778990260209775nteger @ ( uminus1351360451143612070nteger @ A ) @ B ) ) ) ).

% mod_minus_right
thf(fact_5080_psubset__imp__ex__mem,axiom,
    ! [A2: set_real,B3: set_real] :
      ( ( ord_less_set_real @ A2 @ B3 )
     => ? [B5: real] : ( member_real @ B5 @ ( minus_minus_set_real @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5081_psubset__imp__ex__mem,axiom,
    ! [A2: set_complex,B3: set_complex] :
      ( ( ord_less_set_complex @ A2 @ B3 )
     => ? [B5: complex] : ( member_complex @ B5 @ ( minus_811609699411566653omplex @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5082_psubset__imp__ex__mem,axiom,
    ! [A2: set_int,B3: set_int] :
      ( ( ord_less_set_int @ A2 @ B3 )
     => ? [B5: int] : ( member_int @ B5 @ ( minus_minus_set_int @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5083_psubset__imp__ex__mem,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,B3: set_Pr1261947904930325089at_nat] :
      ( ( ord_le7866589430770878221at_nat @ A2 @ B3 )
     => ? [B5: product_prod_nat_nat] : ( member8440522571783428010at_nat @ B5 @ ( minus_1356011639430497352at_nat @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5084_psubset__imp__ex__mem,axiom,
    ! [A2: set_nat,B3: set_nat] :
      ( ( ord_less_set_nat @ A2 @ B3 )
     => ? [B5: nat] : ( member_nat @ B5 @ ( minus_minus_set_nat @ B3 @ A2 ) ) ) ).

% psubset_imp_ex_mem
thf(fact_5085_concat__bit__eq__iff,axiom,
    ! [N2: nat,K: int,L: int,R3: int,S: int] :
      ( ( ( bit_concat_bit @ N2 @ K @ L )
        = ( bit_concat_bit @ N2 @ R3 @ S ) )
      = ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
          = ( bit_se2923211474154528505it_int @ N2 @ R3 ) )
        & ( L = S ) ) ) ).

% concat_bit_eq_iff
thf(fact_5086_concat__bit__take__bit__eq,axiom,
    ! [N2: nat,B: int] :
      ( ( bit_concat_bit @ N2 @ ( bit_se2923211474154528505it_int @ N2 @ B ) )
      = ( bit_concat_bit @ N2 @ B ) ) ).

% concat_bit_take_bit_eq
thf(fact_5087_real__of__nat__div2,axiom,
    ! [N2: nat,X4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X4 ) ) ) ) ).

% real_of_nat_div2
thf(fact_5088_real__of__nat__div3,axiom,
    ! [N2: nat,X4: nat] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X4 ) ) @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X4 ) ) ) @ one_one_real ) ).

% real_of_nat_div3
thf(fact_5089_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_5090_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_5091_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_5092_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_5093_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_rat @ ( semiri681578069525770553at_rat @ M ) @ zero_zero_rat ) ).

% of_nat_less_0_iff
thf(fact_5094_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_real @ ( semiri5074537144036343181t_real @ M ) @ zero_zero_real ) ).

% of_nat_less_0_iff
thf(fact_5095_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ zero_zero_int ) ).

% of_nat_less_0_iff
thf(fact_5096_of__nat__less__0__iff,axiom,
    ! [M: nat] :
      ~ ( ord_less_nat @ ( semiri1316708129612266289at_nat @ M ) @ zero_zero_nat ) ).

% of_nat_less_0_iff
thf(fact_5097_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( suc @ N2 ) )
     != zero_zero_complex ) ).

% of_nat_neq_0
thf(fact_5098_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( suc @ N2 ) )
     != zero_zero_rat ) ).

% of_nat_neq_0
thf(fact_5099_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( suc @ N2 ) )
     != zero_zero_real ) ).

% of_nat_neq_0
thf(fact_5100_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
     != zero_zero_int ) ).

% of_nat_neq_0
thf(fact_5101_of__nat__neq__0,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( suc @ N2 ) )
     != zero_zero_nat ) ).

% of_nat_neq_0
thf(fact_5102_div__mult2__eq_H,axiom,
    ! [A: code_integer,M: nat,N2: nat] :
      ( ( divide6298287555418463151nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) )
      = ( divide6298287555418463151nteger @ ( divide6298287555418463151nteger @ A @ ( semiri4939895301339042750nteger @ M ) ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_5103_div__mult2__eq_H,axiom,
    ! [A: int,M: nat,N2: nat] :
      ( ( divide_divide_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) )
      = ( divide_divide_int @ ( divide_divide_int @ A @ ( semiri1314217659103216013at_int @ M ) ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_5104_div__mult2__eq_H,axiom,
    ! [A: nat,M: nat,N2: nat] :
      ( ( divide_divide_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) )
      = ( divide_divide_nat @ ( divide_divide_nat @ A @ ( semiri1316708129612266289at_nat @ M ) ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% div_mult2_eq'
thf(fact_5105_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_5106_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_5107_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_5108_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_5109_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_5110_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_5111_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_5112_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_5113_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_5114_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ I2 ) @ ( semiri5074537144036343181t_real @ J ) ) ) ).

% of_nat_mono
thf(fact_5115_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ I2 ) @ ( semiri681578069525770553at_rat @ J ) ) ) ).

% of_nat_mono
thf(fact_5116_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_nat @ ( semiri1316708129612266289at_nat @ I2 ) @ ( semiri1316708129612266289at_nat @ J ) ) ) ).

% of_nat_mono
thf(fact_5117_of__nat__mono,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ).

% of_nat_mono
thf(fact_5118_take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% take_bit_int_less_eq_self_iff
thf(fact_5119_take__bit__nonnegative,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ).

% take_bit_nonnegative
thf(fact_5120_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_5121_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_5122_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_5123_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_5124_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_5125_neg__numeral__le__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_le_numeral
thf(fact_5126_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_5127_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_5128_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_5129_not__numeral__le__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_le_neg_numeral
thf(fact_5130_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_5131_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_5132_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5133_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5134_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5135_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_z3403309356797280102nteger
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5136_zero__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( zero_zero_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% zero_neq_neg_numeral
thf(fact_5137_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_5138_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_5139_not__numeral__less__neg__numeral,axiom,
    ! [M: num,N2: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_numeral_less_neg_numeral
thf(fact_5140_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_5141_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_5142_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_5143_neg__numeral__less__numeral,axiom,
    ! [M: num,N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ ( numera6620942414471956472nteger @ N2 ) ) ).

% neg_numeral_less_numeral
thf(fact_5144_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_5145_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_5146_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_5147_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_5148_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_5149_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_5150_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_5151_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_5152_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_5153_zero__neq__neg__one,axiom,
    ( zero_zero_real
   != ( uminus_uminus_real @ one_one_real ) ) ).

% zero_neq_neg_one
thf(fact_5154_zero__neq__neg__one,axiom,
    ( zero_zero_int
   != ( uminus_uminus_int @ one_one_int ) ) ).

% zero_neq_neg_one
thf(fact_5155_zero__neq__neg__one,axiom,
    ( zero_zero_complex
   != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% zero_neq_neg_one
thf(fact_5156_zero__neq__neg__one,axiom,
    ( zero_z3403309356797280102nteger
   != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% zero_neq_neg_one
thf(fact_5157_zero__neq__neg__one,axiom,
    ( zero_zero_rat
   != ( uminus_uminus_rat @ one_one_rat ) ) ).

% zero_neq_neg_one
thf(fact_5158_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_5159_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_5160_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_5161_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_5162_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_5163_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_5164_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_5165_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_5166_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_5167_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_5168_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_5169_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_5170_add_Oinverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( plus_plus_complex @ A @ B )
        = zero_zero_complex )
     => ( ( uminus1482373934393186551omplex @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5171_add_Oinverse__unique,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( plus_p5714425477246183910nteger @ A @ B )
        = zero_z3403309356797280102nteger )
     => ( ( uminus1351360451143612070nteger @ A )
        = B ) ) ).

% add.inverse_unique
thf(fact_5172_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_5173_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_5174_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_5175_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_5176_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_5177_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_5178_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_5179_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_5180_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_5181_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_5182_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_5183_xor__num_Ocases,axiom,
    ! [X4: product_prod_num_num] :
      ( ( X4
       != ( product_Pair_num_num @ one @ one ) )
     => ( ! [N3: num] :
            ( X4
           != ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) )
       => ( ! [N3: num] :
              ( X4
             != ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) )
         => ( ! [M5: num] :
                ( X4
               != ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) )
           => ( ! [M5: num,N3: num] :
                  ( X4
                 != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) )
             => ( ! [M5: num,N3: num] :
                    ( X4
                   != ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) )
               => ( ! [M5: num] :
                      ( X4
                     != ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) )
                 => ( ! [M5: num,N3: num] :
                        ( X4
                       != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) )
                   => ~ ! [M5: num,N3: num] :
                          ( X4
                         != ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.cases
thf(fact_5184_num_Oexhaust,axiom,
    ! [Y: num] :
      ( ( Y != one )
     => ( ! [X23: num] :
            ( Y
           != ( bit0 @ X23 ) )
       => ~ ! [X33: num] :
              ( Y
             != ( bit1 @ X33 ) ) ) ) ).

% num.exhaust
thf(fact_5185_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_5186_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_5187_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_5188_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_5189_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_5190_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_5191_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_5192_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_5193_numeral__times__minus__swap,axiom,
    ! [W: num,X4: real] :
      ( ( times_times_real @ ( numeral_numeral_real @ W ) @ ( uminus_uminus_real @ X4 ) )
      = ( times_times_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5194_numeral__times__minus__swap,axiom,
    ! [W: num,X4: int] :
      ( ( times_times_int @ ( numeral_numeral_int @ W ) @ ( uminus_uminus_int @ X4 ) )
      = ( times_times_int @ X4 @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5195_numeral__times__minus__swap,axiom,
    ! [W: num,X4: complex] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ ( uminus1482373934393186551omplex @ X4 ) )
      = ( times_times_complex @ X4 @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5196_numeral__times__minus__swap,axiom,
    ! [W: num,X4: code_integer] :
      ( ( times_3573771949741848930nteger @ ( numera6620942414471956472nteger @ W ) @ ( uminus1351360451143612070nteger @ X4 ) )
      = ( times_3573771949741848930nteger @ X4 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5197_numeral__times__minus__swap,axiom,
    ! [W: num,X4: rat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ W ) @ ( uminus_uminus_rat @ X4 ) )
      = ( times_times_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% numeral_times_minus_swap
thf(fact_5198_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_5199_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_5200_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_5201_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_5202_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_5203_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_5204_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_real
     != ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5205_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_int
     != ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5206_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_complex
     != ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5207_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_Code_integer
     != ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5208_one__neq__neg__numeral,axiom,
    ! [N2: num] :
      ( one_one_rat
     != ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) ) ).

% one_neq_neg_numeral
thf(fact_5209_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ N2 )
     != ( uminus_uminus_real @ one_one_real ) ) ).

% numeral_neq_neg_one
thf(fact_5210_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ N2 )
     != ( uminus_uminus_int @ one_one_int ) ) ).

% numeral_neq_neg_one
thf(fact_5211_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ N2 )
     != ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% numeral_neq_neg_one
thf(fact_5212_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numera6620942414471956472nteger @ N2 )
     != ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% numeral_neq_neg_one
thf(fact_5213_numeral__neq__neg__one,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ N2 )
     != ( uminus_uminus_rat @ one_one_rat ) ) ).

% numeral_neq_neg_one
thf(fact_5214_square__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( times_times_real @ X4 @ X4 )
        = one_one_real )
      = ( ( X4 = one_one_real )
        | ( X4
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% square_eq_1_iff
thf(fact_5215_square__eq__1__iff,axiom,
    ! [X4: int] :
      ( ( ( times_times_int @ X4 @ X4 )
        = one_one_int )
      = ( ( X4 = one_one_int )
        | ( X4
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% square_eq_1_iff
thf(fact_5216_square__eq__1__iff,axiom,
    ! [X4: complex] :
      ( ( ( times_times_complex @ X4 @ X4 )
        = one_one_complex )
      = ( ( X4 = one_one_complex )
        | ( X4
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% square_eq_1_iff
thf(fact_5217_square__eq__1__iff,axiom,
    ! [X4: code_integer] :
      ( ( ( times_3573771949741848930nteger @ X4 @ X4 )
        = one_one_Code_integer )
      = ( ( X4 = one_one_Code_integer )
        | ( X4
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% square_eq_1_iff
thf(fact_5218_square__eq__1__iff,axiom,
    ! [X4: rat] :
      ( ( ( times_times_rat @ X4 @ X4 )
        = one_one_rat )
      = ( ( X4 = one_one_rat )
        | ( X4
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% square_eq_1_iff
thf(fact_5219_group__cancel_Osub2,axiom,
    ! [B3: real,K: real,B: real,A: real] :
      ( ( B3
        = ( plus_plus_real @ K @ B ) )
     => ( ( minus_minus_real @ A @ B3 )
        = ( plus_plus_real @ ( uminus_uminus_real @ K ) @ ( minus_minus_real @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5220_group__cancel_Osub2,axiom,
    ! [B3: int,K: int,B: int,A: int] :
      ( ( B3
        = ( plus_plus_int @ K @ B ) )
     => ( ( minus_minus_int @ A @ B3 )
        = ( plus_plus_int @ ( uminus_uminus_int @ K ) @ ( minus_minus_int @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5221_group__cancel_Osub2,axiom,
    ! [B3: complex,K: complex,B: complex,A: complex] :
      ( ( B3
        = ( plus_plus_complex @ K @ B ) )
     => ( ( minus_minus_complex @ A @ B3 )
        = ( plus_plus_complex @ ( uminus1482373934393186551omplex @ K ) @ ( minus_minus_complex @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5222_group__cancel_Osub2,axiom,
    ! [B3: code_integer,K: code_integer,B: code_integer,A: code_integer] :
      ( ( B3
        = ( plus_p5714425477246183910nteger @ K @ B ) )
     => ( ( minus_8373710615458151222nteger @ A @ B3 )
        = ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ K ) @ ( minus_8373710615458151222nteger @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5223_group__cancel_Osub2,axiom,
    ! [B3: rat,K: rat,B: rat,A: rat] :
      ( ( B3
        = ( plus_plus_rat @ K @ B ) )
     => ( ( minus_minus_rat @ A @ B3 )
        = ( plus_plus_rat @ ( uminus_uminus_rat @ K ) @ ( minus_minus_rat @ A @ B ) ) ) ) ).

% group_cancel.sub2
thf(fact_5224_diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5225_diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5226_diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5227_diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5228_diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% diff_conv_add_uminus
thf(fact_5229_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_real
    = ( ^ [A3: real,B2: real] : ( plus_plus_real @ A3 @ ( uminus_uminus_real @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5230_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_int
    = ( ^ [A3: int,B2: int] : ( plus_plus_int @ A3 @ ( uminus_uminus_int @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5231_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_complex
    = ( ^ [A3: complex,B2: complex] : ( plus_plus_complex @ A3 @ ( uminus1482373934393186551omplex @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5232_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_8373710615458151222nteger
    = ( ^ [A3: code_integer,B2: code_integer] : ( plus_p5714425477246183910nteger @ A3 @ ( uminus1351360451143612070nteger @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5233_ab__group__add__class_Oab__diff__conv__add__uminus,axiom,
    ( minus_minus_rat
    = ( ^ [A3: rat,B2: rat] : ( plus_plus_rat @ A3 @ ( uminus_uminus_rat @ B2 ) ) ) ) ).

% ab_group_add_class.ab_diff_conv_add_uminus
thf(fact_5234_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide6298287555418463151nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_5235_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_5236_unique__euclidean__semiring__with__nat__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% unique_euclidean_semiring_with_nat_class.of_nat_div
thf(fact_5237_of__nat__dvd__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_Code_integer @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% of_nat_dvd_iff
thf(fact_5238_of__nat__dvd__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% of_nat_dvd_iff
thf(fact_5239_of__nat__dvd__iff,axiom,
    ! [M: nat,N2: nat] :
      ( ( dvd_dvd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) )
      = ( dvd_dvd_nat @ M @ N2 ) ) ).

% of_nat_dvd_iff
thf(fact_5240_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_5241_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_5242_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_5243_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_5244_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_5245_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_5246_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_5247_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_5248_dvd__div__neg,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ A @ ( uminus1482373934393186551omplex @ B ) )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_div_neg
thf(fact_5249_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_5250_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_5251_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_5252_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_5253_dvd__neg__div,axiom,
    ! [B: complex,A: complex] :
      ( ( dvd_dvd_complex @ B @ A )
     => ( ( divide1717551699836669952omplex @ ( uminus1482373934393186551omplex @ A ) @ B )
        = ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ B ) ) ) ) ).

% dvd_neg_div
thf(fact_5254_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_5255_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_5256_real__of__nat__div4,axiom,
    ! [N2: nat,X4: nat] : ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ X4 ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ X4 ) ) ) ).

% real_of_nat_div4
thf(fact_5257_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri4939895301339042750nteger @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo364778990260209775nteger @ ( semiri4939895301339042750nteger @ M ) @ ( semiri4939895301339042750nteger @ N2 ) ) ) ).

% of_nat_mod
thf(fact_5258_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_mod
thf(fact_5259_of__nat__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( modulo_modulo_nat @ M @ N2 ) )
      = ( modulo_modulo_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_mod
thf(fact_5260_real__minus__mult__self__le,axiom,
    ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( times_times_real @ U @ U ) ) @ ( times_times_real @ X4 @ X4 ) ) ).

% real_minus_mult_self_le
thf(fact_5261_real__of__nat__div,axiom,
    ! [D: nat,N2: nat] :
      ( ( dvd_dvd_nat @ D @ N2 )
     => ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ N2 @ D ) )
        = ( divide_divide_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div
thf(fact_5262_zmod__zminus1__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus1_not_zero
thf(fact_5263_zmod__zminus2__not__zero,axiom,
    ! [K: int,L: int] :
      ( ( ( modulo_modulo_int @ K @ ( uminus_uminus_int @ L ) )
       != zero_zero_int )
     => ( ( modulo_modulo_int @ K @ L )
       != zero_zero_int ) ) ).

% zmod_zminus2_not_zero
thf(fact_5264_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_5265_Bernoulli__inequality,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X4 ) ) @ ( power_power_real @ ( plus_plus_real @ one_one_real @ X4 ) @ N2 ) ) ) ).

% Bernoulli_inequality
thf(fact_5266_add__diff__assoc__enat,axiom,
    ! [Z: extended_enat,Y: extended_enat,X4: extended_enat] :
      ( ( ord_le2932123472753598470d_enat @ Z @ Y )
     => ( ( plus_p3455044024723400733d_enat @ X4 @ ( minus_3235023915231533773d_enat @ Y @ Z ) )
        = ( minus_3235023915231533773d_enat @ ( plus_p3455044024723400733d_enat @ X4 @ Y ) @ Z ) ) ) ).

% add_diff_assoc_enat
thf(fact_5267_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_5268_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_5269_neg__numeral__le__zero,axiom,
    ! [N2: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_le_zero
thf(fact_5270_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_5271_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_5272_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_5273_not__zero__le__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_le_neg_numeral
thf(fact_5274_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_5275_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_5276_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_5277_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_5278_not__zero__less__neg__numeral,axiom,
    ! [N2: num] :
      ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% not_zero_less_neg_numeral
thf(fact_5279_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_5280_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_5281_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_5282_neg__numeral__less__zero,axiom,
    ! [N2: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) @ zero_z3403309356797280102nteger ) ).

% neg_numeral_less_zero
thf(fact_5283_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_5284_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_5285_le__minus__one__simps_I3_J,axiom,
    ~ ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% le_minus_one_simps(3)
thf(fact_5286_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_5287_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_5288_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_5289_le__minus__one__simps_I1_J,axiom,
    ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% le_minus_one_simps(1)
thf(fact_5290_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_5291_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_5292_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_5293_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_5294_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_5295_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_5296_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_5297_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_5298_take__bit__decr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
       != zero_zero_int )
     => ( ( bit_se2923211474154528505it_int @ N2 @ ( minus_minus_int @ K @ one_one_int ) )
        = ( minus_minus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) @ one_one_int ) ) ) ).

% take_bit_decr_eq
thf(fact_5299_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_5300_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_5301_less__minus__one__simps_I1_J,axiom,
    ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ zero_z3403309356797280102nteger ).

% less_minus_one_simps(1)
thf(fact_5302_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_5303_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_5304_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_5305_less__minus__one__simps_I3_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% less_minus_one_simps(3)
thf(fact_5306_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_5307_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_5308_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_5309_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_5310_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_5311_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_5312_not__one__le__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_le_neg_numeral
thf(fact_5313_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_5314_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_5315_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_5316_not__numeral__le__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le3102999989581377725nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_le_neg_one
thf(fact_5317_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_5318_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_5319_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_5320_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_5321_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_5322_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_5323_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_5324_neg__one__le__numeral,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_le_numeral
thf(fact_5325_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_5326_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_5327_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_5328_neg__numeral__le__one,axiom,
    ! [M: num] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_le_one
thf(fact_5329_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_5330_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_5331_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_5332_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_5333_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_5334_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_5335_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_5336_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_5337_not__one__less__neg__numeral,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) ) ).

% not_one_less_neg_numeral
thf(fact_5338_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_5339_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_5340_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_5341_not__numeral__less__neg__one,axiom,
    ! [M: num] :
      ~ ( ord_le6747313008572928689nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% not_numeral_less_neg_one
thf(fact_5342_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_5343_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_5344_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_5345_neg__one__less__numeral,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ M ) ) ).

% neg_one_less_numeral
thf(fact_5346_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_5347_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_5348_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_5349_neg__numeral__less__one,axiom,
    ! [M: num] : ( ord_le6747313008572928689nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ M ) ) @ one_one_Code_integer ) ).

% neg_numeral_less_one
thf(fact_5350_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_5351_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_5352_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_5353_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_5354_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_5355_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_5356_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_5357_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_5358_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_5359_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_5360_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_5361_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_5362_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_5363_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_5364_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_5365_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_5366_mult__1s__ring__1_I1_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) @ B )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5367_mult__1s__ring__1_I1_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) @ B )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5368_mult__1s__ring__1_I1_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) @ B )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5369_mult__1s__ring__1_I1_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) @ B )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5370_mult__1s__ring__1_I1_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) @ B )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(1)
thf(fact_5371_mult__1s__ring__1_I2_J,axiom,
    ! [B: real] :
      ( ( times_times_real @ B @ ( uminus_uminus_real @ ( numeral_numeral_real @ one ) ) )
      = ( uminus_uminus_real @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5372_mult__1s__ring__1_I2_J,axiom,
    ! [B: int] :
      ( ( times_times_int @ B @ ( uminus_uminus_int @ ( numeral_numeral_int @ one ) ) )
      = ( uminus_uminus_int @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5373_mult__1s__ring__1_I2_J,axiom,
    ! [B: complex] :
      ( ( times_times_complex @ B @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) ) )
      = ( uminus1482373934393186551omplex @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5374_mult__1s__ring__1_I2_J,axiom,
    ! [B: code_integer] :
      ( ( times_3573771949741848930nteger @ B @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) ) )
      = ( uminus1351360451143612070nteger @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5375_mult__1s__ring__1_I2_J,axiom,
    ! [B: rat] :
      ( ( times_times_rat @ B @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) ) )
      = ( uminus_uminus_rat @ B ) ) ).

% mult_1s_ring_1(2)
thf(fact_5376_uminus__numeral__One,axiom,
    ( ( uminus_uminus_real @ ( numeral_numeral_real @ one ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% uminus_numeral_One
thf(fact_5377_uminus__numeral__One,axiom,
    ( ( uminus_uminus_int @ ( numeral_numeral_int @ one ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% uminus_numeral_One
thf(fact_5378_uminus__numeral__One,axiom,
    ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% uminus_numeral_One
thf(fact_5379_uminus__numeral__One,axiom,
    ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ one ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% uminus_numeral_One
thf(fact_5380_uminus__numeral__One,axiom,
    ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ one ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% uminus_numeral_One
thf(fact_5381_power__minus,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( power_power_real @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5382_power__minus,axiom,
    ! [A: int,N2: nat] :
      ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( power_power_int @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5383_power__minus,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5384_power__minus,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5385_power__minus,axiom,
    ! [A: rat,N2: nat] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( power_power_rat @ A @ N2 ) ) ) ).

% power_minus
thf(fact_5386_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_5387_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_5388_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_5389_cong__exp__iff__simps_I10_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(10)
thf(fact_5390_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit0 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_5391_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_5392_cong__exp__iff__simps_I12_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(12)
thf(fact_5393_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_5394_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_5395_cong__exp__iff__simps_I13_J,axiom,
    ! [M: num,Q3: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) ) ) ) ).

% cong_exp_iff_simps(13)
thf(fact_5396_power__minus__Bit0,axiom,
    ! [X4: real,K: num] :
      ( ( power_power_real @ ( uminus_uminus_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5397_power__minus__Bit0,axiom,
    ! [X4: int,K: num] :
      ( ( power_power_int @ ( uminus_uminus_int @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5398_power__minus__Bit0,axiom,
    ! [X4: complex,K: num] :
      ( ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5399_power__minus__Bit0,axiom,
    ! [X4: code_integer,K: num] :
      ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5400_power__minus__Bit0,axiom,
    ! [X4: rat,K: num] :
      ( ( power_power_rat @ ( uminus_uminus_rat @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% power_minus_Bit0
thf(fact_5401_reals__Archimedean3,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ! [Y4: real] :
        ? [N3: nat] : ( ord_less_real @ Y4 @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X4 ) ) ) ).

% reals_Archimedean3
thf(fact_5402_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_5403_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_5404_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_5405_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_5406_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se1745604003318907178nteger @ ( numeral_numeral_nat @ K ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( minus_8373710615458151222nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_Code_integer ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_5407_take__bit__numeral__minus__1__eq,axiom,
    ! [K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ K ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ K ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_1_eq
thf(fact_5408_real__0__less__add__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y ) )
      = ( ord_less_real @ ( uminus_uminus_real @ X4 ) @ Y ) ) ).

% real_0_less_add_iff
thf(fact_5409_real__add__less__0__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( plus_plus_real @ X4 @ Y ) @ zero_zero_real )
      = ( ord_less_real @ Y @ ( uminus_uminus_real @ X4 ) ) ) ).

% real_add_less_0_iff
thf(fact_5410_real__0__le__add__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( plus_plus_real @ X4 @ Y ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ Y ) ) ).

% real_0_le_add_iff
thf(fact_5411_real__add__le__0__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ X4 @ Y ) @ zero_zero_real )
      = ( ord_less_eq_real @ Y @ ( uminus_uminus_real @ X4 ) ) ) ).

% real_add_le_0_iff
thf(fact_5412_real__of__nat__div__aux,axiom,
    ! [X4: nat,D: nat] :
      ( ( divide_divide_real @ ( semiri5074537144036343181t_real @ X4 ) @ ( semiri5074537144036343181t_real @ D ) )
      = ( plus_plus_real @ ( semiri5074537144036343181t_real @ ( divide_divide_nat @ X4 @ D ) ) @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ X4 @ D ) ) @ ( semiri5074537144036343181t_real @ D ) ) ) ) ).

% real_of_nat_div_aux
thf(fact_5413_zmod__zminus1__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ ( uminus_uminus_int @ A ) @ B )
          = ( minus_minus_int @ B @ ( modulo_modulo_int @ A @ B ) ) ) ) ) ).

% zmod_zminus1_eq_if
thf(fact_5414_zmod__zminus2__eq__if,axiom,
    ! [A: int,B: int] :
      ( ( ( ( modulo_modulo_int @ A @ B )
          = zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = zero_zero_int ) )
      & ( ( ( modulo_modulo_int @ A @ B )
         != zero_zero_int )
       => ( ( modulo_modulo_int @ A @ ( uminus_uminus_int @ B ) )
          = ( minus_minus_int @ ( modulo_modulo_int @ A @ B ) @ B ) ) ) ) ).

% zmod_zminus2_eq_if
thf(fact_5415_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_5416_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_5417_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_5418_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_5419_numeral__Bit1__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( numeral_numeral_nat @ N2 ) ) ).

% numeral_Bit1_div_2
thf(fact_5420_numeral__Bit1__div__2,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% numeral_Bit1_div_2
thf(fact_5421_numeral__Bit1__div__2,axiom,
    ! [N2: num] :
      ( ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% numeral_Bit1_div_2
thf(fact_5422_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_5423_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_5424_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_5425_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_5426_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_5427_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_5428_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_5429_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_5430_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_5431_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_5432_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_5433_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_5434_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
        = ( divide_divide_real @ B @ C ) )
      = ( ( ( C != zero_zero_real )
         => ( ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5435_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: complex,C: complex] :
      ( ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
        = ( divide1717551699836669952omplex @ B @ C ) )
      = ( ( ( C != zero_zero_complex )
         => ( ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5436_eq__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
        = ( divide_divide_rat @ B @ C ) )
      = ( ( ( C != zero_zero_rat )
         => ( ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C )
            = B ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% eq_divide_eq_numeral(2)
thf(fact_5437_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ( divide_divide_real @ B @ C )
        = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( C != zero_zero_real )
         => ( B
            = ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_real )
         => ( ( uminus_uminus_real @ ( numeral_numeral_real @ W ) )
            = zero_zero_real ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5438_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: complex,C: complex,W: num] :
      ( ( ( divide1717551699836669952omplex @ B @ C )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( ( C != zero_zero_complex )
         => ( B
            = ( times_times_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_complex )
         => ( ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) )
            = zero_zero_complex ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5439_divide__eq__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ( divide_divide_rat @ B @ C )
        = ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( C != zero_zero_rat )
         => ( B
            = ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ( C = zero_zero_rat )
         => ( ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) )
            = zero_zero_rat ) ) ) ) ).

% divide_eq_eq_numeral(2)
thf(fact_5440_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_5441_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_5442_odd__numeral,axiom,
    ! [N2: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ).

% odd_numeral
thf(fact_5443_minus__divide__add__eq__iff,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X4 @ Z ) ) @ Y )
        = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5444_minus__divide__add__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X4 @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5445_minus__divide__add__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X4 @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ X4 ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_add_eq_iff
thf(fact_5446_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_real )
       => ( ( plus_plus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( plus_plus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5447_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_complex )
       => ( ( plus_plus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5448_add__divide__eq__if__simps_I3_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = B ) )
      & ( ( Z != zero_zero_rat )
       => ( ( plus_plus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(3)
thf(fact_5449_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
     != zero_zero_nat ) ).

% cong_exp_iff_simps(3)
thf(fact_5450_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
     != zero_zero_int ) ).

% cong_exp_iff_simps(3)
thf(fact_5451_cong__exp__iff__simps_I3_J,axiom,
    ! [N2: num,Q3: num] :
      ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
     != zero_z3403309356797280102nteger ) ).

% cong_exp_iff_simps(3)
thf(fact_5452_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ A @ Z ) ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ A ) @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5453_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ A @ Z ) ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ A ) @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5454_add__divide__eq__if__simps_I6_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ A @ Z ) ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ A ) @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(6)
thf(fact_5455_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: real,A: real,B: real] :
      ( ( ( Z = zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( uminus_uminus_real @ B ) ) )
      & ( ( Z != zero_zero_real )
       => ( ( minus_minus_real @ ( divide_divide_real @ A @ Z ) @ B )
          = ( divide_divide_real @ ( minus_minus_real @ A @ ( times_times_real @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5456_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: complex,A: complex,B: complex] :
      ( ( ( Z = zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( uminus1482373934393186551omplex @ B ) ) )
      & ( ( Z != zero_zero_complex )
       => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ A @ Z ) @ B )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ A @ ( times_times_complex @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5457_add__divide__eq__if__simps_I5_J,axiom,
    ! [Z: rat,A: rat,B: rat] :
      ( ( ( Z = zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( uminus_uminus_rat @ B ) ) )
      & ( ( Z != zero_zero_rat )
       => ( ( minus_minus_rat @ ( divide_divide_rat @ A @ Z ) @ B )
          = ( divide_divide_rat @ ( minus_minus_rat @ A @ ( times_times_rat @ B @ Z ) ) @ Z ) ) ) ) ).

% add_divide_eq_if_simps(5)
thf(fact_5458_minus__divide__diff__eq__iff,axiom,
    ! [Z: real,X4: real,Y: real] :
      ( ( Z != zero_zero_real )
     => ( ( minus_minus_real @ ( uminus_uminus_real @ ( divide_divide_real @ X4 @ Z ) ) @ Y )
        = ( divide_divide_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5459_minus__divide__diff__eq__iff,axiom,
    ! [Z: complex,X4: complex,Y: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( divide1717551699836669952omplex @ X4 @ Z ) ) @ Y )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ ( times_times_complex @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5460_minus__divide__diff__eq__iff,axiom,
    ! [Z: rat,X4: rat,Y: rat] :
      ( ( Z != zero_zero_rat )
     => ( ( minus_minus_rat @ ( uminus_uminus_rat @ ( divide_divide_rat @ X4 @ Z ) ) @ Y )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ X4 ) @ ( times_times_rat @ Y @ Z ) ) @ Z ) ) ) ).

% minus_divide_diff_eq_iff
thf(fact_5461_power3__eq__cube,axiom,
    ! [A: complex] :
      ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_complex @ ( times_times_complex @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_5462_power3__eq__cube,axiom,
    ! [A: real] :
      ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_real @ ( times_times_real @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_5463_power3__eq__cube,axiom,
    ! [A: nat] :
      ( ( power_power_nat @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_nat @ ( times_times_nat @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_5464_power3__eq__cube,axiom,
    ! [A: int] :
      ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit1 @ one ) ) )
      = ( times_times_int @ ( times_times_int @ A @ A ) @ A ) ) ).

% power3_eq_cube
thf(fact_5465_even__minus,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( uminus_uminus_int @ A ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_5466_even__minus,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( uminus1351360451143612070nteger @ A ) )
      = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ).

% even_minus
thf(fact_5467_numeral__3__eq__3,axiom,
    ( ( numeral_numeral_nat @ ( bit1 @ one ) )
    = ( suc @ ( suc @ ( suc @ zero_zero_nat ) ) ) ) ).

% numeral_3_eq_3
thf(fact_5468_even__xor__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3222712562003087583nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_xor_iff
thf(fact_5469_even__xor__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_xor_iff
thf(fact_5470_even__xor__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_xor_iff
thf(fact_5471_power2__eq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5472_power2__eq__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5473_power2__eq__iff,axiom,
    ! [X4: complex,Y: complex] :
      ( ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_complex @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus1482373934393186551omplex @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5474_power2__eq__iff,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5475_power2__eq__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% power2_eq_iff
thf(fact_5476_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_5477_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri681578069525770553at_rat @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri681578069525770553at_rat @ N2 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5478_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri8010041392384452111omplex @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ M ) @ ( semiri8010041392384452111omplex @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri8010041392384452111omplex @ N2 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5479_field__char__0__class_Oof__nat__div,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri5074537144036343181t_real @ ( divide_divide_nat @ M @ N2 ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ ( modulo_modulo_nat @ M @ N2 ) ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ).

% field_char_0_class.of_nat_div
thf(fact_5480_nat__less__real__le,axiom,
    ( ord_less_nat
    = ( ^ [N: nat,M6: nat] : ( ord_less_eq_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N ) @ one_one_real ) @ ( semiri5074537144036343181t_real @ M6 ) ) ) ) ).

% nat_less_real_le
thf(fact_5481_nat__le__real__less,axiom,
    ( ord_less_eq_nat
    = ( ^ [N: nat,M6: nat] : ( ord_less_real @ ( semiri5074537144036343181t_real @ N ) @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M6 ) @ one_one_real ) ) ) ) ).

% nat_le_real_less
thf(fact_5482_verit__less__mono__div__int2,axiom,
    ! [A2: int,B3: int,N2: int] :
      ( ( ord_less_eq_int @ A2 @ B3 )
     => ( ( ord_less_int @ zero_zero_int @ ( uminus_uminus_int @ N2 ) )
       => ( ord_less_eq_int @ ( divide_divide_int @ B3 @ N2 ) @ ( divide_divide_int @ A2 @ N2 ) ) ) ) ).

% verit_less_mono_div_int2
thf(fact_5483_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_5484_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_5485_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_5486_take__bit__eq__mod,axiom,
    ( bit_se1745604003318907178nteger
    = ( ^ [N: nat,A3: code_integer] : ( modulo364778990260209775nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_eq_mod
thf(fact_5487_take__bit__eq__mod,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,A3: int] : ( modulo_modulo_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_eq_mod
thf(fact_5488_take__bit__eq__mod,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,A3: nat] : ( modulo_modulo_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_eq_mod
thf(fact_5489_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_5490_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_5491_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_5492_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_5493_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_5494_take__bit__nat__def,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,M6: nat] : ( modulo_modulo_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_nat_def
thf(fact_5495_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_5496_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_5497_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_5498_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_5499_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_5500_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_5501_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_5502_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_5503_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_5504_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_5505_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_5506_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_5507_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_5508_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_5509_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_5510_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_5511_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_5512_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_5513_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( divide_divide_real @ B @ C ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ 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 @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5514_less__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( divide_divide_rat @ B @ C ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ 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 @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% less_divide_eq_numeral(2)
thf(fact_5515_divide__less__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5516_divide__less__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_less_eq_numeral(2)
thf(fact_5517_take__bit__int__def,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,K3: int] : ( modulo_modulo_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% take_bit_int_def
thf(fact_5518_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ N2 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ N2 ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5519_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ N2 ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5520_cong__exp__iff__simps_I7_J,axiom,
    ! [Q3: num,N2: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ N2 ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(7)
thf(fact_5521_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ ( bit1 @ M ) ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_nat @ ( numeral_numeral_nat @ one ) @ ( numeral_numeral_nat @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ Q3 ) )
        = zero_zero_nat ) ) ).

% cong_exp_iff_simps(11)
thf(fact_5522_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo_modulo_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) )
        = ( modulo_modulo_int @ ( numeral_numeral_int @ one ) @ ( numeral_numeral_int @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ Q3 ) )
        = zero_zero_int ) ) ).

% cong_exp_iff_simps(11)
thf(fact_5523_cong__exp__iff__simps_I11_J,axiom,
    ! [M: num,Q3: num] :
      ( ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ ( bit1 @ M ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) )
        = ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ one ) @ ( numera6620942414471956472nteger @ ( bit0 @ Q3 ) ) ) )
      = ( ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ Q3 ) )
        = zero_z3403309356797280102nteger ) ) ).

% cong_exp_iff_simps(11)
thf(fact_5524_power2__eq__1__iff,axiom,
    ! [A: real] :
      ( ( ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( A = one_one_real )
        | ( A
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5525_power2__eq__1__iff,axiom,
    ! [A: int] :
      ( ( ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( A = one_one_int )
        | ( A
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5526_power2__eq__1__iff,axiom,
    ! [A: complex] :
      ( ( ( power_power_complex @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
      = ( ( A = one_one_complex )
        | ( A
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5527_power2__eq__1__iff,axiom,
    ! [A: code_integer] :
      ( ( ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( A = one_one_Code_integer )
        | ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5528_power2__eq__1__iff,axiom,
    ! [A: rat] :
      ( ( ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( A = one_one_rat )
        | ( A
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% power2_eq_1_iff
thf(fact_5529_real__archimedian__rdiv__eq__0,axiom,
    ! [X4: real,C: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ C )
       => ( ! [M5: nat] :
              ( ( ord_less_nat @ zero_zero_nat @ M5 )
             => ( ord_less_eq_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M5 ) @ X4 ) @ C ) )
         => ( X4 = zero_zero_real ) ) ) ) ).

% real_archimedian_rdiv_eq_0
thf(fact_5530_uminus__power__if,axiom,
    ! [N2: nat,A: real] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
          = ( power_power_real @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 )
          = ( uminus_uminus_real @ ( power_power_real @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5531_uminus__power__if,axiom,
    ! [N2: nat,A: int] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
          = ( power_power_int @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 )
          = ( uminus_uminus_int @ ( power_power_int @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5532_uminus__power__if,axiom,
    ! [N2: nat,A: complex] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
          = ( power_power_complex @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 )
          = ( uminus1482373934393186551omplex @ ( power_power_complex @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5533_uminus__power__if,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
          = ( power_8256067586552552935nteger @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 )
          = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5534_uminus__power__if,axiom,
    ! [N2: nat,A: rat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
          = ( power_power_rat @ A @ N2 ) ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 )
          = ( uminus_uminus_rat @ ( power_power_rat @ A @ N2 ) ) ) ) ) ).

% uminus_power_if
thf(fact_5535_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_5536_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_5537_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_5538_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_5539_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_5540_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_5541_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_5542_realpow__square__minus__le,axiom,
    ! [U: real,X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( power_power_real @ U @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% realpow_square_minus_le
thf(fact_5543_signed__take__bit__int__less__eq__self__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) @ K )
      = ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ K ) ) ).

% signed_take_bit_int_less_eq_self_iff
thf(fact_5544_signed__take__bit__int__greater__eq__minus__exp,axiom,
    ! [N2: nat,K: int] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) @ ( bit_ri631733984087533419it_int @ N2 @ K ) ) ).

% signed_take_bit_int_greater_eq_minus_exp
thf(fact_5545_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_5546_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_5547_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_5548_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_5549_minus__mod__int__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ( modulo_modulo_int @ ( uminus_uminus_int @ K ) @ L )
        = ( minus_minus_int @ ( minus_minus_int @ L @ one_one_int ) @ ( modulo_modulo_int @ ( minus_minus_int @ K @ one_one_int ) @ L ) ) ) ) ).

% minus_mod_int_eq
thf(fact_5550_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_5551_zdiv__zminus1__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus1_eq_if
thf(fact_5552_zdiv__zminus2__eq__if,axiom,
    ! [B: int,A: int] :
      ( ( B != zero_zero_int )
     => ( ( ( ( modulo_modulo_int @ A @ B )
            = zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) ) )
        & ( ( ( modulo_modulo_int @ A @ B )
           != zero_zero_int )
         => ( ( divide_divide_int @ A @ ( uminus_uminus_int @ B ) )
            = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ A @ B ) ) @ one_one_int ) ) ) ) ) ).

% zdiv_zminus2_eq_if
thf(fact_5553_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_5554_zminus1__lemma,axiom,
    ! [A: int,B: int,Q3: int,R3: int] :
      ( ( eucl_rel_int @ A @ B @ ( product_Pair_int_int @ Q3 @ R3 ) )
     => ( ( B != zero_zero_int )
       => ( eucl_rel_int @ ( uminus_uminus_int @ A ) @ B @ ( product_Pair_int_int @ ( if_int @ ( R3 = zero_zero_int ) @ ( uminus_uminus_int @ Q3 ) @ ( minus_minus_int @ ( uminus_uminus_int @ Q3 ) @ one_one_int ) ) @ ( if_int @ ( R3 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ B @ R3 ) ) ) ) ) ) ).

% zminus1_lemma
thf(fact_5555_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_5556_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_5557_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: real,C: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ ( 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 @ W ) ) @ 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 @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ zero_zero_real ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5558_le__divide__eq__numeral_I2_J,axiom,
    ! [W: num,B: rat,C: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ ( 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 @ W ) ) @ 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 @ W ) ) @ C ) ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ zero_zero_rat ) ) ) ) ) ) ).

% le_divide_eq_numeral(2)
thf(fact_5559_divide__le__eq__numeral_I2_J,axiom,
    ! [B: real,C: real,W: num] :
      ( ( ord_less_eq_real @ ( divide_divide_real @ B @ C ) @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( ( ( ord_less_real @ zero_zero_real @ C )
         => ( ord_less_eq_real @ B @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_real @ zero_zero_real @ C )
         => ( ( ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_real @ C @ zero_zero_real )
             => ( ord_less_eq_real @ zero_zero_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5560_divide__le__eq__numeral_I2_J,axiom,
    ! [B: rat,C: rat,W: num] :
      ( ( ord_less_eq_rat @ ( divide_divide_rat @ B @ C ) @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ C )
         => ( ord_less_eq_rat @ B @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) ) )
        & ( ~ ( ord_less_rat @ zero_zero_rat @ C )
         => ( ( ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ ( times_times_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) @ C ) @ B ) )
            & ( ~ ( ord_less_rat @ C @ zero_zero_rat )
             => ( ord_less_eq_rat @ zero_zero_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ) ) ) ) ).

% divide_le_eq_numeral(2)
thf(fact_5561_square__le__1,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ).

% square_le_1
thf(fact_5562_square__le__1,axiom,
    ! [X4: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X4 )
     => ( ( ord_le3102999989581377725nteger @ X4 @ one_one_Code_integer )
       => ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer ) ) ) ).

% square_le_1
thf(fact_5563_square__le__1,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ one_one_rat )
       => ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat ) ) ) ).

% square_le_1
thf(fact_5564_square__le__1,axiom,
    ! [X4: int] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
     => ( ( ord_less_eq_int @ X4 @ one_one_int )
       => ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int ) ) ) ).

% square_le_1
thf(fact_5565_minus__power__mult__self,axiom,
    ! [A: real,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
      = ( power_power_real @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5566_minus__power__mult__self,axiom,
    ! [A: int,N2: nat] :
      ( ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
      = ( power_power_int @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5567_minus__power__mult__self,axiom,
    ! [A: complex,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ A ) @ N2 ) )
      = ( power_power_complex @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5568_minus__power__mult__self,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
      = ( power_8256067586552552935nteger @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5569_minus__power__mult__self,axiom,
    ! [A: rat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
      = ( power_power_rat @ A @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% minus_power_mult_self
thf(fact_5570_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% minus_one_power_iff
thf(fact_5571_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = one_one_int ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 )
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% minus_one_power_iff
thf(fact_5572_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
          = one_one_complex ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 )
          = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ) ).

% minus_one_power_iff
thf(fact_5573_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = one_one_Code_integer ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 )
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% minus_one_power_iff
thf(fact_5574_minus__one__power__iff,axiom,
    ! [N2: nat] :
      ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = one_one_rat ) )
      & ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 )
          = ( uminus_uminus_rat @ one_one_rat ) ) ) ) ).

% minus_one_power_iff
thf(fact_5575_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_5576_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_5577_minus__1__div__exp__eq__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% minus_1_div_exp_eq_int
thf(fact_5578_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_5579_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_5580_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_5581_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_5582_take__bit__incr__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
       != ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ one_one_int ) )
     => ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
        = ( plus_plus_int @ one_one_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).

% take_bit_incr_eq
thf(fact_5583_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_5584_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_5585_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_5586_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_5587_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_5588_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_5589_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_5590_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_5591_int__bit__induct,axiom,
    ! [P: int > $o,K: int] :
      ( ( P @ zero_zero_int )
     => ( ( P @ ( uminus_uminus_int @ one_one_int ) )
       => ( ! [K2: int] :
              ( ( P @ K2 )
             => ( ( K2 != zero_zero_int )
               => ( P @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) )
         => ( ! [K2: int] :
                ( ( P @ K2 )
               => ( ( K2
                   != ( uminus_uminus_int @ one_one_int ) )
                 => ( P @ ( plus_plus_int @ one_one_int @ ( times_times_int @ K2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) )
           => ( P @ K ) ) ) ) ) ).

% int_bit_induct
thf(fact_5592_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_5593_xor__nat__unfold,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( modulo_modulo_nat @ ( plus_plus_nat @ ( modulo_modulo_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( modulo_modulo_nat @ 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 @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% xor_nat_unfold
thf(fact_5594_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_5595_xor__nat__rec,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 ) )
             != ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se6528837805403552850or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_nat_rec
thf(fact_5596_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_5597_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_5598_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_5599_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_5600_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_5601_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_5602_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_5603_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_5604_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_5605_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_5606_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_5607_real__average__minus__first,axiom,
    ! [A: real,B: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_first
thf(fact_5608_real__average__minus__second,axiom,
    ! [B: real,A: real] :
      ( ( minus_minus_real @ ( divide_divide_real @ ( plus_plus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ A )
      = ( divide_divide_real @ ( minus_minus_real @ B @ A ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% real_average_minus_second
thf(fact_5609_linear__plus__1__le__power,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X4 ) @ one_one_real ) @ ( power_power_real @ ( plus_plus_real @ X4 @ one_one_real ) @ N2 ) ) ) ).

% linear_plus_1_le_power
thf(fact_5610_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_5611_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_5612_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_5613_compl__le__compl__iff,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X4 ) @ ( uminus1532241313380277803et_int @ Y ) )
      = ( ord_less_eq_set_int @ Y @ X4 ) ) ).

% compl_le_compl_iff
thf(fact_5614_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5615_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5616_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5617_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5618_dbl__dec__simps_I4_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ) ).

% dbl_dec_simps(4)
thf(fact_5619_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_5620_xor__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        = ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% xor_nonnegative_int_iff
thf(fact_5621_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_5622_negative__zle,axiom,
    ! [N2: nat,M: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zle
thf(fact_5623_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ one_one_complex )
    = one_one_complex ) ).

% dbl_dec_simps(3)
thf(fact_5624_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ one_one_real )
    = one_one_real ) ).

% dbl_dec_simps(3)
thf(fact_5625_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ one_one_rat )
    = one_one_rat ) ).

% dbl_dec_simps(3)
thf(fact_5626_dbl__dec__simps_I3_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ one_one_int )
    = one_one_int ) ).

% dbl_dec_simps(3)
thf(fact_5627_negative__zless,axiom,
    ! [N2: nat,M: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ ( semiri1314217659103216013at_int @ M ) ) ).

% negative_zless
thf(fact_5628_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6075765906172075777c_real @ zero_zero_real )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_dec_simps(2)
thf(fact_5629_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3811975205180677377ec_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_dec_simps(2)
thf(fact_5630_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu6511756317524482435omplex @ zero_zero_complex )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_dec_simps(2)
thf(fact_5631_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu7757733837767384882nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_dec_simps(2)
thf(fact_5632_dbl__dec__simps_I2_J,axiom,
    ( ( neg_nu3179335615603231917ec_rat @ zero_zero_rat )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_dec_simps(2)
thf(fact_5633_take__bit__minus,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ K ) ) ) ).

% take_bit_minus
thf(fact_5634_int__cases,axiom,
    ! [Z: int] :
      ( ! [N3: nat] :
          ( Z
         != ( semiri1314217659103216013at_int @ N3 ) )
     => ~ ! [N3: nat] :
            ( Z
           != ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% int_cases
thf(fact_5635_int__of__nat__induct,axiom,
    ! [P: int > $o,Z: int] :
      ( ! [N3: nat] : ( P @ ( semiri1314217659103216013at_int @ N3 ) )
     => ( ! [N3: nat] : ( P @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) )
       => ( P @ Z ) ) ) ).

% int_of_nat_induct
thf(fact_5636_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_5637_XOR__lower,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se6526347334894502574or_int @ X4 @ Y ) ) ) ) ).

% XOR_lower
thf(fact_5638_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_5639_int__ops_I3_J,axiom,
    ! [N2: num] :
      ( ( semiri1314217659103216013at_int @ ( numeral_numeral_nat @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% int_ops(3)
thf(fact_5640_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_5641_nat__int__comparison_I2_J,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(2)
thf(fact_5642_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_5643_nat__int__comparison_I3_J,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_int_comparison(3)
thf(fact_5644_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_5645_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_5646_negative__zle__0,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ zero_zero_int ) ).

% negative_zle_0
thf(fact_5647_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_5648_zadd__int__left,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( plus_plus_int @ ( semiri1314217659103216013at_int @ M ) @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ Z ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ M @ N2 ) ) @ Z ) ) ).

% zadd_int_left
thf(fact_5649_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_5650_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_5651_int__ops_I2_J,axiom,
    ( ( semiri1314217659103216013at_int @ one_one_nat )
    = one_one_int ) ).

% int_ops(2)
thf(fact_5652_zle__iff__zadd,axiom,
    ( ord_less_eq_int
    = ( ^ [W3: int,Z5: int] :
        ? [N: nat] :
          ( Z5
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% zle_iff_zadd
thf(fact_5653_zdiv__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( divide_divide_nat @ A @ B ) )
      = ( divide_divide_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zdiv_int
thf(fact_5654_zmod__int,axiom,
    ! [A: nat,B: nat] :
      ( ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ A @ B ) )
      = ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% zmod_int
thf(fact_5655_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_5656_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_5657_negD,axiom,
    ! [X4: int] :
      ( ( ord_less_int @ X4 @ zero_zero_int )
     => ? [N3: nat] :
          ( X4
          = ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N3 ) ) ) ) ) ).

% negD
thf(fact_5658_negative__zless__0,axiom,
    ! [N2: nat] : ( ord_less_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) ) @ zero_zero_int ) ).

% negative_zless_0
thf(fact_5659_int__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( suc @ N2 ) )
      = ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) ) ).

% int_Suc
thf(fact_5660_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_5661_zless__iff__Suc__zadd,axiom,
    ( ord_less_int
    = ( ^ [W3: int,Z5: int] :
        ? [N: nat] :
          ( Z5
          = ( plus_plus_int @ W3 @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) ) ) ) ) ).

% zless_iff_Suc_zadd
thf(fact_5662_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_5663_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_5664_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_5665_zmult__zless__mono2__lemma,axiom,
    ! [I2: int,J: int,K: nat] :
      ( ( ord_less_int @ I2 @ J )
     => ( ( ord_less_nat @ zero_zero_nat @ K )
       => ( ord_less_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ I2 ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ K ) @ J ) ) ) ) ).

% zmult_zless_mono2_lemma
thf(fact_5666_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_5667_zdiff__int__split,axiom,
    ! [P: int > $o,X4: nat,Y: nat] :
      ( ( P @ ( semiri1314217659103216013at_int @ ( minus_minus_nat @ X4 @ Y ) ) )
      = ( ( ( ord_less_eq_nat @ Y @ X4 )
         => ( P @ ( minus_minus_int @ ( semiri1314217659103216013at_int @ X4 ) @ ( semiri1314217659103216013at_int @ Y ) ) ) )
        & ( ( ord_less_nat @ X4 @ Y )
         => ( P @ zero_zero_int ) ) ) ) ).

% zdiff_int_split
thf(fact_5668_dbl__dec__def,axiom,
    ( neg_nu6511756317524482435omplex
    = ( ^ [X: complex] : ( minus_minus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).

% dbl_dec_def
thf(fact_5669_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_5670_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_5671_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_5672_compl__le__swap2,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ X4 )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ X4 ) @ Y ) ) ).

% compl_le_swap2
thf(fact_5673_compl__le__swap1,axiom,
    ! [Y: set_int,X4: set_int] :
      ( ( ord_less_eq_set_int @ Y @ ( uminus1532241313380277803et_int @ X4 ) )
     => ( ord_less_eq_set_int @ X4 @ ( uminus1532241313380277803et_int @ Y ) ) ) ).

% compl_le_swap1
thf(fact_5674_compl__mono,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_less_eq_set_int @ X4 @ Y )
     => ( ord_less_eq_set_int @ ( uminus1532241313380277803et_int @ Y ) @ ( uminus1532241313380277803et_int @ X4 ) ) ) ).

% compl_mono
thf(fact_5675_XOR__upper,axiom,
    ! [X4: int,N2: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_int @ X4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se6526347334894502574or_int @ X4 @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% XOR_upper
thf(fact_5676_real__arch__simple,axiom,
    ! [X4: real] :
    ? [N3: nat] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% real_arch_simple
thf(fact_5677_real__arch__simple,axiom,
    ! [X4: rat] :
    ? [N3: nat] : ( ord_less_eq_rat @ X4 @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% real_arch_simple
thf(fact_5678_reals__Archimedean2,axiom,
    ! [X4: rat] :
    ? [N3: nat] : ( ord_less_rat @ X4 @ ( semiri681578069525770553at_rat @ N3 ) ) ).

% reals_Archimedean2
thf(fact_5679_reals__Archimedean2,axiom,
    ! [X4: real] :
    ? [N3: nat] : ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ N3 ) ) ).

% reals_Archimedean2
thf(fact_5680_exists__least__lemma,axiom,
    ! [P: nat > $o] :
      ( ~ ( P @ zero_zero_nat )
     => ( ? [X_1: nat] : ( P @ X_1 )
       => ? [N3: nat] :
            ( ~ ( P @ N3 )
            & ( P @ ( suc @ N3 ) ) ) ) ) ).

% exists_least_lemma
thf(fact_5681_xor__int__rec,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( 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_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% xor_int_rec
thf(fact_5682_Bolzano,axiom,
    ! [A: real,B: real,P: real > real > $o] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [A5: real,B5: real,C3: real] :
            ( ( P @ A5 @ B5 )
           => ( ( P @ B5 @ C3 )
             => ( ( ord_less_eq_real @ A5 @ B5 )
               => ( ( ord_less_eq_real @ B5 @ C3 )
                 => ( P @ A5 @ C3 ) ) ) ) )
       => ( ! [X5: real] :
              ( ( ord_less_eq_real @ A @ X5 )
             => ( ( ord_less_eq_real @ X5 @ B )
               => ? [D6: real] :
                    ( ( ord_less_real @ zero_zero_real @ D6 )
                    & ! [A5: real,B5: real] :
                        ( ( ( ord_less_eq_real @ A5 @ X5 )
                          & ( ord_less_eq_real @ X5 @ B5 )
                          & ( ord_less_real @ ( minus_minus_real @ B5 @ A5 ) @ D6 ) )
                       => ( P @ A5 @ B5 ) ) ) ) )
         => ( P @ A @ B ) ) ) ) ).

% Bolzano
thf(fact_5683_ex__less__of__nat__mult,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ? [N3: nat] : ( ord_less_rat @ Y @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N3 ) @ X4 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_5684_ex__less__of__nat__mult,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [N3: nat] : ( ord_less_real @ Y @ ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ X4 ) ) ) ).

% ex_less_of_nat_mult
thf(fact_5685_signed__take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( minus_minus_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) @ one_one_int ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_minus_bit1
thf(fact_5686_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_5687_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_5688_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_5689_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_5690_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_5691_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_5692_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_5693_lemma__termdiff3,axiom,
    ! [H: real,Z: real,K5: real,N2: nat] :
      ( ( H != zero_zero_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ Z @ H ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( 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 @ H ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5694_lemma__termdiff3,axiom,
    ! [H: complex,Z: complex,K5: real,N2: nat] :
      ( ( H != zero_zero_complex )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ K5 )
       => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ Z @ H ) ) @ K5 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( 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 @ H ) ) ) ) ) ) ).

% lemma_termdiff3
thf(fact_5695_signed__take__bit__numeral__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit1 @ K ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% signed_take_bit_numeral_bit1
thf(fact_5696_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ one_one_rat )
    = ( numeral_numeral_rat @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_5697_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ one_one_complex )
    = ( numera6690914467698888265omplex @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_5698_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ one_one_real )
    = ( numeral_numeral_real @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_5699_dbl__inc__simps_I3_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ one_one_int )
    = ( numeral_numeral_int @ ( bit1 @ one ) ) ) ).

% dbl_inc_simps(3)
thf(fact_5700_pred__numeral__simps_I1_J,axiom,
    ( ( pred_numeral @ one )
    = zero_zero_nat ) ).

% pred_numeral_simps(1)
thf(fact_5701_Suc__eq__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( ( suc @ N2 )
        = ( numeral_numeral_nat @ K ) )
      = ( N2
        = ( pred_numeral @ K ) ) ) ).

% Suc_eq_numeral
thf(fact_5702_eq__numeral__Suc,axiom,
    ! [K: num,N2: nat] :
      ( ( ( numeral_numeral_nat @ K )
        = ( suc @ N2 ) )
      = ( ( pred_numeral @ K )
        = N2 ) ) ).

% eq_numeral_Suc
thf(fact_5703_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ zero_zero_complex )
    = one_one_complex ) ).

% dbl_inc_simps(2)
thf(fact_5704_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ zero_zero_real )
    = one_one_real ) ).

% dbl_inc_simps(2)
thf(fact_5705_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ zero_zero_rat )
    = one_one_rat ) ).

% dbl_inc_simps(2)
thf(fact_5706_dbl__inc__simps_I2_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ zero_zero_int )
    = one_one_int ) ).

% dbl_inc_simps(2)
thf(fact_5707_pred__numeral__inc,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( inc @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% pred_numeral_inc
thf(fact_5708_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% dbl_inc_simps(4)
thf(fact_5709_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% dbl_inc_simps(4)
thf(fact_5710_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% dbl_inc_simps(4)
thf(fact_5711_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% dbl_inc_simps(4)
thf(fact_5712_dbl__inc__simps_I4_J,axiom,
    ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% dbl_inc_simps(4)
thf(fact_5713_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_5714_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_5715_dbl__inc__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bit1 @ K ) ) ) ).

% dbl_inc_simps(5)
thf(fact_5716_pred__numeral__simps_I3_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit1 @ K ) )
      = ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ).

% pred_numeral_simps(3)
thf(fact_5717_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_5718_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_5719_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_5720_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_5721_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_5722_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_5723_minus__numeral__div__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% minus_numeral_div_numeral
thf(fact_5724_numeral__div__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( divide_divide_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% numeral_div_minus_numeral
thf(fact_5725_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8295874005876285629c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5726_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5851722552734809277nc_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5727_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu8557863876264182079omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5728_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5831290666863070958nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu7757733837767384882nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5729_dbl__inc__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu5219082963157363817nc_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_nu3179335615603231917ec_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_inc_simps(1)
thf(fact_5730_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6075765906172075777c_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ K ) ) )
      = ( uminus_uminus_real @ ( neg_nu8295874005876285629c_real @ ( numeral_numeral_real @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5731_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus_uminus_int @ ( neg_nu5851722552734809277nc_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5732_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( neg_nu8557863876264182079omplex @ ( numera6690914467698888265omplex @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5733_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu7757733837767384882nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( uminus1351360451143612070nteger @ ( neg_nu5831290666863070958nteger @ ( numera6620942414471956472nteger @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5734_dbl__dec__simps_I1_J,axiom,
    ! [K: num] :
      ( ( neg_nu3179335615603231917ec_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ K ) ) )
      = ( uminus_uminus_rat @ ( neg_nu5219082963157363817nc_rat @ ( numeral_numeral_rat @ K ) ) ) ) ).

% dbl_dec_simps(1)
thf(fact_5735_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ M ) @ ( numeral_numeral_int @ N2 ) )
      = ( unique6319869463603278526ux_int @ ( unique5052692396658037445od_int @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5736_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ M ) @ ( numeral_numeral_nat @ N2 ) )
      = ( unique6322359934112328802ux_nat @ ( unique5055182867167087721od_nat @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5737_dvd__numeral__simp,axiom,
    ! [M: num,N2: num] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ M ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( unique5706413561485394159nteger @ ( unique3479559517661332726nteger @ N2 @ M ) ) ) ).

% dvd_numeral_simp
thf(fact_5738_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_5739_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_5740_divmod__algorithm__code_I2_J,axiom,
    ! [M: num] :
      ( ( unique3479559517661332726nteger @ M @ one )
      = ( produc1086072967326762835nteger @ ( numera6620942414471956472nteger @ M ) @ zero_z3403309356797280102nteger ) ) ).

% divmod_algorithm_code(2)
thf(fact_5741_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_5742_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_5743_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_5744_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_5745_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_5746_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_5747_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_5748_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_5749_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_5750_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_5751_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_real @ ( numeral_numeral_real @ M ) @ ( uminus_uminus_real @ one_one_real ) )
      = ( numeral_numeral_real @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5752_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5753_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_complex @ ( numera6690914467698888265omplex @ M ) @ ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( numera6690914467698888265omplex @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5754_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_8373710615458151222nteger @ ( numera6620942414471956472nteger @ M ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( numera6620942414471956472nteger @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5755_diff__numeral__special_I6_J,axiom,
    ! [M: num] :
      ( ( minus_minus_rat @ ( numeral_numeral_rat @ M ) @ ( uminus_uminus_rat @ one_one_rat ) )
      = ( numeral_numeral_rat @ ( inc @ M ) ) ) ).

% diff_numeral_special(6)
thf(fact_5756_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_real @ ( uminus_uminus_real @ one_one_real ) @ ( numeral_numeral_real @ N2 ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_5757_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_5758_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ ( numera6690914467698888265omplex @ N2 ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_5759_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ ( numera6620942414471956472nteger @ N2 ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_5760_diff__numeral__special_I5_J,axiom,
    ! [N2: num] :
      ( ( minus_minus_rat @ ( uminus_uminus_rat @ one_one_rat ) @ ( numeral_numeral_rat @ N2 ) )
      = ( uminus_uminus_rat @ ( numeral_numeral_rat @ ( inc @ N2 ) ) ) ) ).

% diff_numeral_special(5)
thf(fact_5761_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_5762_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_5763_divmod__algorithm__code_I3_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit0 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(3)
thf(fact_5764_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_5765_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_5766_divmod__algorithm__code_I4_J,axiom,
    ! [N2: num] :
      ( ( unique3479559517661332726nteger @ one @ ( bit1 @ N2 ) )
      = ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ one ) ) ) ).

% divmod_algorithm_code(4)
thf(fact_5767_one__div__minus__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% one_div_minus_numeral
thf(fact_5768_minus__one__div__numeral,axiom,
    ! [N2: num] :
      ( ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( adjust_div @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% minus_one_div_numeral
thf(fact_5769_signed__take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_bit0
thf(fact_5770_signed__take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_ri631733984087533419it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_ri631733984087533419it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% signed_take_bit_numeral_minus_bit0
thf(fact_5771_num__induct,axiom,
    ! [P: num > $o,X4: num] :
      ( ( P @ one )
     => ( ! [X5: num] :
            ( ( P @ X5 )
           => ( P @ ( inc @ X5 ) ) )
       => ( P @ X4 ) ) ) ).

% num_induct
thf(fact_5772_add__inc,axiom,
    ! [X4: num,Y: num] :
      ( ( plus_plus_num @ X4 @ ( inc @ Y ) )
      = ( inc @ ( plus_plus_num @ X4 @ Y ) ) ) ).

% add_inc
thf(fact_5773_numeral__eq__Suc,axiom,
    ( numeral_numeral_nat
    = ( ^ [K3: num] : ( suc @ ( pred_numeral @ K3 ) ) ) ) ).

% numeral_eq_Suc
thf(fact_5774_inc_Osimps_I1_J,axiom,
    ( ( inc @ one )
    = ( bit0 @ one ) ) ).

% inc.simps(1)
thf(fact_5775_inc_Osimps_I2_J,axiom,
    ! [X4: num] :
      ( ( inc @ ( bit0 @ X4 ) )
      = ( bit1 @ X4 ) ) ).

% inc.simps(2)
thf(fact_5776_inc_Osimps_I3_J,axiom,
    ! [X4: num] :
      ( ( inc @ ( bit1 @ X4 ) )
      = ( bit0 @ ( inc @ X4 ) ) ) ).

% inc.simps(3)
thf(fact_5777_add__One,axiom,
    ! [X4: num] :
      ( ( plus_plus_num @ X4 @ one )
      = ( inc @ X4 ) ) ).

% add_One
thf(fact_5778_mult__inc,axiom,
    ! [X4: num,Y: num] :
      ( ( times_times_num @ X4 @ ( inc @ Y ) )
      = ( plus_plus_num @ ( times_times_num @ X4 @ Y ) @ X4 ) ) ).

% mult_inc
thf(fact_5779_pred__numeral__def,axiom,
    ( pred_numeral
    = ( ^ [K3: num] : ( minus_minus_nat @ ( numeral_numeral_nat @ K3 ) @ one_one_nat ) ) ) ).

% pred_numeral_def
thf(fact_5780_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numeral_numeral_rat @ ( inc @ X4 ) )
      = ( plus_plus_rat @ ( numeral_numeral_rat @ X4 ) @ one_one_rat ) ) ).

% numeral_inc
thf(fact_5781_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numera1916890842035813515d_enat @ ( inc @ X4 ) )
      = ( plus_p3455044024723400733d_enat @ ( numera1916890842035813515d_enat @ X4 ) @ one_on7984719198319812577d_enat ) ) ).

% numeral_inc
thf(fact_5782_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numera6690914467698888265omplex @ ( inc @ X4 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ X4 ) @ one_one_complex ) ) ).

% numeral_inc
thf(fact_5783_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numeral_numeral_real @ ( inc @ X4 ) )
      = ( plus_plus_real @ ( numeral_numeral_real @ X4 ) @ one_one_real ) ) ).

% numeral_inc
thf(fact_5784_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numeral_numeral_nat @ ( inc @ X4 ) )
      = ( plus_plus_nat @ ( numeral_numeral_nat @ X4 ) @ one_one_nat ) ) ).

% numeral_inc
thf(fact_5785_numeral__inc,axiom,
    ! [X4: num] :
      ( ( numeral_numeral_int @ ( inc @ X4 ) )
      = ( plus_plus_int @ ( numeral_numeral_int @ X4 ) @ one_one_int ) ) ).

% numeral_inc
thf(fact_5786_dbl__inc__def,axiom,
    ( neg_nu8557863876264182079omplex
    = ( ^ [X: complex] : ( plus_plus_complex @ ( plus_plus_complex @ X @ X ) @ one_one_complex ) ) ) ).

% dbl_inc_def
thf(fact_5787_dbl__inc__def,axiom,
    ( neg_nu8295874005876285629c_real
    = ( ^ [X: real] : ( plus_plus_real @ ( plus_plus_real @ X @ X ) @ one_one_real ) ) ) ).

% dbl_inc_def
thf(fact_5788_dbl__inc__def,axiom,
    ( neg_nu5219082963157363817nc_rat
    = ( ^ [X: rat] : ( plus_plus_rat @ ( plus_plus_rat @ X @ X ) @ one_one_rat ) ) ) ).

% dbl_inc_def
thf(fact_5789_dbl__inc__def,axiom,
    ( neg_nu5851722552734809277nc_int
    = ( ^ [X: int] : ( plus_plus_int @ ( plus_plus_int @ X @ X ) @ one_one_int ) ) ) ).

% dbl_inc_def
thf(fact_5790_divmod__int__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_int_def
thf(fact_5791_divmod__def,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N: num] : ( product_Pair_int_int @ ( divide_divide_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) @ ( modulo_modulo_int @ ( numeral_numeral_int @ M6 ) @ ( numeral_numeral_int @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5792_divmod__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5793_divmod__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M6: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_def
thf(fact_5794_divmod_H__nat__def,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N: num] : ( product_Pair_nat_nat @ ( divide_divide_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) @ ( modulo_modulo_nat @ ( numeral_numeral_nat @ M6 ) @ ( numeral_numeral_nat @ N ) ) ) ) ) ).

% divmod'_nat_def
thf(fact_5795_take__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( plus_plus_int @ ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ one_one_int ) ) ).

% take_bit_numeral_minus_bit1
thf(fact_5796_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ K ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_5797_take__bit__numeral__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) )
      = ( times_times_nat @ ( bit_se2925701944663578781it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ K ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_bit0
thf(fact_5798_take__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( times_times_int @ ( bit_se2923211474154528505it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% take_bit_numeral_minus_bit0
thf(fact_5799_divmod__divmod__step,axiom,
    ( unique5055182867167087721od_nat
    = ( ^ [M6: num,N: num] : ( if_Pro6206227464963214023at_nat @ ( ord_less_num @ M6 @ N ) @ ( product_Pair_nat_nat @ zero_zero_nat @ ( numeral_numeral_nat @ M6 ) ) @ ( unique5026877609467782581ep_nat @ N @ ( unique5055182867167087721od_nat @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5800_divmod__divmod__step,axiom,
    ( unique5052692396658037445od_int
    = ( ^ [M6: num,N: num] : ( if_Pro3027730157355071871nt_int @ ( ord_less_num @ M6 @ N ) @ ( product_Pair_int_int @ zero_zero_int @ ( numeral_numeral_int @ M6 ) ) @ ( unique5024387138958732305ep_int @ N @ ( unique5052692396658037445od_int @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5801_divmod__divmod__step,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M6: num,N: num] : ( if_Pro6119634080678213985nteger @ ( ord_less_num @ M6 @ N ) @ ( produc1086072967326762835nteger @ zero_z3403309356797280102nteger @ ( numera6620942414471956472nteger @ M6 ) ) @ ( unique4921790084139445826nteger @ N @ ( unique3479559517661332726nteger @ M6 @ ( bit0 @ N ) ) ) ) ) ) ).

% divmod_divmod_step
thf(fact_5802_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_5803_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_5804_norm__divide__numeral,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_5805_norm__divide__numeral,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_divide_numeral
thf(fact_5806_norm__mult__numeral2,axiom,
    ! [A: real,W: num] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ A @ ( numeral_numeral_real @ W ) ) )
      = ( times_times_real @ ( real_V7735802525324610683m_real @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_5807_norm__mult__numeral2,axiom,
    ! [A: complex,W: num] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ A @ ( numera6690914467698888265omplex @ W ) ) )
      = ( times_times_real @ ( real_V1022390504157884413omplex @ A ) @ ( numeral_numeral_real @ W ) ) ) ).

% norm_mult_numeral2
thf(fact_5808_norm__mult__numeral1,axiom,
    ! [W: num,A: real] :
      ( ( real_V7735802525324610683m_real @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V7735802525324610683m_real @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_5809_norm__mult__numeral1,axiom,
    ! [W: num,A: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( times_times_real @ ( numeral_numeral_real @ W ) @ ( real_V1022390504157884413omplex @ A ) ) ) ).

% norm_mult_numeral1
thf(fact_5810_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_5811_norm__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_neg_numeral
thf(fact_5812_norm__le__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X4 ) @ zero_zero_real )
      = ( X4 = zero_zero_real ) ) ).

% norm_le_zero_iff
thf(fact_5813_norm__le__zero__iff,axiom,
    ! [X4: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ zero_zero_real )
      = ( X4 = zero_zero_complex ) ) ).

% norm_le_zero_iff
thf(fact_5814_zero__less__norm__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V7735802525324610683m_real @ X4 ) )
      = ( X4 != zero_zero_real ) ) ).

% zero_less_norm_iff
thf(fact_5815_zero__less__norm__iff,axiom,
    ! [X4: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X4 ) )
      = ( X4 != zero_zero_complex ) ) ).

% zero_less_norm_iff
thf(fact_5816_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V7735802525324610683m_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_5817_norm__numeral,axiom,
    ! [W: num] :
      ( ( real_V1022390504157884413omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% norm_numeral
thf(fact_5818_norm__one,axiom,
    ( ( real_V7735802525324610683m_real @ one_one_real )
    = one_one_real ) ).

% norm_one
thf(fact_5819_norm__one,axiom,
    ( ( real_V1022390504157884413omplex @ one_one_complex )
    = one_one_real ) ).

% norm_one
thf(fact_5820_norm__not__less__zero,axiom,
    ! [X4: complex] :
      ~ ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ zero_zero_real ) ).

% norm_not_less_zero
thf(fact_5821_norm__ge__zero,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ zero_zero_real @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% norm_ge_zero
thf(fact_5822_norm__divide,axiom,
    ! [A: real,B: real] :
      ( ( real_V7735802525324610683m_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_divide
thf(fact_5823_norm__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( real_V1022390504157884413omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide_divide_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_divide
thf(fact_5824_norm__power,axiom,
    ! [X4: real,N2: nat] :
      ( ( real_V7735802525324610683m_real @ ( power_power_real @ X4 @ N2 ) )
      = ( power_power_real @ ( real_V7735802525324610683m_real @ X4 ) @ N2 ) ) ).

% norm_power
thf(fact_5825_norm__power,axiom,
    ! [X4: complex,N2: nat] :
      ( ( real_V1022390504157884413omplex @ ( power_power_complex @ X4 @ N2 ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ X4 ) @ N2 ) ) ).

% norm_power
thf(fact_5826_norm__uminus__minus,axiom,
    ! [X4: real,Y: real] :
      ( ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ Y ) )
      = ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_5827_norm__uminus__minus,axiom,
    ! [X4: complex,Y: complex] :
      ( ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ X4 ) @ Y ) )
      = ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y ) ) ) ).

% norm_uminus_minus
thf(fact_5828_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_5829_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_5830_power__eq__imp__eq__norm,axiom,
    ! [W: real,N2: nat,Z: real] :
      ( ( ( power_power_real @ W @ N2 )
        = ( power_power_real @ Z @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V7735802525324610683m_real @ W )
          = ( real_V7735802525324610683m_real @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_5831_power__eq__imp__eq__norm,axiom,
    ! [W: complex,N2: nat,Z: complex] :
      ( ( ( power_power_complex @ W @ N2 )
        = ( power_power_complex @ Z @ N2 ) )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( real_V1022390504157884413omplex @ W )
          = ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% power_eq_imp_eq_norm
thf(fact_5832_norm__mult__less,axiom,
    ! [X4: real,R3: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ R3 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y ) ) @ ( times_times_real @ R3 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_5833_norm__mult__less,axiom,
    ! [X4: complex,R3: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ R3 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y ) ) @ ( times_times_real @ R3 @ S ) ) ) ) ).

% norm_mult_less
thf(fact_5834_norm__mult__ineq,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( times_times_real @ X4 @ Y ) ) @ ( times_times_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_5835_norm__mult__ineq,axiom,
    ! [X4: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( times_times_complex @ X4 @ Y ) ) @ ( times_times_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_mult_ineq
thf(fact_5836_norm__triangle__lt,axiom,
    ! [X4: real,Y: real,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
     => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_5837_norm__triangle__lt,axiom,
    ! [X4: complex,Y: complex,E2: real] :
      ( ( ord_less_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
     => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y ) ) @ E2 ) ) ).

% norm_triangle_lt
thf(fact_5838_norm__add__less,axiom,
    ! [X4: real,R3: real,Y: real,S: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ R3 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Y ) @ S )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y ) ) @ ( plus_plus_real @ R3 @ S ) ) ) ) ).

% norm_add_less
thf(fact_5839_norm__add__less,axiom,
    ! [X4: complex,R3: real,Y: complex,S: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ R3 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Y ) @ S )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y ) ) @ ( plus_plus_real @ R3 @ S ) ) ) ) ).

% norm_add_less
thf(fact_5840_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_5841_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_5842_norm__triangle__le,axiom,
    ! [X4: real,Y: real,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_5843_norm__triangle__le,axiom,
    ! [X4: complex,Y: complex,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y ) ) @ E2 ) ) ).

% norm_triangle_le
thf(fact_5844_norm__triangle__ineq,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ X4 @ Y ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_5845_norm__triangle__ineq,axiom,
    ! [X4: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ X4 @ Y ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y ) ) ) ).

% norm_triangle_ineq
thf(fact_5846_norm__triangle__mono,axiom,
    ! [A: real,R3: real,B: real,S: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ A ) @ R3 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ A @ B ) ) @ ( plus_plus_real @ R3 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_5847_norm__triangle__mono,axiom,
    ! [A: complex,R3: real,B: complex,S: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ A ) @ R3 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ B ) @ S )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ A @ B ) ) @ ( plus_plus_real @ R3 @ S ) ) ) ) ).

% norm_triangle_mono
thf(fact_5848_norm__diff__triangle__less,axiom,
    ! [X4: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_5849_norm__diff__triangle__less,axiom,
    ! [X4: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y ) ) @ E1 )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_less
thf(fact_5850_norm__power__ineq,axiom,
    ! [X4: real,N2: nat] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( power_power_real @ X4 @ N2 ) ) @ ( power_power_real @ ( real_V7735802525324610683m_real @ X4 ) @ N2 ) ) ).

% norm_power_ineq
thf(fact_5851_norm__power__ineq,axiom,
    ! [X4: complex,N2: nat] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( power_power_complex @ X4 @ N2 ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ X4 ) @ N2 ) ) ).

% norm_power_ineq
thf(fact_5852_norm__triangle__sub,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ Y ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_5853_norm__triangle__sub,axiom,
    ! [X4: complex,Y: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Y ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y ) ) ) ) ).

% norm_triangle_sub
thf(fact_5854_norm__triangle__ineq4,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_5855_norm__triangle__ineq4,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) ).

% norm_triangle_ineq4
thf(fact_5856_norm__diff__triangle__le,axiom,
    ! [X4: real,Y: real,E1: real,Z: real,E22: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_5857_norm__diff__triangle__le,axiom,
    ! [X4: complex,Y: complex,E1: real,Z: complex,E22: real] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y ) ) @ E1 )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Y @ Z ) ) @ E22 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Z ) ) @ ( plus_plus_real @ E1 @ E22 ) ) ) ) ).

% norm_diff_triangle_le
thf(fact_5858_norm__triangle__le__diff,axiom,
    ! [X4: real,Y: real,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ X4 @ Y ) ) @ E2 ) ) ).

% norm_triangle_le_diff
thf(fact_5859_norm__triangle__le__diff,axiom,
    ! [X4: complex,Y: complex,E2: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y ) ) @ E2 )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ X4 @ Y ) ) @ E2 ) ) ).

% norm_triangle_le_diff
thf(fact_5860_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_5861_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_5862_norm__triangle__ineq2,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_5863_norm__triangle__ineq2,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq2
thf(fact_5864_power__eq__1__iff,axiom,
    ! [W: real,N2: nat] :
      ( ( ( power_power_real @ W @ N2 )
        = one_one_real )
     => ( ( ( real_V7735802525324610683m_real @ W )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_5865_power__eq__1__iff,axiom,
    ! [W: complex,N2: nat] :
      ( ( ( power_power_complex @ W @ N2 )
        = one_one_complex )
     => ( ( ( real_V1022390504157884413omplex @ W )
          = one_one_real )
        | ( N2 = zero_zero_nat ) ) ) ).

% power_eq_1_iff
thf(fact_5866_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_5867_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_5868_square__norm__one,axiom,
    ! [X4: real] :
      ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
     => ( ( real_V7735802525324610683m_real @ X4 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_5869_square__norm__one,axiom,
    ! [X4: complex] :
      ( ( ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_complex )
     => ( ( real_V1022390504157884413omplex @ X4 )
        = one_one_real ) ) ).

% square_norm_one
thf(fact_5870_norm__power__diff,axiom,
    ! [Z: real,W: real,M: nat] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( power_power_real @ Z @ M ) @ ( power_power_real @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_5871_norm__power__diff,axiom,
    ! [Z: complex,W: complex,M: nat] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ W ) @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( power_power_complex @ Z @ M ) @ ( power_power_complex @ W @ M ) ) ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ Z @ W ) ) ) ) ) ) ).

% norm_power_diff
thf(fact_5872_arcosh__1,axiom,
    ( ( arcosh_real @ one_one_real )
    = zero_zero_real ) ).

% arcosh_1
thf(fact_5873_pochhammer__double,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ Z ) @ ( 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 @ Z @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_5874_pochhammer__double,axiom,
    ! [Z: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) @ ( 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 @ Z @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_5875_pochhammer__double,axiom,
    ! [Z: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) @ ( 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 @ Z @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ N2 ) ) ) ).

% pochhammer_double
thf(fact_5876_ln__one__minus__pos__lower__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( uminus_uminus_real @ X4 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) ) ) ) ).

% ln_one_minus_pos_lower_bound
thf(fact_5877_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_5878_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_5879_ln__one,axiom,
    ( ( ln_ln_real @ one_one_real )
    = zero_zero_real ) ).

% ln_one
thf(fact_5880_ln__inj__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ( ln_ln_real @ X4 )
            = ( ln_ln_real @ Y ) )
          = ( X4 = Y ) ) ) ) ).

% ln_inj_iff
thf(fact_5881_ln__less__cancel__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_real @ X4 @ Y ) ) ) ) ).

% ln_less_cancel_iff
thf(fact_5882_pochhammer__0,axiom,
    ! [A: complex] :
      ( ( comm_s2602460028002588243omplex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% pochhammer_0
thf(fact_5883_pochhammer__0,axiom,
    ! [A: real] :
      ( ( comm_s7457072308508201937r_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% pochhammer_0
thf(fact_5884_pochhammer__0,axiom,
    ! [A: rat] :
      ( ( comm_s4028243227959126397er_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% pochhammer_0
thf(fact_5885_pochhammer__0,axiom,
    ! [A: nat] :
      ( ( comm_s4663373288045622133er_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% pochhammer_0
thf(fact_5886_pochhammer__0,axiom,
    ! [A: int] :
      ( ( comm_s4660882817536571857er_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% pochhammer_0
thf(fact_5887_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu6511756317524482435omplex @ ( numera6690914467698888265omplex @ K ) )
      = ( numera6690914467698888265omplex @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_5888_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu6075765906172075777c_real @ ( numeral_numeral_real @ K ) )
      = ( numeral_numeral_real @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_5889_dbl__dec__simps_I5_J,axiom,
    ! [K: num] :
      ( ( neg_nu3811975205180677377ec_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ ( bitM @ K ) ) ) ).

% dbl_dec_simps(5)
thf(fact_5890_ln__le__cancel__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) )
          = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ).

% ln_le_cancel_iff
thf(fact_5891_ln__less__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
        = ( ord_less_real @ X4 @ one_one_real ) ) ) ).

% ln_less_zero_iff
thf(fact_5892_ln__gt__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
        = ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% ln_gt_zero_iff
thf(fact_5893_ln__eq__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ln_ln_real @ X4 )
          = zero_zero_real )
        = ( X4 = one_one_real ) ) ) ).

% ln_eq_zero_iff
thf(fact_5894_pred__numeral__simps_I2_J,axiom,
    ! [K: num] :
      ( ( pred_numeral @ ( bit0 @ K ) )
      = ( numeral_numeral_nat @ ( bitM @ K ) ) ) ).

% pred_numeral_simps(2)
thf(fact_5895_ln__ge__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
        = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% ln_ge_zero_iff
thf(fact_5896_ln__le__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).

% ln_le_zero_iff
thf(fact_5897_semiring__norm_I26_J,axiom,
    ( ( bitM @ one )
    = one ) ).

% semiring_norm(26)
thf(fact_5898_ln__less__self,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).

% ln_less_self
thf(fact_5899_pochhammer__pos,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_5900_pochhammer__pos,axiom,
    ! [X4: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ord_less_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_5901_pochhammer__pos,axiom,
    ! [X4: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X4 )
     => ( ord_less_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_5902_pochhammer__pos,axiom,
    ! [X4: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ord_less_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N2 ) ) ) ).

% pochhammer_pos
thf(fact_5903_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_5904_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_5905_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_5906_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_5907_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_5908_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_5909_semiring__norm_I27_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit0 @ N2 ) )
      = ( bit1 @ ( bitM @ N2 ) ) ) ).

% semiring_norm(27)
thf(fact_5910_semiring__norm_I28_J,axiom,
    ! [N2: num] :
      ( ( bitM @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ N2 ) ) ) ).

% semiring_norm(28)
thf(fact_5911_inc__BitM__eq,axiom,
    ! [N2: num] :
      ( ( inc @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% inc_BitM_eq
thf(fact_5912_BitM__inc__eq,axiom,
    ! [N2: num] :
      ( ( bitM @ ( inc @ N2 ) )
      = ( bit1 @ N2 ) ) ).

% BitM_inc_eq
thf(fact_5913_ln__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ).

% ln_bound
thf(fact_5914_ln__gt__zero__imp__gt__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% ln_gt_zero_imp_gt_one
thf(fact_5915_ln__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ord_less_real @ ( ln_ln_real @ X4 ) @ zero_zero_real ) ) ) ).

% ln_less_zero
thf(fact_5916_ln__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ord_less_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).

% ln_gt_zero
thf(fact_5917_ln__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) ) ) ).

% ln_ge_zero
thf(fact_5918_pochhammer__nonneg,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( comm_s7457072308508201937r_real @ X4 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_5919_pochhammer__nonneg,axiom,
    ! [X4: rat,N2: nat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( comm_s4028243227959126397er_rat @ X4 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_5920_pochhammer__nonneg,axiom,
    ! [X4: nat,N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ X4 )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( comm_s4663373288045622133er_nat @ X4 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_5921_pochhammer__nonneg,axiom,
    ! [X4: int,N2: nat] :
      ( ( ord_less_int @ zero_zero_int @ X4 )
     => ( ord_less_eq_int @ zero_zero_int @ ( comm_s4660882817536571857er_int @ X4 @ N2 ) ) ) ).

% pochhammer_nonneg
thf(fact_5922_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_5923_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_5924_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_5925_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_5926_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_5927_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_5928_ln__ge__zero__imp__ge__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ln_ln_real @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% ln_ge_zero_imp_ge_one
thf(fact_5929_one__plus__BitM,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ one @ ( bitM @ N2 ) )
      = ( bit0 @ N2 ) ) ).

% one_plus_BitM
thf(fact_5930_BitM__plus__one,axiom,
    ! [N2: num] :
      ( ( plus_plus_num @ ( bitM @ N2 ) @ one )
      = ( bit0 @ N2 ) ) ).

% BitM_plus_one
thf(fact_5931_ln__add__one__self__le__self,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).

% ln_add_one_self_le_self
thf(fact_5932_ln__mult,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( times_times_real @ X4 @ Y ) )
          = ( plus_plus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_mult
thf(fact_5933_ln__eq__minus__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ln_ln_real @ X4 )
          = ( minus_minus_real @ X4 @ one_one_real ) )
       => ( X4 = one_one_real ) ) ) ).

% ln_eq_minus_one
thf(fact_5934_ln__div,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ( ln_ln_real @ ( divide_divide_real @ X4 @ Y ) )
          = ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) ) ) ) ) ).

% ln_div
thf(fact_5935_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_5936_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_5937_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_5938_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_5939_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_5940_pochhammer__rec_H,axiom,
    ! [Z: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) )
      = ( times_times_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_5941_pochhammer__rec_H,axiom,
    ! [Z: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) )
      = ( times_times_complex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ ( comm_s2602460028002588243omplex @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_5942_pochhammer__rec_H,axiom,
    ! [Z: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) )
      = ( times_times_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ ( comm_s7457072308508201937r_real @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_5943_pochhammer__rec_H,axiom,
    ! [Z: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( suc @ N2 ) )
      = ( times_times_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ ( comm_s4660882817536571857er_int @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_5944_pochhammer__rec_H,axiom,
    ! [Z: nat,N2: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( suc @ N2 ) )
      = ( times_times_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) ) ) ).

% pochhammer_rec'
thf(fact_5945_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_5946_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_5947_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_5948_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_5949_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_5950_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_5951_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_5952_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_5953_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_5954_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_5955_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_5956_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_5957_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_5958_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_5959_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_5960_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_5961_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_5962_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_5963_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_5964_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_5965_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_5966_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_5967_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_5968_pochhammer__product_H,axiom,
    ! [Z: rat,N2: nat,M: nat] :
      ( ( comm_s4028243227959126397er_rat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ N2 ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_5969_pochhammer__product_H,axiom,
    ! [Z: complex,N2: nat,M: nat] :
      ( ( comm_s2602460028002588243omplex @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ N2 ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_5970_pochhammer__product_H,axiom,
    ! [Z: real,N2: nat,M: nat] :
      ( ( comm_s7457072308508201937r_real @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ N2 ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_5971_pochhammer__product_H,axiom,
    ! [Z: int,N2: nat,M: nat] :
      ( ( comm_s4660882817536571857er_int @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ N2 ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_5972_pochhammer__product_H,axiom,
    ! [Z: nat,N2: nat,M: nat] :
      ( ( comm_s4663373288045622133er_nat @ Z @ ( plus_plus_nat @ N2 @ M ) )
      = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ N2 ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ N2 ) ) @ M ) ) ) ).

% pochhammer_product'
thf(fact_5973_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_5974_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_5975_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_5976_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_rat @ ( bitM @ N2 ) )
      = ( minus_minus_rat @ ( numeral_numeral_rat @ ( bit0 @ N2 ) ) @ one_one_rat ) ) ).

% numeral_BitM
thf(fact_5977_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bitM @ N2 ) )
      = ( minus_minus_complex @ ( numera6690914467698888265omplex @ ( bit0 @ N2 ) ) @ one_one_complex ) ) ).

% numeral_BitM
thf(fact_5978_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_real @ ( bitM @ N2 ) )
      = ( minus_minus_real @ ( numeral_numeral_real @ ( bit0 @ N2 ) ) @ one_one_real ) ) ).

% numeral_BitM
thf(fact_5979_numeral__BitM,axiom,
    ! [N2: num] :
      ( ( numeral_numeral_int @ ( bitM @ N2 ) )
      = ( minus_minus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) @ one_one_int ) ) ).

% numeral_BitM
thf(fact_5980_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_5981_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_5982_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numera6620942414471956472nteger @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_5983_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_5984_odd__numeral__BitM,axiom,
    ! [W: num] :
      ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_int @ ( bitM @ W ) ) ) ).

% odd_numeral_BitM
thf(fact_5985_ln__le__minus__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ).

% ln_le_minus_one
thf(fact_5986_ln__diff__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( minus_minus_real @ ( ln_ln_real @ X4 ) @ ( ln_ln_real @ Y ) ) @ ( divide_divide_real @ ( minus_minus_real @ X4 @ Y ) @ Y ) ) ) ) ).

% ln_diff_le
thf(fact_5987_ln__add__one__self__le__self2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ord_less_eq_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) ).

% ln_add_one_self_le_self2
thf(fact_5988_ln__realpow,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ln_ln_real @ ( power_power_real @ X4 @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ln_ln_real @ X4 ) ) ) ) ).

% ln_realpow
thf(fact_5989_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4028243227959126397er_rat @ Z @ N2 )
        = ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ M ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( semiri681578069525770553at_rat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_5990_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s2602460028002588243omplex @ Z @ N2 )
        = ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ M ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( semiri8010041392384452111omplex @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_5991_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s7457072308508201937r_real @ Z @ N2 )
        = ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ M ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( semiri5074537144036343181t_real @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_5992_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4660882817536571857er_int @ Z @ N2 )
        = ( times_times_int @ ( comm_s4660882817536571857er_int @ Z @ M ) @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_5993_pochhammer__product,axiom,
    ! [M: nat,N2: nat,Z: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( comm_s4663373288045622133er_nat @ Z @ N2 )
        = ( times_times_nat @ ( comm_s4663373288045622133er_nat @ Z @ M ) @ ( comm_s4663373288045622133er_nat @ ( plus_plus_nat @ Z @ ( semiri1316708129612266289at_nat @ M ) ) @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ).

% pochhammer_product
thf(fact_5994_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_5995_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_5996_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_5997_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_5998_ln__one__minus__pos__upper__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( ln_ln_real @ ( minus_minus_real @ one_one_real @ X4 ) ) @ ( uminus_uminus_real @ X4 ) ) ) ) ).

% ln_one_minus_pos_upper_bound
thf(fact_5999_pochhammer__absorb__comp,axiom,
    ! [R3: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ R3 @ ( semiri8010041392384452111omplex @ K ) ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ R3 ) @ K ) )
      = ( times_times_complex @ R3 @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ ( uminus1482373934393186551omplex @ R3 ) @ one_one_complex ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6000_pochhammer__absorb__comp,axiom,
    ! [R3: code_integer,K: nat] :
      ( ( times_3573771949741848930nteger @ ( minus_8373710615458151222nteger @ R3 @ ( semiri4939895301339042750nteger @ K ) ) @ ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ R3 ) @ K ) )
      = ( times_3573771949741848930nteger @ R3 @ ( comm_s8582702949713902594nteger @ ( plus_p5714425477246183910nteger @ ( uminus1351360451143612070nteger @ R3 ) @ one_one_Code_integer ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6001_pochhammer__absorb__comp,axiom,
    ! [R3: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ R3 @ ( semiri681578069525770553at_rat @ K ) ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ R3 ) @ K ) )
      = ( times_times_rat @ R3 @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ R3 ) @ one_one_rat ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6002_pochhammer__absorb__comp,axiom,
    ! [R3: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ R3 @ ( semiri5074537144036343181t_real @ K ) ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ R3 ) @ K ) )
      = ( times_times_real @ R3 @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ ( uminus_uminus_real @ R3 ) @ one_one_real ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6003_pochhammer__absorb__comp,axiom,
    ! [R3: int,K: nat] :
      ( ( times_times_int @ ( minus_minus_int @ R3 @ ( semiri1314217659103216013at_int @ K ) ) @ ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ R3 ) @ K ) )
      = ( times_times_int @ R3 @ ( comm_s4660882817536571857er_int @ ( plus_plus_int @ ( uminus_uminus_int @ R3 ) @ one_one_int ) @ K ) ) ) ).

% pochhammer_absorb_comp
thf(fact_6004_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_6005_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_6006_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_6007_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_6008_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_6009_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_6010_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_6011_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_6012_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_6013_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_6014_ln__one__plus__pos__lower__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( minus_minus_real @ X4 @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) ) ) ) ).

% ln_one_plus_pos_lower_bound
thf(fact_6015_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_6016_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_6017_choose__two,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( divide_divide_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% choose_two
thf(fact_6018_binomial__n__0,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ zero_zero_nat )
      = one_one_nat ) ).

% binomial_n_0
thf(fact_6019_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_6020_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_6021_binomial__Suc__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ ( suc @ N2 ) @ N2 )
      = ( suc @ N2 ) ) ).

% binomial_Suc_n
thf(fact_6022_binomial__n__n,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ N2 )
      = one_one_nat ) ).

% binomial_n_n
thf(fact_6023_binomial__0__Suc,axiom,
    ! [K: nat] :
      ( ( binomial @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% binomial_0_Suc
thf(fact_6024_binomial__1,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ ( suc @ zero_zero_nat ) )
      = N2 ) ).

% binomial_1
thf(fact_6025_choose__one,axiom,
    ! [N2: nat] :
      ( ( binomial @ N2 @ one_one_nat )
      = N2 ) ).

% choose_one
thf(fact_6026_binomial__eq__0,axiom,
    ! [N2: nat,K: nat] :
      ( ( ord_less_nat @ N2 @ K )
     => ( ( binomial @ N2 @ K )
        = zero_zero_nat ) ) ).

% binomial_eq_0
thf(fact_6027_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_6028_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_6029_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_6030_choose__mult__lemma,axiom,
    ! [M: nat,R3: nat,K: nat] :
      ( ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ ( plus_plus_nat @ M @ K ) ) @ ( binomial @ ( plus_plus_nat @ M @ K ) @ K ) )
      = ( times_times_nat @ ( binomial @ ( plus_plus_nat @ ( plus_plus_nat @ M @ R3 ) @ K ) @ K ) @ ( binomial @ ( plus_plus_nat @ M @ R3 ) @ M ) ) ) ).

% choose_mult_lemma
thf(fact_6031_binomial__le__pow,axiom,
    ! [R3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R3 @ N2 )
     => ( ord_less_eq_nat @ ( binomial @ N2 @ R3 ) @ ( power_power_nat @ N2 @ R3 ) ) ) ).

% binomial_le_pow
thf(fact_6032_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_6033_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_6034_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_6035_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_6036_binomial__absorb__comp,axiom,
    ! [N2: nat,K: nat] :
      ( ( times_times_nat @ ( minus_minus_nat @ N2 @ K ) @ ( binomial @ N2 @ K ) )
      = ( times_times_nat @ N2 @ ( binomial @ ( minus_minus_nat @ N2 @ one_one_nat ) @ K ) ) ) ).

% binomial_absorb_comp
thf(fact_6037_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_6038_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_6039_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_6040_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_6041_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_6042_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_6043_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_6044_abs__ln__one__plus__x__minus__x__bound__nonpos,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonpos
thf(fact_6045_tanh__ln__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( tanh_real @ ( ln_ln_real @ X4 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% tanh_ln_real
thf(fact_6046_abs__ln__one__plus__x__minus__x__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound
thf(fact_6047_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_6048_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri773545260158071498ct_rat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% fact_double
thf(fact_6049_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_double
thf(fact_6050_fact__double,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
      = ( times_times_real @ ( times_times_real @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_double
thf(fact_6051_abs__ln__one__plus__x__minus__x__bound__nonneg,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ln_ln_real @ ( plus_plus_real @ one_one_real @ X4 ) ) @ X4 ) ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% abs_ln_one_plus_x_minus_x_bound_nonneg
thf(fact_6052_abs__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_abs
thf(fact_6053_abs__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_abs
thf(fact_6054_abs__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_abs
thf(fact_6055_abs__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_abs
thf(fact_6056_abs__idempotent,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( abs_abs_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_idempotent
thf(fact_6057_abs__idempotent,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( abs_abs_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_idempotent
thf(fact_6058_abs__idempotent,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_idempotent
thf(fact_6059_abs__idempotent,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( abs_abs_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_idempotent
thf(fact_6060_abs__0,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_0
thf(fact_6061_abs__0,axiom,
    ( ( abs_abs_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% abs_0
thf(fact_6062_abs__0,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_0
thf(fact_6063_abs__0,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_0
thf(fact_6064_abs__0,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_0
thf(fact_6065_abs__0__eq,axiom,
    ! [A: code_integer] :
      ( ( zero_z3403309356797280102nteger
        = ( abs_abs_Code_integer @ A ) )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_0_eq
thf(fact_6066_abs__0__eq,axiom,
    ! [A: real] :
      ( ( zero_zero_real
        = ( abs_abs_real @ A ) )
      = ( A = zero_zero_real ) ) ).

% abs_0_eq
thf(fact_6067_abs__0__eq,axiom,
    ! [A: rat] :
      ( ( zero_zero_rat
        = ( abs_abs_rat @ A ) )
      = ( A = zero_zero_rat ) ) ).

% abs_0_eq
thf(fact_6068_abs__0__eq,axiom,
    ! [A: int] :
      ( ( zero_zero_int
        = ( abs_abs_int @ A ) )
      = ( A = zero_zero_int ) ) ).

% abs_0_eq
thf(fact_6069_abs__eq__0,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0
thf(fact_6070_abs__eq__0,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0
thf(fact_6071_abs__eq__0,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0
thf(fact_6072_abs__eq__0,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0
thf(fact_6073_abs__zero,axiom,
    ( ( abs_abs_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% abs_zero
thf(fact_6074_abs__zero,axiom,
    ( ( abs_abs_real @ zero_zero_real )
    = zero_zero_real ) ).

% abs_zero
thf(fact_6075_abs__zero,axiom,
    ( ( abs_abs_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% abs_zero
thf(fact_6076_abs__zero,axiom,
    ( ( abs_abs_int @ zero_zero_int )
    = zero_zero_int ) ).

% abs_zero
thf(fact_6077_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( numera6620942414471956472nteger @ N2 ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_numeral
thf(fact_6078_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_rat @ ( numeral_numeral_rat @ N2 ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% abs_numeral
thf(fact_6079_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% abs_numeral
thf(fact_6080_abs__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_int @ ( numeral_numeral_int @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% abs_numeral
thf(fact_6081_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_6082_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_6083_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_6084_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_6085_abs__1,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_1
thf(fact_6086_abs__1,axiom,
    ( ( abs_abs_complex @ one_one_complex )
    = one_one_complex ) ).

% abs_1
thf(fact_6087_abs__1,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_1
thf(fact_6088_abs__1,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_1
thf(fact_6089_abs__1,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_1
thf(fact_6090_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_6091_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_6092_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_6093_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_6094_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_6095_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_6096_abs__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( abs_abs_complex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( abs_abs_complex @ A ) @ ( abs_abs_complex @ B ) ) ) ).

% abs_divide
thf(fact_6097_abs__minus,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus
thf(fact_6098_abs__minus,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus
thf(fact_6099_abs__minus,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( abs_abs_complex @ A ) ) ).

% abs_minus
thf(fact_6100_abs__minus,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus
thf(fact_6101_abs__minus,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus
thf(fact_6102_abs__minus__cancel,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ A ) )
      = ( abs_abs_real @ A ) ) ).

% abs_minus_cancel
thf(fact_6103_abs__minus__cancel,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ A ) )
      = ( abs_abs_int @ A ) ) ).

% abs_minus_cancel
thf(fact_6104_abs__minus__cancel,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( abs_abs_Code_integer @ A ) ) ).

% abs_minus_cancel
thf(fact_6105_abs__minus__cancel,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ A ) )
      = ( abs_abs_rat @ A ) ) ).

% abs_minus_cancel
thf(fact_6106_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_6107_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_6108_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_6109_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_6110_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_6111_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_6112_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_6113_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_6114_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( semiri4939895301339042750nteger @ N2 ) ) ).

% abs_of_nat
thf(fact_6115_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( semiri681578069525770553at_rat @ N2 ) ) ).

% abs_of_nat
thf(fact_6116_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( semiri5074537144036343181t_real @ N2 ) ) ).

% abs_of_nat
thf(fact_6117_abs__of__nat,axiom,
    ! [N2: nat] :
      ( ( abs_abs_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( semiri1314217659103216013at_int @ N2 ) ) ).

% abs_of_nat
thf(fact_6118_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_real @ ( zero_n3304061248610475627l_real @ P ) )
      = ( zero_n3304061248610475627l_real @ P ) ) ).

% abs_bool_eq
thf(fact_6119_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_rat @ ( zero_n2052037380579107095ol_rat @ P ) )
      = ( zero_n2052037380579107095ol_rat @ P ) ) ).

% abs_bool_eq
thf(fact_6120_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_int @ ( zero_n2684676970156552555ol_int @ P ) )
      = ( zero_n2684676970156552555ol_int @ P ) ) ).

% abs_bool_eq
thf(fact_6121_abs__bool__eq,axiom,
    ! [P: $o] :
      ( ( abs_abs_Code_integer @ ( zero_n356916108424825756nteger @ P ) )
      = ( zero_n356916108424825756nteger @ P ) ) ).

% abs_bool_eq
thf(fact_6122_tanh__real__less__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y ) )
      = ( ord_less_real @ X4 @ Y ) ) ).

% tanh_real_less_iff
thf(fact_6123_tanh__real__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% tanh_real_le_iff
thf(fact_6124_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_6125_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_6126_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_6127_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_6128_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_6129_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_6130_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_6131_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_6132_abs__of__nonneg,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6133_abs__of__nonneg,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6134_abs__of__nonneg,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6135_abs__of__nonneg,axiom,
    ! [A: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_nonneg
thf(fact_6136_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_6137_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_6138_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_6139_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_6140_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( numeral_numeral_real @ N2 ) ) ).

% abs_neg_numeral
thf(fact_6141_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% abs_neg_numeral
thf(fact_6142_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ N2 ) ) ).

% abs_neg_numeral
thf(fact_6143_abs__neg__numeral,axiom,
    ! [N2: num] :
      ( ( abs_abs_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( numeral_numeral_rat @ N2 ) ) ).

% abs_neg_numeral
thf(fact_6144_abs__neg__one,axiom,
    ( ( abs_abs_real @ ( uminus_uminus_real @ one_one_real ) )
    = one_one_real ) ).

% abs_neg_one
thf(fact_6145_abs__neg__one,axiom,
    ( ( abs_abs_int @ ( uminus_uminus_int @ one_one_int ) )
    = one_one_int ) ).

% abs_neg_one
thf(fact_6146_abs__neg__one,axiom,
    ( ( abs_abs_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = one_one_Code_integer ) ).

% abs_neg_one
thf(fact_6147_abs__neg__one,axiom,
    ( ( abs_abs_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = one_one_rat ) ).

% abs_neg_one
thf(fact_6148_abs__power__minus,axiom,
    ! [A: real,N2: nat] :
      ( ( abs_abs_real @ ( power_power_real @ ( uminus_uminus_real @ A ) @ N2 ) )
      = ( abs_abs_real @ ( power_power_real @ A @ N2 ) ) ) ).

% abs_power_minus
thf(fact_6149_abs__power__minus,axiom,
    ! [A: int,N2: nat] :
      ( ( abs_abs_int @ ( power_power_int @ ( uminus_uminus_int @ A ) @ N2 ) )
      = ( abs_abs_int @ ( power_power_int @ A @ N2 ) ) ) ).

% abs_power_minus
thf(fact_6150_abs__power__minus,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ A ) @ N2 ) )
      = ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% abs_power_minus
thf(fact_6151_abs__power__minus,axiom,
    ! [A: rat,N2: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ ( uminus_uminus_rat @ A ) @ N2 ) )
      = ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) ) ) ).

% abs_power_minus
thf(fact_6152_fact__0,axiom,
    ( ( semiri5044797733671781792omplex @ zero_zero_nat )
    = one_one_complex ) ).

% fact_0
thf(fact_6153_fact__0,axiom,
    ( ( semiri773545260158071498ct_rat @ zero_zero_nat )
    = one_one_rat ) ).

% fact_0
thf(fact_6154_fact__0,axiom,
    ( ( semiri1406184849735516958ct_int @ zero_zero_nat )
    = one_one_int ) ).

% fact_0
thf(fact_6155_fact__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ zero_zero_nat )
    = one_one_nat ) ).

% fact_0
thf(fact_6156_fact__0,axiom,
    ( ( semiri2265585572941072030t_real @ zero_zero_nat )
    = one_one_real ) ).

% fact_0
thf(fact_6157_fact__1,axiom,
    ( ( semiri5044797733671781792omplex @ one_one_nat )
    = one_one_complex ) ).

% fact_1
thf(fact_6158_fact__1,axiom,
    ( ( semiri773545260158071498ct_rat @ one_one_nat )
    = one_one_rat ) ).

% fact_1
thf(fact_6159_fact__1,axiom,
    ( ( semiri1406184849735516958ct_int @ one_one_nat )
    = one_one_int ) ).

% fact_1
thf(fact_6160_fact__1,axiom,
    ( ( semiri1408675320244567234ct_nat @ one_one_nat )
    = one_one_nat ) ).

% fact_1
thf(fact_6161_fact__1,axiom,
    ( ( semiri2265585572941072030t_real @ one_one_nat )
    = one_one_real ) ).

% fact_1
thf(fact_6162_tanh__real__neg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( tanh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% tanh_real_neg_iff
thf(fact_6163_tanh__real__pos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( tanh_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% tanh_real_pos_iff
thf(fact_6164_tanh__real__nonneg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( tanh_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% tanh_real_nonneg_iff
thf(fact_6165_tanh__real__nonpos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( tanh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% tanh_real_nonpos_iff
thf(fact_6166_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_6167_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_6168_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_6169_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_6170_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_6171_abs__of__nonpos,axiom,
    ! [A: code_integer] :
      ( ( ord_le3102999989581377725nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_nonpos
thf(fact_6172_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_6173_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_6174_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_6175_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_6176_fact__Suc__0,axiom,
    ( ( semiri5044797733671781792omplex @ ( suc @ zero_zero_nat ) )
    = one_one_complex ) ).

% fact_Suc_0
thf(fact_6177_fact__Suc__0,axiom,
    ( ( semiri773545260158071498ct_rat @ ( suc @ zero_zero_nat ) )
    = one_one_rat ) ).

% fact_Suc_0
thf(fact_6178_fact__Suc__0,axiom,
    ( ( semiri1406184849735516958ct_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% fact_Suc_0
thf(fact_6179_fact__Suc__0,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% fact_Suc_0
thf(fact_6180_fact__Suc__0,axiom,
    ( ( semiri2265585572941072030t_real @ ( suc @ zero_zero_nat ) )
    = one_one_real ) ).

% fact_Suc_0
thf(fact_6181_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_6182_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_6183_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri5044797733671781792omplex @ ( suc @ N2 ) )
      = ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N2 ) ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% fact_Suc
thf(fact_6184_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1406184849735516958ct_int @ ( suc @ N2 ) )
      = ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N2 ) ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% fact_Suc
thf(fact_6185_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri1408675320244567234ct_nat @ ( suc @ N2 ) )
      = ( times_times_nat @ ( semiri1316708129612266289at_nat @ ( suc @ N2 ) ) @ ( semiri1408675320244567234ct_nat @ N2 ) ) ) ).

% fact_Suc
thf(fact_6186_fact__Suc,axiom,
    ! [N2: nat] :
      ( ( semiri2265585572941072030t_real @ ( suc @ N2 ) )
      = ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% fact_Suc
thf(fact_6187_artanh__minus__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( artanh_real @ ( uminus_uminus_real @ X4 ) )
        = ( uminus_uminus_real @ ( artanh_real @ X4 ) ) ) ) ).

% artanh_minus_real
thf(fact_6188_signed__take__bit__nonnegative__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_ri631733984087533419it_int @ N2 @ K ) )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% signed_take_bit_nonnegative_iff
thf(fact_6189_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_6190_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_6191_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_6192_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_6193_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_6194_power2__abs,axiom,
    ! [A: code_integer] :
      ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_6195_power2__abs,axiom,
    ! [A: rat] :
      ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_6196_power2__abs,axiom,
    ! [A: real] :
      ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_6197_power2__abs,axiom,
    ! [A: int] :
      ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% power2_abs
thf(fact_6198_abs__power2,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_6199_abs__power2,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_rat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_6200_abs__power2,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_6201_abs__power2,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_int @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% abs_power2
thf(fact_6202_fact__2,axiom,
    ( ( semiri4449623510593786356d_enat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6203_fact__2,axiom,
    ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6204_fact__2,axiom,
    ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_int @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6205_fact__2,axiom,
    ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6206_fact__2,axiom,
    ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% fact_2
thf(fact_6207_bit__numeral__simps_I2_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ).

% bit_numeral_simps(2)
thf(fact_6208_bit__numeral__simps_I2_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit0 @ W ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N2 ) ) ) ).

% bit_numeral_simps(2)
thf(fact_6209_bit__minus__numeral__Bit0__Suc__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( suc @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ N2 ) ) ).

% bit_minus_numeral_Bit0_Suc_iff
thf(fact_6210_bit__numeral__simps_I3_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ).

% bit_numeral_simps(3)
thf(fact_6211_bit__numeral__simps_I3_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ ( bit1 @ W ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ W ) @ ( pred_numeral @ N2 ) ) ) ).

% bit_numeral_simps(3)
thf(fact_6212_bit__minus__numeral__Bit1__Suc__iff,axiom,
    ! [W: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( suc @ N2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ N2 ) ) ) ).

% bit_minus_numeral_Bit1_Suc_iff
thf(fact_6213_bit__0,axiom,
    ! [A: code_integer] :
      ( ( bit_se9216721137139052372nteger @ A @ zero_zero_nat )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% bit_0
thf(fact_6214_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_6215_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_6216_power__even__abs__numeral,axiom,
    ! [W: num,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_8256067586552552935nteger @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_6217_power__even__abs__numeral,axiom,
    ! [W: num,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_rat @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_6218_power__even__abs__numeral,axiom,
    ! [W: num,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_real @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_6219_power__even__abs__numeral,axiom,
    ! [W: num,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ W ) )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ ( numeral_numeral_nat @ W ) )
        = ( power_power_int @ A @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_even_abs_numeral
thf(fact_6220_bit__minus__numeral__int_I1_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ W ) ) @ ( pred_numeral @ N2 ) ) ) ).

% bit_minus_numeral_int(1)
thf(fact_6221_bit__minus__numeral__int_I2_J,axiom,
    ! [W: num,N2: num] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ W ) ) ) @ ( numeral_numeral_nat @ N2 ) )
      = ( ~ ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ W ) @ ( pred_numeral @ N2 ) ) ) ) ).

% bit_minus_numeral_int(2)
thf(fact_6222_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_6223_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_6224_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_6225_bit__numeral__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( numeral_numeral_int @ M ) @ N2 )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% bit_numeral_iff
thf(fact_6226_bit__numeral__iff,axiom,
    ! [M: num,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 )
      = ( bit_se1148574629649215175it_nat @ ( numeral_numeral_nat @ M ) @ N2 ) ) ).

% bit_numeral_iff
thf(fact_6227_bit__of__nat__iff__bit,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( semiri1314217659103216013at_int @ M ) @ N2 )
      = ( bit_se1148574629649215175it_nat @ M @ N2 ) ) ).

% bit_of_nat_iff_bit
thf(fact_6228_bit__of__nat__iff__bit,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( semiri1316708129612266289at_nat @ M ) @ N2 )
      = ( bit_se1148574629649215175it_nat @ M @ N2 ) ) ).

% bit_of_nat_iff_bit
thf(fact_6229_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_6230_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_6231_abs__ge__self,axiom,
    ! [A: real] : ( ord_less_eq_real @ A @ ( abs_abs_real @ A ) ) ).

% abs_ge_self
thf(fact_6232_abs__ge__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ A @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_self
thf(fact_6233_abs__ge__self,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ A @ ( abs_abs_rat @ A ) ) ).

% abs_ge_self
thf(fact_6234_abs__ge__self,axiom,
    ! [A: int] : ( ord_less_eq_int @ A @ ( abs_abs_int @ A ) ) ).

% abs_ge_self
thf(fact_6235_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_6236_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_6237_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_6238_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_6239_abs__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( abs_abs_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% abs_eq_0_iff
thf(fact_6240_abs__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( abs_abs_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% abs_eq_0_iff
thf(fact_6241_abs__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( abs_abs_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% abs_eq_0_iff
thf(fact_6242_abs__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( abs_abs_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% abs_eq_0_iff
thf(fact_6243_abs__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( abs_abs_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% abs_eq_0_iff
thf(fact_6244_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_6245_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_6246_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_6247_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_6248_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_6249_abs__one,axiom,
    ( ( abs_abs_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% abs_one
thf(fact_6250_abs__one,axiom,
    ( ( abs_abs_real @ one_one_real )
    = one_one_real ) ).

% abs_one
thf(fact_6251_abs__one,axiom,
    ( ( abs_abs_rat @ one_one_rat )
    = one_one_rat ) ).

% abs_one
thf(fact_6252_abs__one,axiom,
    ( ( abs_abs_int @ one_one_int )
    = one_one_int ) ).

% abs_one
thf(fact_6253_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_6254_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_6255_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_6256_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_6257_abs__eq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( abs_abs_real @ X4 )
        = ( abs_abs_real @ Y ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus_uminus_real @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6258_abs__eq__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ( abs_abs_int @ X4 )
        = ( abs_abs_int @ Y ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus_uminus_int @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6259_abs__eq__iff,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( ( abs_abs_Code_integer @ X4 )
        = ( abs_abs_Code_integer @ Y ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus1351360451143612070nteger @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6260_abs__eq__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( abs_abs_rat @ X4 )
        = ( abs_abs_rat @ Y ) )
      = ( ( X4 = Y )
        | ( X4
          = ( uminus_uminus_rat @ Y ) ) ) ) ).

% abs_eq_iff
thf(fact_6261_fact__ge__self,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_self
thf(fact_6262_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_6263_power__abs,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( abs_abs_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) )
      = ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 ) ) ).

% power_abs
thf(fact_6264_power__abs,axiom,
    ! [A: rat,N2: nat] :
      ( ( abs_abs_rat @ ( power_power_rat @ A @ N2 ) )
      = ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 ) ) ).

% power_abs
thf(fact_6265_power__abs,axiom,
    ! [A: real,N2: nat] :
      ( ( abs_abs_real @ ( power_power_real @ A @ N2 ) )
      = ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) ) ).

% power_abs
thf(fact_6266_power__abs,axiom,
    ! [A: int,N2: nat] :
      ( ( abs_abs_int @ ( power_power_int @ A @ N2 ) )
      = ( power_power_int @ ( abs_abs_int @ A ) @ N2 ) ) ).

% power_abs
thf(fact_6267_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_6268_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_6269_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_6270_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_6271_bit__xor__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se6528837805403552850or_nat @ A @ B ) @ N2 )
      = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
       != ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ).

% bit_xor_iff
thf(fact_6272_bit__xor__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ A @ B ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ A @ N2 )
       != ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ).

% bit_xor_iff
thf(fact_6273_bit__unset__bit__iff,axiom,
    ! [M: nat,A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se4203085406695923979it_int @ M @ A ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ A @ N2 )
        & ( M != N2 ) ) ) ).

% bit_unset_bit_iff
thf(fact_6274_bit__unset__bit__iff,axiom,
    ! [M: nat,A: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se4205575877204974255it_nat @ M @ A ) @ N2 )
      = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
        & ( M != N2 ) ) ) ).

% bit_unset_bit_iff
thf(fact_6275_bit__xor__int__iff,axiom,
    ! [K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se6526347334894502574or_int @ K @ L ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
       != ( bit_se1146084159140164899it_int @ L @ N2 ) ) ) ).

% bit_xor_int_iff
thf(fact_6276_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_6277_not__bit__1__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( suc @ N2 ) ) ).

% not_bit_1_Suc
thf(fact_6278_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1146084159140164899it_int @ one_one_int @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_6279_bit__1__iff,axiom,
    ! [N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ one_one_nat @ N2 )
      = ( N2 = zero_zero_nat ) ) ).

% bit_1_iff
thf(fact_6280_abs__ge__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_zero
thf(fact_6281_abs__ge__zero,axiom,
    ! [A: real] : ( ord_less_eq_real @ zero_zero_real @ ( abs_abs_real @ A ) ) ).

% abs_ge_zero
thf(fact_6282_abs__ge__zero,axiom,
    ! [A: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( abs_abs_rat @ A ) ) ).

% abs_ge_zero
thf(fact_6283_abs__ge__zero,axiom,
    ! [A: int] : ( ord_less_eq_int @ zero_zero_int @ ( abs_abs_int @ A ) ) ).

% abs_ge_zero
thf(fact_6284_bit__numeral__simps_I1_J,axiom,
    ! [N2: num] :
      ~ ( bit_se1146084159140164899it_int @ one_one_int @ ( numeral_numeral_nat @ N2 ) ) ).

% bit_numeral_simps(1)
thf(fact_6285_bit__numeral__simps_I1_J,axiom,
    ! [N2: num] :
      ~ ( bit_se1148574629649215175it_nat @ one_one_nat @ ( numeral_numeral_nat @ N2 ) ) ).

% bit_numeral_simps(1)
thf(fact_6286_abs__of__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( abs_abs_Code_integer @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6287_abs__of__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( abs_abs_real @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6288_abs__of__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( abs_abs_rat @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6289_abs__of__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( abs_abs_int @ A )
        = A ) ) ).

% abs_of_pos
thf(fact_6290_abs__not__less__zero,axiom,
    ! [A: code_integer] :
      ~ ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ A ) @ zero_z3403309356797280102nteger ) ).

% abs_not_less_zero
thf(fact_6291_abs__not__less__zero,axiom,
    ! [A: real] :
      ~ ( ord_less_real @ ( abs_abs_real @ A ) @ zero_zero_real ) ).

% abs_not_less_zero
thf(fact_6292_abs__not__less__zero,axiom,
    ! [A: rat] :
      ~ ( ord_less_rat @ ( abs_abs_rat @ A ) @ zero_zero_rat ) ).

% abs_not_less_zero
thf(fact_6293_abs__not__less__zero,axiom,
    ! [A: int] :
      ~ ( ord_less_int @ ( abs_abs_int @ A ) @ zero_zero_int ) ).

% abs_not_less_zero
thf(fact_6294_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_6295_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_6296_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_6297_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_6298_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_6299_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_6300_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_6301_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_6302_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_6303_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_6304_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_6305_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_6306_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_6307_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_6308_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_6309_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_6310_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_6311_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_6312_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_6313_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_6314_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_6315_abs__ge__minus__self,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ A ) @ ( abs_abs_Code_integer @ A ) ) ).

% abs_ge_minus_self
thf(fact_6316_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_6317_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_6318_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_6319_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_6320_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_6321_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_6322_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_6323_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_6324_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_6325_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_6326_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_6327_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_6328_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_6329_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_6330_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_6331_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_6332_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_6333_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_6334_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_6335_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_6336_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_6337_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_6338_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_6339_bit__of__bool__iff,axiom,
    ! [B: $o,N2: nat] :
      ( ( bit_se9216721137139052372nteger @ ( zero_n356916108424825756nteger @ B ) @ N2 )
      = ( B
        & ( N2 = zero_zero_nat ) ) ) ).

% bit_of_bool_iff
thf(fact_6340_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_6341_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_6342_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_zero
thf(fact_6343_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_zero
thf(fact_6344_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_zero
thf(fact_6345_fact__ge__zero,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_zero
thf(fact_6346_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_rat @ ( semiri773545260158071498ct_rat @ N2 ) @ zero_zero_rat ) ).

% fact_not_neg
thf(fact_6347_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( semiri1406184849735516958ct_int @ N2 ) @ zero_zero_int ) ).

% fact_not_neg
thf(fact_6348_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ zero_zero_nat ) ).

% fact_not_neg
thf(fact_6349_fact__not__neg,axiom,
    ! [N2: nat] :
      ~ ( ord_less_real @ ( semiri2265585572941072030t_real @ N2 ) @ zero_zero_real ) ).

% fact_not_neg
thf(fact_6350_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_rat @ zero_zero_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_gt_zero
thf(fact_6351_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_int @ zero_zero_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_gt_zero
thf(fact_6352_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_nat @ zero_zero_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_gt_zero
thf(fact_6353_fact__gt__zero,axiom,
    ! [N2: nat] : ( ord_less_real @ zero_zero_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_gt_zero
thf(fact_6354_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_rat @ one_one_rat @ ( semiri773545260158071498ct_rat @ N2 ) ) ).

% fact_ge_1
thf(fact_6355_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ one_one_int @ ( semiri1406184849735516958ct_int @ N2 ) ) ).

% fact_ge_1
thf(fact_6356_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ one_one_nat @ ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% fact_ge_1
thf(fact_6357_fact__ge__1,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ one_one_real @ ( semiri2265585572941072030t_real @ N2 ) ) ).

% fact_ge_1
thf(fact_6358_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_6359_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_6360_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_6361_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_6362_signed__take__bit__eq__if__positive,axiom,
    ! [A: int,N2: nat] :
      ( ~ ( bit_se1146084159140164899it_int @ A @ N2 )
     => ( ( bit_ri631733984087533419it_int @ N2 @ A )
        = ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ).

% signed_take_bit_eq_if_positive
thf(fact_6363_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_6364_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_6365_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_6366_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_6367_pochhammer__fact,axiom,
    ( semiri5044797733671781792omplex
    = ( comm_s2602460028002588243omplex @ one_one_complex ) ) ).

% pochhammer_fact
thf(fact_6368_pochhammer__fact,axiom,
    ( semiri773545260158071498ct_rat
    = ( comm_s4028243227959126397er_rat @ one_one_rat ) ) ).

% pochhammer_fact
thf(fact_6369_pochhammer__fact,axiom,
    ( semiri1406184849735516958ct_int
    = ( comm_s4660882817536571857er_int @ one_one_int ) ) ).

% pochhammer_fact
thf(fact_6370_pochhammer__fact,axiom,
    ( semiri1408675320244567234ct_nat
    = ( comm_s4663373288045622133er_nat @ one_one_nat ) ) ).

% pochhammer_fact
thf(fact_6371_pochhammer__fact,axiom,
    ( semiri2265585572941072030t_real
    = ( comm_s7457072308508201937r_real @ one_one_real ) ) ).

% pochhammer_fact
thf(fact_6372_tanh__real__lt__1,axiom,
    ! [X4: real] : ( ord_less_real @ ( tanh_real @ X4 ) @ one_one_real ) ).

% tanh_real_lt_1
thf(fact_6373_dense__eq0__I,axiom,
    ! [X4: real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ E ) )
     => ( X4 = zero_zero_real ) ) ).

% dense_eq0_I
thf(fact_6374_dense__eq0__I,axiom,
    ! [X4: rat] :
      ( ! [E: rat] :
          ( ( ord_less_rat @ zero_zero_rat @ E )
         => ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ E ) )
     => ( X4 = zero_zero_rat ) ) ).

% dense_eq0_I
thf(fact_6375_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_6376_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_6377_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_6378_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_6379_abs__mult__pos,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X4 )
     => ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ Y ) @ X4 )
        = ( abs_abs_Code_integer @ ( times_3573771949741848930nteger @ Y @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_6380_abs__mult__pos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( times_times_real @ ( abs_abs_real @ Y ) @ X4 )
        = ( abs_abs_real @ ( times_times_real @ Y @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_6381_abs__mult__pos,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
     => ( ( times_times_rat @ ( abs_abs_rat @ Y ) @ X4 )
        = ( abs_abs_rat @ ( times_times_rat @ Y @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_6382_abs__mult__pos,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( times_times_int @ ( abs_abs_int @ Y ) @ X4 )
        = ( abs_abs_int @ ( times_times_int @ Y @ X4 ) ) ) ) ).

% abs_mult_pos
thf(fact_6383_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_6384_abs__minus__le__zero,axiom,
    ! [A: code_integer] : ( ord_le3102999989581377725nteger @ ( uminus1351360451143612070nteger @ ( abs_abs_Code_integer @ A ) ) @ zero_z3403309356797280102nteger ) ).

% abs_minus_le_zero
thf(fact_6385_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_6386_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_6387_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_6388_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_6389_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_6390_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_6391_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_6392_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_6393_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_6394_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_6395_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_6396_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_6397_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_6398_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_6399_abs__div__pos,axiom,
    ! [Y: rat,X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Y )
     => ( ( divide_divide_rat @ ( abs_abs_rat @ X4 ) @ Y )
        = ( abs_abs_rat @ ( divide_divide_rat @ X4 @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_6400_abs__div__pos,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ( ( divide_divide_real @ ( abs_abs_real @ X4 ) @ Y )
        = ( abs_abs_real @ ( divide_divide_real @ X4 @ Y ) ) ) ) ).

% abs_div_pos
thf(fact_6401_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_6402_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_6403_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_6404_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_6405_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_6406_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_6407_abs__of__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ A ) ) ) ).

% abs_of_neg
thf(fact_6408_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_6409_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_6410_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_6411_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_6412_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_6413_abs__diff__le__iff,axiom,
    ! [X4: code_integer,A: code_integer,R3: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R3 )
      = ( ( ord_le3102999989581377725nteger @ ( minus_8373710615458151222nteger @ A @ R3 ) @ X4 )
        & ( ord_le3102999989581377725nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6414_abs__diff__le__iff,axiom,
    ! [X4: real,A: real,R3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_eq_real @ ( minus_minus_real @ A @ R3 ) @ X4 )
        & ( ord_less_eq_real @ X4 @ ( plus_plus_real @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6415_abs__diff__le__iff,axiom,
    ! [X4: rat,A: rat,R3: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_eq_rat @ ( minus_minus_rat @ A @ R3 ) @ X4 )
        & ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6416_abs__diff__le__iff,axiom,
    ! [X4: int,A: int,R3: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_eq_int @ ( minus_minus_int @ A @ R3 ) @ X4 )
        & ( ord_less_eq_int @ X4 @ ( plus_plus_int @ A @ R3 ) ) ) ) ).

% abs_diff_le_iff
thf(fact_6417_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_6418_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_6419_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_6420_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_6421_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_6422_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_6423_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_6424_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_6425_abs__diff__less__iff,axiom,
    ! [X4: code_integer,A: code_integer,R3: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ A ) ) @ R3 )
      = ( ( ord_le6747313008572928689nteger @ ( minus_8373710615458151222nteger @ A @ R3 ) @ X4 )
        & ( ord_le6747313008572928689nteger @ X4 @ ( plus_p5714425477246183910nteger @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6426_abs__diff__less__iff,axiom,
    ! [X4: real,A: real,R3: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_real @ ( minus_minus_real @ A @ R3 ) @ X4 )
        & ( ord_less_real @ X4 @ ( plus_plus_real @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6427_abs__diff__less__iff,axiom,
    ! [X4: rat,A: rat,R3: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_rat @ ( minus_minus_rat @ A @ R3 ) @ X4 )
        & ( ord_less_rat @ X4 @ ( plus_plus_rat @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6428_abs__diff__less__iff,axiom,
    ! [X4: int,A: int,R3: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ A ) ) @ R3 )
      = ( ( ord_less_int @ ( minus_minus_int @ A @ R3 ) @ X4 )
        & ( ord_less_int @ X4 @ ( plus_plus_int @ A @ R3 ) ) ) ) ).

% abs_diff_less_iff
thf(fact_6429_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_6430_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_6431_bit__not__int__iff_H,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( minus_minus_int @ ( uminus_uminus_int @ K ) @ one_one_int ) @ N2 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_not_int_iff'
thf(fact_6432_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_6433_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_6434_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_6435_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_6436_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_6437_lemma__interval__lt,axiom,
    ! [A: real,X4: real,B: real] :
      ( ( ord_less_real @ A @ X4 )
     => ( ( ord_less_real @ X4 @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D3 )
               => ( ( ord_less_real @ A @ Y4 )
                  & ( ord_less_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval_lt
thf(fact_6438_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_6439_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_6440_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_6441_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_6442_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_6443_fact__mod,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( modulo364778990260209775nteger @ ( semiri3624122377584611663nteger @ N2 ) @ ( semiri3624122377584611663nteger @ M ) )
        = zero_z3403309356797280102nteger ) ) ).

% fact_mod
thf(fact_6444_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_6445_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_6446_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_6447_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_6448_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_6449_sin__bound__lemma,axiom,
    ! [X4: real,Y: real,U: real,V: real] :
      ( ( X4 = Y )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ U ) @ V )
       => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( plus_plus_real @ X4 @ U ) @ Y ) ) @ V ) ) ) ).

% sin_bound_lemma
thf(fact_6450_flip__bit__eq__if,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N: nat,A3: int] : ( if_nat_int_int @ ( bit_se1146084159140164899it_int @ A3 @ N ) @ bit_se4203085406695923979it_int @ bit_se7879613467334960850it_int @ N @ A3 ) ) ) ).

% flip_bit_eq_if
thf(fact_6451_flip__bit__eq__if,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [N: nat,A3: nat] : ( if_nat_nat_nat @ ( bit_se1148574629649215175it_nat @ A3 @ N ) @ bit_se4205575877204974255it_nat @ bit_se7882103937844011126it_nat @ N @ A3 ) ) ) ).

% flip_bit_eq_if
thf(fact_6452_tanh__real__gt__neg1,axiom,
    ! [X4: real] : ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ ( tanh_real @ X4 ) ) ).

% tanh_real_gt_neg1
thf(fact_6453_abs__add__one__gt__zero,axiom,
    ! [X4: code_integer] : ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( plus_p5714425477246183910nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6454_abs__add__one__gt__zero,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ one_one_real @ ( abs_abs_real @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6455_abs__add__one__gt__zero,axiom,
    ! [X4: rat] : ( ord_less_rat @ zero_zero_rat @ ( plus_plus_rat @ one_one_rat @ ( abs_abs_rat @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6456_abs__add__one__gt__zero,axiom,
    ! [X4: int] : ( ord_less_int @ zero_zero_int @ ( plus_plus_int @ one_one_int @ ( abs_abs_int @ X4 ) ) ) ).

% abs_add_one_gt_zero
thf(fact_6457_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_6458_fact__div__fact__le__pow,axiom,
    ! [R3: nat,N2: nat] :
      ( ( ord_less_eq_nat @ R3 @ N2 )
     => ( ord_less_eq_nat @ ( divide_divide_nat @ ( semiri1408675320244567234ct_nat @ N2 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ N2 @ R3 ) ) ) @ ( power_power_nat @ N2 @ R3 ) ) ) ).

% fact_div_fact_le_pow
thf(fact_6459_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_6460_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_6461_lemma__interval,axiom,
    ! [A: real,X4: real,B: real] :
      ( ( ord_less_real @ A @ X4 )
     => ( ( ord_less_real @ X4 @ B )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [Y4: real] :
                ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y4 ) ) @ D3 )
               => ( ( ord_less_eq_real @ A @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ B ) ) ) ) ) ) ).

% lemma_interval
thf(fact_6462_norm__triangle__ineq3,axiom,
    ! [A: real,B: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V7735802525324610683m_real @ A ) @ ( real_V7735802525324610683m_real @ B ) ) ) @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ A @ B ) ) ) ).

% norm_triangle_ineq3
thf(fact_6463_norm__triangle__ineq3,axiom,
    ! [A: complex,B: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ A ) @ ( real_V1022390504157884413omplex @ B ) ) ) @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ A @ B ) ) ) ).

% norm_triangle_ineq3
thf(fact_6464_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_6465_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_6466_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_6467_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_6468_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_6469_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri4449623510593786356d_enat @ ( numeral_numeral_nat @ K ) )
      = ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ K ) @ ( semiri4449623510593786356d_enat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6470_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri5044797733671781792omplex @ ( numeral_numeral_nat @ K ) )
      = ( times_times_complex @ ( numera6690914467698888265omplex @ K ) @ ( semiri5044797733671781792omplex @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6471_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1406184849735516958ct_int @ ( numeral_numeral_nat @ K ) )
      = ( times_times_int @ ( numeral_numeral_int @ K ) @ ( semiri1406184849735516958ct_int @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6472_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri1408675320244567234ct_nat @ ( numeral_numeral_nat @ K ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ K ) @ ( semiri1408675320244567234ct_nat @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6473_fact__numeral,axiom,
    ! [K: num] :
      ( ( semiri2265585572941072030t_real @ ( numeral_numeral_nat @ K ) )
      = ( times_times_real @ ( numeral_numeral_real @ K ) @ ( semiri2265585572941072030t_real @ ( pred_numeral @ K ) ) ) ) ).

% fact_numeral
thf(fact_6474_signed__take__bit__eq__concat__bit,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,K3: int] : ( bit_concat_bit @ N @ K3 @ ( uminus_uminus_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) ) ) ) ) ).

% signed_take_bit_eq_concat_bit
thf(fact_6475_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_6476_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_6477_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_6478_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_6479_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_6480_stable__imp__bit__iff__odd,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A )
     => ( ( bit_se9216721137139052372nteger @ A @ N2 )
        = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_6481_stable__imp__bit__iff__odd,axiom,
    ! [A: int,N2: nat] :
      ( ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A )
     => ( ( bit_se1146084159140164899it_int @ A @ N2 )
        = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_6482_stable__imp__bit__iff__odd,axiom,
    ! [A: nat,N2: nat] :
      ( ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A )
     => ( ( bit_se1148574629649215175it_nat @ A @ N2 )
        = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ) ).

% stable_imp_bit_iff_odd
thf(fact_6483_bit__iff__idd__imp__stable,axiom,
    ! [A: code_integer] :
      ( ! [N3: nat] :
          ( ( bit_se9216721137139052372nteger @ A @ N3 )
          = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) )
     => ( ( divide6298287555418463151nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) )
        = A ) ) ).

% bit_iff_idd_imp_stable
thf(fact_6484_bit__iff__idd__imp__stable,axiom,
    ! [A: int] :
      ( ! [N3: nat] :
          ( ( bit_se1146084159140164899it_int @ A @ N3 )
          = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) )
     => ( ( divide_divide_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
        = A ) ) ).

% bit_iff_idd_imp_stable
thf(fact_6485_bit__iff__idd__imp__stable,axiom,
    ! [A: nat] :
      ( ! [N3: nat] :
          ( ( bit_se1148574629649215175it_nat @ A @ N3 )
          = ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) )
     => ( ( divide_divide_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = A ) ) ).

% bit_iff_idd_imp_stable
thf(fact_6486_abs__le__square__iff,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ ( abs_abs_Code_integer @ Y ) )
      = ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6487_abs__le__square__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ Y ) )
      = ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6488_abs__le__square__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ ( abs_abs_rat @ Y ) )
      = ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6489_abs__le__square__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ ( abs_abs_int @ Y ) )
      = ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_le_square_iff
thf(fact_6490_abs__square__eq__1,axiom,
    ! [X4: code_integer] :
      ( ( ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_Code_integer )
      = ( ( abs_abs_Code_integer @ X4 )
        = one_one_Code_integer ) ) ).

% abs_square_eq_1
thf(fact_6491_abs__square__eq__1,axiom,
    ! [X4: rat] :
      ( ( ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_rat )
      = ( ( abs_abs_rat @ X4 )
        = one_one_rat ) ) ).

% abs_square_eq_1
thf(fact_6492_abs__square__eq__1,axiom,
    ! [X4: real] :
      ( ( ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_real )
      = ( ( abs_abs_real @ X4 )
        = one_one_real ) ) ).

% abs_square_eq_1
thf(fact_6493_abs__square__eq__1,axiom,
    ! [X4: int] :
      ( ( ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = one_one_int )
      = ( ( abs_abs_int @ X4 )
        = one_one_int ) ) ).

% abs_square_eq_1
thf(fact_6494_power__even__abs,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_8256067586552552935nteger @ ( abs_abs_Code_integer @ A ) @ N2 )
        = ( power_8256067586552552935nteger @ A @ N2 ) ) ) ).

% power_even_abs
thf(fact_6495_power__even__abs,axiom,
    ! [N2: nat,A: rat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_rat @ ( abs_abs_rat @ A ) @ N2 )
        = ( power_power_rat @ A @ N2 ) ) ) ).

% power_even_abs
thf(fact_6496_power__even__abs,axiom,
    ! [N2: nat,A: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( abs_abs_real @ A ) @ N2 )
        = ( power_power_real @ A @ N2 ) ) ) ).

% power_even_abs
thf(fact_6497_power__even__abs,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_int @ ( abs_abs_int @ A ) @ N2 )
        = ( power_power_int @ A @ N2 ) ) ) ).

% power_even_abs
thf(fact_6498_int__bit__bound,axiom,
    ! [K: int] :
      ~ ! [N3: nat] :
          ( ! [M2: nat] :
              ( ( ord_less_eq_nat @ N3 @ M2 )
             => ( ( bit_se1146084159140164899it_int @ K @ M2 )
                = ( 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_6499_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_6500_bit__iff__odd,axiom,
    ( bit_se9216721137139052372nteger
    = ( ^ [A3: code_integer,N: nat] :
          ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( divide6298287555418463151nteger @ A3 @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_iff_odd
thf(fact_6501_bit__iff__odd,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A3: int,N: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_iff_odd
thf(fact_6502_bit__iff__odd,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A3: nat,N: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_iff_odd
thf(fact_6503_power2__le__iff__abs__le,axiom,
    ! [Y: code_integer,X4: code_integer] :
      ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ Y )
     => ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_8256067586552552935nteger @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6504_power2__le__iff__abs__le,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6505_power2__le__iff__abs__le,axiom,
    ! [Y: rat,X4: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ Y )
     => ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_rat @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6506_power2__le__iff__abs__le,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_int @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ Y ) ) ) ).

% power2_le_iff_abs_le
thf(fact_6507_abs__square__le__1,axiom,
    ! [X4: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer ) ) ).

% abs_square_le_1
thf(fact_6508_abs__square__le__1,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real ) ) ).

% abs_square_le_1
thf(fact_6509_abs__square__le__1,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_eq_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat ) ) ).

% abs_square_le_1
thf(fact_6510_abs__square__le__1,axiom,
    ! [X4: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_eq_int @ ( abs_abs_int @ X4 ) @ one_one_int ) ) ).

% abs_square_le_1
thf(fact_6511_abs__square__less__1,axiom,
    ! [X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer ) ) ).

% abs_square_less_1
thf(fact_6512_abs__square__less__1,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real )
      = ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real ) ) ).

% abs_square_less_1
thf(fact_6513_abs__square__less__1,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_rat )
      = ( ord_less_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat ) ) ).

% abs_square_less_1
thf(fact_6514_abs__square__less__1,axiom,
    ! [X4: int] :
      ( ( ord_less_int @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_int )
      = ( ord_less_int @ ( abs_abs_int @ X4 ) @ one_one_int ) ) ).

% abs_square_less_1
thf(fact_6515_square__fact__le__2__fact,axiom,
    ! [N2: nat] : ( ord_less_eq_real @ ( times_times_real @ ( semiri2265585572941072030t_real @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% square_fact_le_2_fact
thf(fact_6516_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_6517_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_6518_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_6519_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_6520_bit__int__def,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [K3: int,N: nat] :
          ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_int_def
thf(fact_6521_fact__num__eq__if,axiom,
    ( semiri773545260158071498ct_rat
    = ( ^ [M6: nat] : ( if_rat @ ( M6 = zero_zero_nat ) @ one_one_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ M6 ) @ ( semiri773545260158071498ct_rat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6522_fact__num__eq__if,axiom,
    ( semiri5044797733671781792omplex
    = ( ^ [M6: nat] : ( if_complex @ ( M6 = zero_zero_nat ) @ one_one_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ M6 ) @ ( semiri5044797733671781792omplex @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6523_fact__num__eq__if,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [M6: nat] : ( if_int @ ( M6 = zero_zero_nat ) @ one_one_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1406184849735516958ct_int @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6524_fact__num__eq__if,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [M6: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ one_one_nat @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ M6 ) @ ( semiri1408675320244567234ct_nat @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6525_fact__num__eq__if,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [M6: nat] : ( if_real @ ( M6 = zero_zero_nat ) @ one_one_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M6 ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ M6 @ one_one_nat ) ) ) ) ) ) ).

% fact_num_eq_if
thf(fact_6526_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_6527_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_6528_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_6529_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_6530_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ N2 )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( semiri5044797733671781792omplex @ N2 ) ) ) ).

% pochhammer_same
thf(fact_6531_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s8582702949713902594nteger @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ N2 ) ) @ N2 )
      = ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ N2 ) @ ( semiri3624122377584611663nteger @ N2 ) ) ) ).

% pochhammer_same
thf(fact_6532_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ N2 )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( semiri773545260158071498ct_rat @ N2 ) ) ) ).

% pochhammer_same
thf(fact_6533_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ N2 ) ) @ N2 )
      = ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ N2 ) @ ( semiri1406184849735516958ct_int @ N2 ) ) ) ).

% pochhammer_same
thf(fact_6534_pochhammer__same,axiom,
    ! [N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ N2 )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ).

% pochhammer_same
thf(fact_6535_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_6536_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_6537_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_6538_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_6539_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_6540_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_6541_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_6542_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_6543_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_6544_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_6545_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_6546_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_6547_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_6548_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_6549_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_6550_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_6551_set__bit__eq,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N: nat,K3: int] :
          ( plus_plus_int @ K3
          @ ( times_times_int
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( bit_se1146084159140164899it_int @ K3 @ N ) )
            @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% set_bit_eq
thf(fact_6552_unset__bit__eq,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N: nat,K3: int] : ( minus_minus_int @ K3 @ ( times_times_int @ ( zero_n2684676970156552555ol_int @ ( bit_se1146084159140164899it_int @ K3 @ N ) ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% unset_bit_eq
thf(fact_6553_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_6554_abs__sqrt__wlog,axiom,
    ! [P: code_integer > code_integer > $o,X4: code_integer] :
      ( ! [X5: code_integer] :
          ( ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ X5 )
         => ( P @ X5 @ ( power_8256067586552552935nteger @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_Code_integer @ X4 ) @ ( power_8256067586552552935nteger @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6555_abs__sqrt__wlog,axiom,
    ! [P: real > real > $o,X4: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
         => ( P @ X5 @ ( power_power_real @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_real @ X4 ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6556_abs__sqrt__wlog,axiom,
    ! [P: rat > rat > $o,X4: rat] :
      ( ! [X5: rat] :
          ( ( ord_less_eq_rat @ zero_zero_rat @ X5 )
         => ( P @ X5 @ ( power_power_rat @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_rat @ X4 ) @ ( power_power_rat @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6557_abs__sqrt__wlog,axiom,
    ! [P: int > int > $o,X4: int] :
      ( ! [X5: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X5 )
         => ( P @ X5 @ ( power_power_int @ X5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
     => ( P @ ( abs_abs_int @ X4 ) @ ( power_power_int @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% abs_sqrt_wlog
thf(fact_6558_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_6559_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_6560_cos__coeff__def,axiom,
    ( cos_coeff
    = ( ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri2265585572941072030t_real @ N ) ) @ zero_zero_real ) ) ) ).

% cos_coeff_def
thf(fact_6561_arctan__double,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ X4 ) )
        = ( arctan @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% arctan_double
thf(fact_6562_fact__code,axiom,
    ( semiri1406184849735516958ct_int
    = ( ^ [N: nat] : ( semiri1314217659103216013at_int @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6563_fact__code,axiom,
    ( semiri1408675320244567234ct_nat
    = ( ^ [N: nat] : ( semiri1316708129612266289at_nat @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6564_fact__code,axiom,
    ( semiri2265585572941072030t_real
    = ( ^ [N: nat] : ( semiri5074537144036343181t_real @ ( set_fo2584398358068434914at_nat @ times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N @ one_one_nat ) ) ) ) ).

% fact_code
thf(fact_6565_zabs__less__one__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( abs_abs_int @ Z ) @ one_one_int )
      = ( Z = zero_zero_int ) ) ).

% zabs_less_one_iff
thf(fact_6566_zero__less__arctan__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( arctan @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% zero_less_arctan_iff
thf(fact_6567_arctan__less__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( arctan @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% arctan_less_zero_iff
thf(fact_6568_zero__le__arctan__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( arctan @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% zero_le_arctan_iff
thf(fact_6569_arctan__le__zero__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( arctan @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% arctan_le_zero_iff
thf(fact_6570_cos__coeff__0,axiom,
    ( ( cos_coeff @ zero_zero_nat )
    = one_one_real ) ).

% cos_coeff_0
thf(fact_6571_arctan__less__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( arctan @ X4 ) @ ( arctan @ Y ) )
      = ( ord_less_real @ X4 @ Y ) ) ).

% arctan_less_iff
thf(fact_6572_arctan__monotone,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ord_less_real @ ( arctan @ X4 ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone
thf(fact_6573_arctan__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% arctan_le_iff
thf(fact_6574_arctan__monotone_H,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ord_less_eq_real @ ( arctan @ X4 ) @ ( arctan @ Y ) ) ) ).

% arctan_monotone'
thf(fact_6575_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_6576_not__bit__Suc__0__Suc,axiom,
    ! [N2: nat] :
      ~ ( bit_se1148574629649215175it_nat @ ( suc @ zero_zero_nat ) @ ( suc @ N2 ) ) ).

% not_bit_Suc_0_Suc
thf(fact_6577_abs__div,axiom,
    ! [Y: int,X4: int] :
      ( ( dvd_dvd_int @ Y @ X4 )
     => ( ( abs_abs_int @ ( divide_divide_int @ X4 @ Y ) )
        = ( divide_divide_int @ ( abs_abs_int @ X4 ) @ ( abs_abs_int @ Y ) ) ) ) ).

% abs_div
thf(fact_6578_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_6579_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_6580_zabs__def,axiom,
    ( abs_abs_int
    = ( ^ [I3: int] : ( if_int @ ( ord_less_int @ I3 @ zero_zero_int ) @ ( uminus_uminus_int @ I3 ) @ I3 ) ) ) ).

% zabs_def
thf(fact_6581_dvd__imp__le__int,axiom,
    ! [I2: int,D: int] :
      ( ( I2 != zero_zero_int )
     => ( ( dvd_dvd_int @ D @ I2 )
       => ( ord_less_eq_int @ ( abs_abs_int @ D ) @ ( abs_abs_int @ I2 ) ) ) ) ).

% dvd_imp_le_int
thf(fact_6582_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_6583_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_6584_fold__atLeastAtMost__nat_Oelims,axiom,
    ! [X4: nat > nat > nat,Xa: nat,Xb: nat,Xc: nat,Y: nat] :
      ( ( ( set_fo2584398358068434914at_nat @ X4 @ Xa @ Xb @ Xc )
        = Y )
     => ( ( ( ord_less_nat @ Xb @ Xa )
         => ( Y = Xc ) )
        & ( ~ ( ord_less_nat @ Xb @ Xa )
         => ( Y
            = ( set_fo2584398358068434914at_nat @ X4 @ ( plus_plus_nat @ Xa @ one_one_nat ) @ Xb @ ( X4 @ Xa @ Xc ) ) ) ) ) ) ).

% fold_atLeastAtMost_nat.elims
thf(fact_6585_fold__atLeastAtMost__nat_Osimps,axiom,
    ( set_fo2584398358068434914at_nat
    = ( ^ [F3: nat > nat > nat,A3: nat,B2: nat,Acc2: nat] : ( if_nat @ ( ord_less_nat @ B2 @ A3 ) @ Acc2 @ ( set_fo2584398358068434914at_nat @ F3 @ ( plus_plus_nat @ A3 @ one_one_nat ) @ B2 @ ( F3 @ A3 @ Acc2 ) ) ) ) ) ).

% fold_atLeastAtMost_nat.simps
thf(fact_6586_even__add__abs__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ ( abs_abs_int @ L ) ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_add_abs_iff
thf(fact_6587_even__abs__add__iff,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ ( abs_abs_int @ K ) @ L ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( plus_plus_int @ K @ L ) ) ) ).

% even_abs_add_iff
thf(fact_6588_bit__nat__def,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [M6: nat,N: nat] :
          ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ).

% bit_nat_def
thf(fact_6589_nat__intermed__int__val,axiom,
    ! [M: nat,N2: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ( ord_less_eq_nat @ M @ I4 )
            & ( ord_less_nat @ I4 @ N2 ) )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( ord_less_eq_int @ ( F @ M ) @ K )
         => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
           => ? [I4: nat] :
                ( ( ord_less_eq_nat @ M @ I4 )
                & ( ord_less_eq_nat @ I4 @ N2 )
                & ( ( F @ I4 )
                  = K ) ) ) ) ) ) ).

% nat_intermed_int_val
thf(fact_6590_incr__lemma,axiom,
    ! [D: int,Z: int,X4: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ Z @ ( plus_plus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) ) ) ).

% incr_lemma
thf(fact_6591_decr__lemma,axiom,
    ! [D: int,X4: int,Z: int] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ord_less_int @ ( minus_minus_int @ X4 @ ( times_times_int @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Z ) ) @ one_one_int ) @ D ) ) @ Z ) ) ).

% decr_lemma
thf(fact_6592_nat__ivt__aux,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( suc @ I4 ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I4: nat] :
              ( ( ord_less_eq_nat @ I4 @ N2 )
              & ( ( F @ I4 )
                = K ) ) ) ) ) ).

% nat_ivt_aux
thf(fact_6593_complex__mod__minus__le__complex__mod,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% complex_mod_minus_le_complex_mod
thf(fact_6594_complex__mod__triangle__ineq2,axiom,
    ! [B: complex,A: complex] : ( ord_less_eq_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ B @ A ) ) @ ( real_V1022390504157884413omplex @ B ) ) @ ( real_V1022390504157884413omplex @ A ) ) ).

% complex_mod_triangle_ineq2
thf(fact_6595_nat0__intermed__int__val,axiom,
    ! [N2: nat,F: nat > int,K: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ord_less_eq_int @ ( abs_abs_int @ ( minus_minus_int @ ( F @ ( plus_plus_nat @ I4 @ one_one_nat ) ) @ ( F @ I4 ) ) ) @ one_one_int ) )
     => ( ( ord_less_eq_int @ ( F @ zero_zero_nat ) @ K )
       => ( ( ord_less_eq_int @ K @ ( F @ N2 ) )
         => ? [I4: nat] :
              ( ( ord_less_eq_nat @ I4 @ N2 )
              & ( ( F @ I4 )
                = K ) ) ) ) ) ).

% nat0_intermed_int_val
thf(fact_6596_arctan__add,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( plus_plus_real @ ( arctan @ X4 ) @ ( arctan @ Y ) )
          = ( arctan @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ X4 @ Y ) ) ) ) ) ) ) ).

% arctan_add
thf(fact_6597_xor__int__unfold,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( K3
            = ( uminus_uminus_int @ one_one_int ) )
          @ ( bit_ri7919022796975470100ot_int @ L2 )
          @ ( if_int
            @ ( L2
              = ( uminus_uminus_int @ one_one_int ) )
            @ ( bit_ri7919022796975470100ot_int @ K3 )
            @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( abs_abs_int @ ( minus_minus_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se6526347334894502574or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_int_unfold
thf(fact_6598_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_6599_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_6600_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_6601_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_6602_tanh__real__altdef,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) @ ( plus_plus_real @ one_one_real @ ( exp_real @ ( times_times_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ) ) ).

% tanh_real_altdef
thf(fact_6603_and__int__unfold,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3 = zero_zero_int )
            | ( L2 = zero_zero_int ) )
          @ zero_zero_int
          @ ( if_int
            @ ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            @ L2
            @ ( if_int
              @ ( L2
                = ( uminus_uminus_int @ one_one_int ) )
              @ K3
              @ ( plus_plus_int @ ( times_times_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% and_int_unfold
thf(fact_6604_and_Oidem,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ A )
      = A ) ).

% and.idem
thf(fact_6605_and_Oidem,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ A )
      = A ) ).

% and.idem
thf(fact_6606_and_Oleft__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( bit_se725231765392027082nd_int @ A @ B ) ) ).

% and.left_idem
thf(fact_6607_and_Oleft__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).

% and.left_idem
thf(fact_6608_and_Oright__idem,axiom,
    ! [A: int,B: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ B )
      = ( bit_se725231765392027082nd_int @ A @ B ) ) ).

% and.right_idem
thf(fact_6609_and_Oright__idem,axiom,
    ! [A: nat,B: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ B )
      = ( bit_se727722235901077358nd_nat @ A @ B ) ) ).

% and.right_idem
thf(fact_6610_sgn__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( sgn_sgn_int @ A ) )
      = ( sgn_sgn_int @ A ) ) ).

% sgn_sgn
thf(fact_6611_sgn__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( sgn_sgn_real @ A ) )
      = ( sgn_sgn_real @ A ) ) ).

% sgn_sgn
thf(fact_6612_sgn__sgn,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( sgn_sgn_complex @ A ) )
      = ( sgn_sgn_complex @ A ) ) ).

% sgn_sgn
thf(fact_6613_sgn__sgn,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
      = ( sgn_sgn_Code_integer @ A ) ) ).

% sgn_sgn
thf(fact_6614_sgn__sgn,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( sgn_sgn_rat @ A ) )
      = ( sgn_sgn_rat @ A ) ) ).

% sgn_sgn
thf(fact_6615_bit_Odouble__compl,axiom,
    ! [X4: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( bit_ri7919022796975470100ot_int @ X4 ) )
      = X4 ) ).

% bit.double_compl
thf(fact_6616_bit_Ocompl__eq__compl__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ( bit_ri7919022796975470100ot_int @ X4 )
        = ( bit_ri7919022796975470100ot_int @ Y ) )
      = ( X4 = Y ) ) ).

% bit.compl_eq_compl_iff
thf(fact_6617_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_6618_and__zero__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ zero_zero_int )
      = zero_zero_int ) ).

% and_zero_eq
thf(fact_6619_and__zero__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ zero_zero_nat )
      = zero_zero_nat ) ).

% and_zero_eq
thf(fact_6620_zero__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ A )
      = zero_zero_int ) ).

% zero_and_eq
thf(fact_6621_zero__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ zero_zero_nat @ A )
      = zero_zero_nat ) ).

% zero_and_eq
thf(fact_6622_bit_Oconj__zero__left,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ zero_zero_int @ X4 )
      = zero_zero_int ) ).

% bit.conj_zero_left
thf(fact_6623_bit_Oconj__zero__right,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ X4 @ zero_zero_int )
      = zero_zero_int ) ).

% bit.conj_zero_right
thf(fact_6624_sgn__0,axiom,
    ( ( sgn_sgn_Code_integer @ zero_z3403309356797280102nteger )
    = zero_z3403309356797280102nteger ) ).

% sgn_0
thf(fact_6625_sgn__0,axiom,
    ( ( sgn_sgn_complex @ zero_zero_complex )
    = zero_zero_complex ) ).

% sgn_0
thf(fact_6626_sgn__0,axiom,
    ( ( sgn_sgn_real @ zero_zero_real )
    = zero_zero_real ) ).

% sgn_0
thf(fact_6627_sgn__0,axiom,
    ( ( sgn_sgn_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% sgn_0
thf(fact_6628_sgn__0,axiom,
    ( ( sgn_sgn_int @ zero_zero_int )
    = zero_zero_int ) ).

% sgn_0
thf(fact_6629_sgn__1,axiom,
    ( ( sgn_sgn_int @ one_one_int )
    = one_one_int ) ).

% sgn_1
thf(fact_6630_sgn__1,axiom,
    ( ( sgn_sgn_real @ one_one_real )
    = one_one_real ) ).

% sgn_1
thf(fact_6631_sgn__1,axiom,
    ( ( sgn_sgn_complex @ one_one_complex )
    = one_one_complex ) ).

% sgn_1
thf(fact_6632_sgn__1,axiom,
    ( ( sgn_sgn_Code_integer @ one_one_Code_integer )
    = one_one_Code_integer ) ).

% sgn_1
thf(fact_6633_sgn__1,axiom,
    ( ( sgn_sgn_rat @ one_one_rat )
    = one_one_rat ) ).

% sgn_1
thf(fact_6634_sgn__one,axiom,
    ( ( sgn_sgn_real @ one_one_real )
    = one_one_real ) ).

% sgn_one
thf(fact_6635_sgn__one,axiom,
    ( ( sgn_sgn_complex @ one_one_complex )
    = one_one_complex ) ).

% sgn_one
thf(fact_6636_sgn__divide,axiom,
    ! [A: rat,B: rat] :
      ( ( sgn_sgn_rat @ ( divide_divide_rat @ A @ B ) )
      = ( divide_divide_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).

% sgn_divide
thf(fact_6637_sgn__divide,axiom,
    ! [A: real,B: real] :
      ( ( sgn_sgn_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).

% sgn_divide
thf(fact_6638_sgn__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( sgn_sgn_complex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).

% sgn_divide
thf(fact_6639_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( uminus_uminus_real @ A ) )
      = ( uminus_uminus_real @ ( sgn_sgn_real @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6640_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( uminus_uminus_int @ A ) )
      = ( uminus_uminus_int @ ( sgn_sgn_int @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6641_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ A ) )
      = ( uminus1482373934393186551omplex @ ( sgn_sgn_complex @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6642_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ A ) )
      = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6643_idom__abs__sgn__class_Osgn__minus,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ A ) )
      = ( uminus_uminus_rat @ ( sgn_sgn_rat @ A ) ) ) ).

% idom_abs_sgn_class.sgn_minus
thf(fact_6644_power__sgn,axiom,
    ! [A: code_integer,N2: nat] :
      ( ( sgn_sgn_Code_integer @ ( power_8256067586552552935nteger @ A @ N2 ) )
      = ( power_8256067586552552935nteger @ ( sgn_sgn_Code_integer @ A ) @ N2 ) ) ).

% power_sgn
thf(fact_6645_power__sgn,axiom,
    ! [A: rat,N2: nat] :
      ( ( sgn_sgn_rat @ ( power_power_rat @ A @ N2 ) )
      = ( power_power_rat @ ( sgn_sgn_rat @ A ) @ N2 ) ) ).

% power_sgn
thf(fact_6646_power__sgn,axiom,
    ! [A: real,N2: nat] :
      ( ( sgn_sgn_real @ ( power_power_real @ A @ N2 ) )
      = ( power_power_real @ ( sgn_sgn_real @ A ) @ N2 ) ) ).

% power_sgn
thf(fact_6647_power__sgn,axiom,
    ! [A: int,N2: nat] :
      ( ( sgn_sgn_int @ ( power_power_int @ A @ N2 ) )
      = ( power_power_int @ ( sgn_sgn_int @ A ) @ N2 ) ) ).

% power_sgn
thf(fact_6648_exp__less__cancel__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y ) )
      = ( ord_less_real @ X4 @ Y ) ) ).

% exp_less_cancel_iff
thf(fact_6649_exp__less__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y ) ) ) ).

% exp_less_mono
thf(fact_6650_take__bit__and,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) @ ( bit_se2923211474154528505it_int @ N2 @ B ) ) ) ).

% take_bit_and
thf(fact_6651_take__bit__and,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se2925701944663578781it_nat @ N2 @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( bit_se727722235901077358nd_nat @ ( bit_se2925701944663578781it_nat @ N2 @ A ) @ ( bit_se2925701944663578781it_nat @ N2 @ B ) ) ) ).

% take_bit_and
thf(fact_6652_exp__le__cancel__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( exp_real @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% exp_le_cancel_iff
thf(fact_6653_bit_Oxor__compl__left,axiom,
    ! [X4: int,Y: int] :
      ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ Y )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X4 @ Y ) ) ) ).

% bit.xor_compl_left
thf(fact_6654_bit_Oxor__compl__right,axiom,
    ! [X4: int,Y: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_ri7919022796975470100ot_int @ Y ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ X4 @ Y ) ) ) ).

% bit.xor_compl_right
thf(fact_6655_sgn__less,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( sgn_sgn_Code_integer @ A ) @ zero_z3403309356797280102nteger )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_less
thf(fact_6656_sgn__less,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( sgn_sgn_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_less
thf(fact_6657_sgn__less,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( sgn_sgn_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% sgn_less
thf(fact_6658_sgn__less,axiom,
    ! [A: int] :
      ( ( ord_less_int @ ( sgn_sgn_int @ A ) @ zero_zero_int )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_less
thf(fact_6659_sgn__greater,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ ( sgn_sgn_Code_integer @ A ) )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_greater
thf(fact_6660_sgn__greater,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sgn_sgn_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_greater
thf(fact_6661_sgn__greater,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( sgn_sgn_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% sgn_greater
thf(fact_6662_sgn__greater,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ ( sgn_sgn_int @ A ) )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_greater
thf(fact_6663_exp__zero,axiom,
    ( ( exp_complex @ zero_zero_complex )
    = one_one_complex ) ).

% exp_zero
thf(fact_6664_exp__zero,axiom,
    ( ( exp_real @ zero_zero_real )
    = one_one_real ) ).

% exp_zero
thf(fact_6665_bit_Oconj__one__right,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3949692690581998587nteger @ X4 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = X4 ) ).

% bit.conj_one_right
thf(fact_6666_bit_Oconj__one__right,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
      = X4 ) ).

% bit.conj_one_right
thf(fact_6667_and_Oright__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = A ) ).

% and.right_neutral
thf(fact_6668_and_Oright__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ ( uminus_uminus_int @ one_one_int ) )
      = A ) ).

% and.right_neutral
thf(fact_6669_and_Oleft__neutral,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_6670_and_Oleft__neutral,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ one_one_int ) @ A )
      = A ) ).

% and.left_neutral
thf(fact_6671_divide__sgn,axiom,
    ! [A: rat,B: rat] :
      ( ( divide_divide_rat @ A @ ( sgn_sgn_rat @ B ) )
      = ( times_times_rat @ A @ ( sgn_sgn_rat @ B ) ) ) ).

% divide_sgn
thf(fact_6672_divide__sgn,axiom,
    ! [A: real,B: real] :
      ( ( divide_divide_real @ A @ ( sgn_sgn_real @ B ) )
      = ( times_times_real @ A @ ( sgn_sgn_real @ B ) ) ) ).

% divide_sgn
thf(fact_6673_mask__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ zero_zero_nat )
    = zero_zero_nat ) ).

% mask_0
thf(fact_6674_mask__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ zero_zero_nat )
    = zero_zero_int ) ).

% mask_0
thf(fact_6675_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2002935070580805687sk_nat @ N2 )
        = zero_zero_nat )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6676_mask__eq__0__iff,axiom,
    ! [N2: nat] :
      ( ( ( bit_se2000444600071755411sk_int @ N2 )
        = zero_zero_int )
      = ( N2 = zero_zero_nat ) ) ).

% mask_eq_0_iff
thf(fact_6677_bit_Oconj__cancel__left,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ X4 )
      = zero_zero_int ) ).

% bit.conj_cancel_left
thf(fact_6678_bit_Oconj__cancel__right,axiom,
    ! [X4: int] :
      ( ( bit_se725231765392027082nd_int @ X4 @ ( bit_ri7919022796975470100ot_int @ X4 ) )
      = zero_zero_int ) ).

% bit.conj_cancel_right
thf(fact_6679_exp__eq__one__iff,axiom,
    ! [X4: real] :
      ( ( ( exp_real @ X4 )
        = one_one_real )
      = ( X4 = zero_zero_real ) ) ).

% exp_eq_one_iff
thf(fact_6680_and__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        | ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% and_nonnegative_int_iff
thf(fact_6681_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_6682_sgn__pos,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A )
     => ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer ) ) ).

% sgn_pos
thf(fact_6683_sgn__pos,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( sgn_sgn_real @ A )
        = one_one_real ) ) ).

% sgn_pos
thf(fact_6684_sgn__pos,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( sgn_sgn_rat @ A )
        = one_one_rat ) ) ).

% sgn_pos
thf(fact_6685_sgn__pos,axiom,
    ! [A: int] :
      ( ( ord_less_int @ zero_zero_int @ A )
     => ( ( sgn_sgn_int @ A )
        = one_one_int ) ) ).

% sgn_pos
thf(fact_6686_bit_Ocompl__one,axiom,
    ( ( bit_ri7632146776885996613nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = zero_z3403309356797280102nteger ) ).

% bit.compl_one
thf(fact_6687_bit_Ocompl__one,axiom,
    ( ( bit_ri7919022796975470100ot_int @ ( uminus_uminus_int @ one_one_int ) )
    = zero_zero_int ) ).

% bit.compl_one
thf(fact_6688_bit_Ocompl__zero,axiom,
    ( ( bit_ri7632146776885996613nteger @ zero_z3403309356797280102nteger )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.compl_zero
thf(fact_6689_bit_Ocompl__zero,axiom,
    ( ( bit_ri7919022796975470100ot_int @ zero_zero_int )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.compl_zero
thf(fact_6690_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = one_one_int ) ).

% and_numerals(2)
thf(fact_6691_and__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_numerals(2)
thf(fact_6692_and__numerals_I8_J,axiom,
    ! [X4: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ one_one_int )
      = one_one_int ) ).

% and_numerals(8)
thf(fact_6693_and__numerals_I8_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ one_one_nat )
      = one_one_nat ) ).

% and_numerals(8)
thf(fact_6694_abs__sgn__eq__1,axiom,
    ! [A: code_integer] :
      ( ( A != zero_z3403309356797280102nteger )
     => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
        = one_one_Code_integer ) ) ).

% abs_sgn_eq_1
thf(fact_6695_abs__sgn__eq__1,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
        = one_one_real ) ) ).

% abs_sgn_eq_1
thf(fact_6696_abs__sgn__eq__1,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
        = one_one_rat ) ) ).

% abs_sgn_eq_1
thf(fact_6697_abs__sgn__eq__1,axiom,
    ! [A: int] :
      ( ( A != zero_zero_int )
     => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
        = one_one_int ) ) ).

% abs_sgn_eq_1
thf(fact_6698_sgn__mult__self__eq,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).

% sgn_mult_self_eq
thf(fact_6699_sgn__mult__self__eq,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% sgn_mult_self_eq
thf(fact_6700_sgn__mult__self__eq,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_mult_self_eq
thf(fact_6701_sgn__mult__self__eq,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
      = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).

% sgn_mult_self_eq
thf(fact_6702_mask__Suc__0,axiom,
    ( ( bit_se2002935070580805687sk_nat @ ( suc @ zero_zero_nat ) )
    = one_one_nat ) ).

% mask_Suc_0
thf(fact_6703_mask__Suc__0,axiom,
    ( ( bit_se2000444600071755411sk_int @ ( suc @ zero_zero_nat ) )
    = one_one_int ) ).

% mask_Suc_0
thf(fact_6704_bit_Oxor__one__left,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ X4 )
      = ( bit_ri7632146776885996613nteger @ X4 ) ) ).

% bit.xor_one_left
thf(fact_6705_bit_Oxor__one__left,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ one_one_int ) @ X4 )
      = ( bit_ri7919022796975470100ot_int @ X4 ) ) ).

% bit.xor_one_left
thf(fact_6706_bit_Oxor__one__right,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X4 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ X4 ) ) ).

% bit.xor_one_right
thf(fact_6707_bit_Oxor__one__right,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ X4 ) ) ).

% bit.xor_one_right
thf(fact_6708_bit_Oxor__cancel__left,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3222712562003087583nteger @ ( bit_ri7632146776885996613nteger @ X4 ) @ X4 )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_left
thf(fact_6709_bit_Oxor__cancel__left,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ ( bit_ri7919022796975470100ot_int @ X4 ) @ X4 )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_left
thf(fact_6710_bit_Oxor__cancel__right,axiom,
    ! [X4: code_integer] :
      ( ( bit_se3222712562003087583nteger @ X4 @ ( bit_ri7632146776885996613nteger @ X4 ) )
      = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% bit.xor_cancel_right
thf(fact_6711_bit_Oxor__cancel__right,axiom,
    ! [X4: int] :
      ( ( bit_se6526347334894502574or_int @ X4 @ ( bit_ri7919022796975470100ot_int @ X4 ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% bit.xor_cancel_right
thf(fact_6712_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( abs_abs_complex @ A ) )
      = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6713_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( abs_abs_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6714_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( abs_abs_rat @ A ) )
      = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6715_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: int] :
      ( ( sgn_sgn_int @ ( abs_abs_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6716_idom__abs__sgn__class_Oabs__sgn,axiom,
    ! [A: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( abs_abs_Code_integer @ A ) )
      = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).

% idom_abs_sgn_class.abs_sgn
thf(fact_6717_sgn__abs,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( sgn_sgn_complex @ A ) )
      = ( zero_n1201886186963655149omplex @ ( A != zero_zero_complex ) ) ) ).

% sgn_abs
thf(fact_6718_sgn__abs,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
      = ( zero_n3304061248610475627l_real @ ( A != zero_zero_real ) ) ) ).

% sgn_abs
thf(fact_6719_sgn__abs,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
      = ( zero_n2052037380579107095ol_rat @ ( A != zero_zero_rat ) ) ) ).

% sgn_abs
thf(fact_6720_sgn__abs,axiom,
    ! [A: int] :
      ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
      = ( zero_n2684676970156552555ol_int @ ( A != zero_zero_int ) ) ) ).

% sgn_abs
thf(fact_6721_sgn__abs,axiom,
    ! [A: code_integer] :
      ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
      = ( zero_n356916108424825756nteger @ ( A != zero_z3403309356797280102nteger ) ) ) ).

% sgn_abs
thf(fact_6722_one__less__exp__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% one_less_exp_iff
thf(fact_6723_exp__less__one__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( exp_real @ X4 ) @ one_one_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% exp_less_one_iff
thf(fact_6724_exp__le__one__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( exp_real @ X4 ) @ one_one_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% exp_le_one_iff
thf(fact_6725_one__le__exp__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( exp_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% one_le_exp_iff
thf(fact_6726_take__bit__minus__one__eq__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se1745604003318907178nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_se2119862282449309892nteger @ N2 ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_6727_take__bit__minus__one__eq__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% take_bit_minus_one_eq_mask
thf(fact_6728_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_6729_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_6730_exp__ln,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( exp_real @ ( ln_ln_real @ X4 ) )
        = X4 ) ) ).

% exp_ln
thf(fact_6731_exp__ln__iff,axiom,
    ! [X4: real] :
      ( ( ( exp_real @ ( ln_ln_real @ X4 ) )
        = X4 )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% exp_ln_iff
thf(fact_6732_minus__not__numeral__eq,axiom,
    ! [N2: num] :
      ( ( uminus1351360451143612070nteger @ ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N2 ) ) )
      = ( numera6620942414471956472nteger @ ( inc @ N2 ) ) ) ).

% minus_not_numeral_eq
thf(fact_6733_minus__not__numeral__eq,axiom,
    ! [N2: num] :
      ( ( uminus_uminus_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( numeral_numeral_int @ ( inc @ N2 ) ) ) ).

% minus_not_numeral_eq
thf(fact_6734_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = zero_zero_int ) ).

% and_numerals(1)
thf(fact_6735_and__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_numerals(1)
thf(fact_6736_and__numerals_I5_J,axiom,
    ! [X4: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_numerals(5)
thf(fact_6737_and__numerals_I5_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ one_one_nat )
      = zero_zero_nat ) ).

% and_numerals(5)
thf(fact_6738_sgn__neg,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% sgn_neg
thf(fact_6739_sgn__neg,axiom,
    ! [A: int] :
      ( ( ord_less_int @ A @ zero_zero_int )
     => ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) ) ) ).

% sgn_neg
thf(fact_6740_sgn__neg,axiom,
    ! [A: code_integer] :
      ( ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger )
     => ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ).

% sgn_neg
thf(fact_6741_sgn__neg,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% sgn_neg
thf(fact_6742_and__numerals_I3_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_6743_and__numerals_I3_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(3)
thf(fact_6744_even__not__iff,axiom,
    ! [A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_ri7632146776885996613nteger @ A ) )
      = ( ~ ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_not_iff
thf(fact_6745_even__not__iff,axiom,
    ! [A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ A ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_not_iff
thf(fact_6746_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_rat @ ( semiri681578069525770553at_rat @ N2 ) )
      = ( zero_n2052037380579107095ol_rat @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_6747_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_real @ ( semiri5074537144036343181t_real @ N2 ) )
      = ( zero_n3304061248610475627l_real @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_6748_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_int @ ( semiri1314217659103216013at_int @ N2 ) )
      = ( zero_n2684676970156552555ol_int @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_6749_sgn__of__nat,axiom,
    ! [N2: nat] :
      ( ( sgn_sgn_Code_integer @ ( semiri4939895301339042750nteger @ N2 ) )
      = ( zero_n356916108424825756nteger @ ( ord_less_nat @ zero_zero_nat @ N2 ) ) ) ).

% sgn_of_nat
thf(fact_6750_and__minus__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = one_one_int ) ).

% and_minus_numerals(2)
thf(fact_6751_and__minus__numerals_I6_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
      = one_one_int ) ).

% and_minus_numerals(6)
thf(fact_6752_not__one__eq,axiom,
    ( ( bit_ri7632146776885996613nteger @ one_one_Code_integer )
    = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_6753_not__one__eq,axiom,
    ( ( bit_ri7919022796975470100ot_int @ one_one_int )
    = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% not_one_eq
thf(fact_6754_and__numerals_I4_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_6755_and__numerals_I4_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(4)
thf(fact_6756_and__numerals_I6_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit0 @ Y ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_6757_and__numerals_I6_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ).

% and_numerals(6)
thf(fact_6758_and__minus__numerals_I1_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = zero_zero_int ) ).

% and_minus_numerals(1)
thf(fact_6759_and__minus__numerals_I5_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
      = zero_zero_int ) ).

% and_minus_numerals(5)
thf(fact_6760_and__numerals_I7_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ X4 ) ) @ ( numeral_numeral_int @ ( bit1 @ Y ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ X4 ) @ ( numeral_numeral_int @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6761_and__numerals_I7_J,axiom,
    ! [X4: num,Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( plus_plus_nat @ one_one_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ X4 ) @ ( numeral_numeral_nat @ Y ) ) ) ) ) ).

% and_numerals(7)
thf(fact_6762_disjunctive__diff,axiom,
    ! [B: int,A: int] :
      ( ! [N3: nat] :
          ( ( bit_se1146084159140164899it_int @ B @ N3 )
         => ( bit_se1146084159140164899it_int @ A @ N3 ) )
     => ( ( minus_minus_int @ A @ B )
        = ( bit_se725231765392027082nd_int @ A @ ( bit_ri7919022796975470100ot_int @ B ) ) ) ) ).

% disjunctive_diff
thf(fact_6763_norm__exp,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ X4 ) ) @ ( exp_real @ ( real_V7735802525324610683m_real @ X4 ) ) ) ).

% norm_exp
thf(fact_6764_norm__exp,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ X4 ) ) @ ( exp_real @ ( real_V1022390504157884413omplex @ X4 ) ) ) ).

% norm_exp
thf(fact_6765_and__not__numerals_I1_J,axiom,
    ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = zero_zero_int ) ).

% and_not_numerals(1)
thf(fact_6766_take__bit__eq__mask,axiom,
    ( bit_se2923211474154528505it_int
    = ( ^ [N: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_se2000444600071755411sk_int @ N ) ) ) ) ).

% take_bit_eq_mask
thf(fact_6767_take__bit__eq__mask,axiom,
    ( bit_se2925701944663578781it_nat
    = ( ^ [N: nat,A3: nat] : ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se2002935070580805687sk_nat @ N ) ) ) ) ).

% take_bit_eq_mask
thf(fact_6768_and_Oassoc,axiom,
    ! [A: int,B: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ C )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).

% and.assoc
thf(fact_6769_and_Oassoc,axiom,
    ! [A: nat,B: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ C )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).

% and.assoc
thf(fact_6770_and_Ocommute,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [A3: int,B2: int] : ( bit_se725231765392027082nd_int @ B2 @ A3 ) ) ) ).

% and.commute
thf(fact_6771_and_Ocommute,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [A3: nat,B2: nat] : ( bit_se727722235901077358nd_nat @ B2 @ A3 ) ) ) ).

% and.commute
thf(fact_6772_and_Oleft__commute,axiom,
    ! [B: int,A: int,C: int] :
      ( ( bit_se725231765392027082nd_int @ B @ ( bit_se725231765392027082nd_int @ A @ C ) )
      = ( bit_se725231765392027082nd_int @ A @ ( bit_se725231765392027082nd_int @ B @ C ) ) ) ).

% and.left_commute
thf(fact_6773_and_Oleft__commute,axiom,
    ! [B: nat,A: nat,C: nat] :
      ( ( bit_se727722235901077358nd_nat @ B @ ( bit_se727722235901077358nd_nat @ A @ C ) )
      = ( bit_se727722235901077358nd_nat @ A @ ( bit_se727722235901077358nd_nat @ B @ C ) ) ) ).

% and.left_commute
thf(fact_6774_of__nat__mask__eq,axiom,
    ! [N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% of_nat_mask_eq
thf(fact_6775_of__nat__mask__eq,axiom,
    ! [N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se2002935070580805687sk_nat @ N2 ) )
      = ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% of_nat_mask_eq
thf(fact_6776_of__nat__and__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se727722235901077358nd_nat @ M @ N2 ) )
      = ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M ) @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_and_eq
thf(fact_6777_of__nat__and__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se727722235901077358nd_nat @ M @ N2 ) )
      = ( bit_se727722235901077358nd_nat @ ( semiri1316708129612266289at_nat @ M ) @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_and_eq
thf(fact_6778_exp__less__cancel,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( exp_real @ X4 ) @ ( exp_real @ Y ) )
     => ( ord_less_real @ X4 @ Y ) ) ).

% exp_less_cancel
thf(fact_6779_sgn__0__0,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_0_0
thf(fact_6780_sgn__0__0,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_0_0
thf(fact_6781_sgn__0__0,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_0_0
thf(fact_6782_sgn__0__0,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_0_0
thf(fact_6783_sgn__eq__0__iff,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = zero_z3403309356797280102nteger )
      = ( A = zero_z3403309356797280102nteger ) ) ).

% sgn_eq_0_iff
thf(fact_6784_sgn__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( ( sgn_sgn_complex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% sgn_eq_0_iff
thf(fact_6785_sgn__eq__0__iff,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% sgn_eq_0_iff
thf(fact_6786_sgn__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% sgn_eq_0_iff
thf(fact_6787_sgn__eq__0__iff,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% sgn_eq_0_iff
thf(fact_6788_sgn__mult,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( sgn_sgn_Code_integer @ ( times_3573771949741848930nteger @ A @ B ) )
      = ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ B ) ) ) ).

% sgn_mult
thf(fact_6789_sgn__mult,axiom,
    ! [A: rat,B: rat] :
      ( ( sgn_sgn_rat @ ( times_times_rat @ A @ B ) )
      = ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( sgn_sgn_rat @ B ) ) ) ).

% sgn_mult
thf(fact_6790_sgn__mult,axiom,
    ! [A: complex,B: complex] :
      ( ( sgn_sgn_complex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( sgn_sgn_complex @ B ) ) ) ).

% sgn_mult
thf(fact_6791_sgn__mult,axiom,
    ! [A: real,B: real] :
      ( ( sgn_sgn_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( sgn_sgn_real @ A ) @ ( sgn_sgn_real @ B ) ) ) ).

% sgn_mult
thf(fact_6792_sgn__mult,axiom,
    ! [A: int,B: int] :
      ( ( sgn_sgn_int @ ( times_times_int @ A @ B ) )
      = ( times_times_int @ ( sgn_sgn_int @ A ) @ ( sgn_sgn_int @ B ) ) ) ).

% sgn_mult
thf(fact_6793_same__sgn__sgn__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( sgn_sgn_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( sgn_sgn_Code_integer @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_6794_same__sgn__sgn__add,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
        = ( sgn_sgn_real @ A ) )
     => ( ( sgn_sgn_real @ ( plus_plus_real @ A @ B ) )
        = ( sgn_sgn_real @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_6795_same__sgn__sgn__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( sgn_sgn_rat @ ( plus_plus_rat @ A @ B ) )
        = ( sgn_sgn_rat @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_6796_same__sgn__sgn__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( sgn_sgn_int @ ( plus_plus_int @ A @ B ) )
        = ( sgn_sgn_int @ A ) ) ) ).

% same_sgn_sgn_add
thf(fact_6797_take__bit__not__eq__mask__diff,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ A ) )
      = ( minus_minus_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) ) ).

% take_bit_not_eq_mask_diff
thf(fact_6798_bit__and__iff,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ A @ B ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ A @ N2 )
        & ( bit_se1146084159140164899it_int @ B @ N2 ) ) ) ).

% bit_and_iff
thf(fact_6799_bit__and__iff,axiom,
    ! [A: nat,B: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se727722235901077358nd_nat @ A @ B ) @ N2 )
      = ( ( bit_se1148574629649215175it_nat @ A @ N2 )
        & ( bit_se1148574629649215175it_nat @ B @ N2 ) ) ) ).

% bit_and_iff
thf(fact_6800_bit_Oconj__xor__distrib,axiom,
    ! [X4: int,Y: int,Z: int] :
      ( ( bit_se725231765392027082nd_int @ X4 @ ( bit_se6526347334894502574or_int @ Y @ Z ) )
      = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ ( bit_se725231765392027082nd_int @ X4 @ Z ) ) ) ).

% bit.conj_xor_distrib
thf(fact_6801_bit_Oconj__xor__distrib2,axiom,
    ! [Y: int,Z: int,X4: int] :
      ( ( bit_se725231765392027082nd_int @ ( bit_se6526347334894502574or_int @ Y @ Z ) @ X4 )
      = ( bit_se6526347334894502574or_int @ ( bit_se725231765392027082nd_int @ Y @ X4 ) @ ( bit_se725231765392027082nd_int @ Z @ X4 ) ) ) ).

% bit.conj_xor_distrib2
thf(fact_6802_div__eq__sgn__abs,axiom,
    ! [K: int,L: int] :
      ( ( ( sgn_sgn_int @ K )
        = ( sgn_sgn_int @ L ) )
     => ( ( divide_divide_int @ K @ L )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_eq_sgn_abs
thf(fact_6803_take__bit__not__iff,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ A ) )
        = ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ B ) ) )
      = ( ( bit_se2923211474154528505it_int @ N2 @ A )
        = ( bit_se2923211474154528505it_int @ N2 @ B ) ) ) ).

% take_bit_not_iff
thf(fact_6804_take__bit__not__take__bit,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ ( bit_se2923211474154528505it_int @ N2 @ A ) ) )
      = ( bit_se2923211474154528505it_int @ N2 @ ( bit_ri7919022796975470100ot_int @ A ) ) ) ).

% take_bit_not_take_bit
thf(fact_6805_bit__not__int__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ K ) @ N2 )
      = ( ~ ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_not_int_iff
thf(fact_6806_bit__and__int__iff,axiom,
    ! [K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se725231765392027082nd_int @ K @ L ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
        & ( bit_se1146084159140164899it_int @ L @ N2 ) ) ) ).

% bit_and_int_iff
thf(fact_6807_less__eq__mask,axiom,
    ! [N2: nat] : ( ord_less_eq_nat @ N2 @ ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% less_eq_mask
thf(fact_6808_and__not__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = one_one_int ) ).

% and_not_numerals(2)
thf(fact_6809_and__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(4)
thf(fact_6810_take__bit__not__mask__eq__0,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( bit_se2923211474154528505it_int @ M @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) )
        = zero_zero_int ) ) ).

% take_bit_not_mask_eq_0
thf(fact_6811_and__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% and_not_numerals(5)
thf(fact_6812_and__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( numeral_numeral_int @ ( bit0 @ M ) ) ) ).

% and_not_numerals(7)
thf(fact_6813_and__not__numerals_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = zero_zero_int ) ).

% and_not_numerals(3)
thf(fact_6814_and__eq__minus__1__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ( bit_se3949692690581998587nteger @ A @ B )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ( A
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
        & ( B
          = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_6815_and__eq__minus__1__iff,axiom,
    ! [A: int,B: int] :
      ( ( ( bit_se725231765392027082nd_int @ A @ B )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ( A
          = ( uminus_uminus_int @ one_one_int ) )
        & ( B
          = ( uminus_uminus_int @ one_one_int ) ) ) ) ).

% and_eq_minus_1_iff
thf(fact_6816_exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X5: real] :
          ( ( exp_real @ X5 )
          = Y ) ) ).

% exp_total
thf(fact_6817_exp__gt__zero,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).

% exp_gt_zero
thf(fact_6818_not__exp__less__zero,axiom,
    ! [X4: real] :
      ~ ( ord_less_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).

% not_exp_less_zero
thf(fact_6819_exp__ge__zero,axiom,
    ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( exp_real @ X4 ) ) ).

% exp_ge_zero
thf(fact_6820_not__exp__le__zero,axiom,
    ! [X4: real] :
      ~ ( ord_less_eq_real @ ( exp_real @ X4 ) @ zero_zero_real ) ).

% not_exp_le_zero
thf(fact_6821_sgn__not__eq__imp,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
       != ( sgn_sgn_real @ A ) )
     => ( ( ( sgn_sgn_real @ A )
         != zero_zero_real )
       => ( ( ( sgn_sgn_real @ B )
           != zero_zero_real )
         => ( ( sgn_sgn_real @ A )
            = ( uminus_uminus_real @ ( sgn_sgn_real @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6822_sgn__not__eq__imp,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
       != ( sgn_sgn_int @ A ) )
     => ( ( ( sgn_sgn_int @ A )
         != zero_zero_int )
       => ( ( ( sgn_sgn_int @ B )
           != zero_zero_int )
         => ( ( sgn_sgn_int @ A )
            = ( uminus_uminus_int @ ( sgn_sgn_int @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6823_sgn__not__eq__imp,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
       != ( sgn_sgn_Code_integer @ A ) )
     => ( ( ( sgn_sgn_Code_integer @ A )
         != zero_z3403309356797280102nteger )
       => ( ( ( sgn_sgn_Code_integer @ B )
           != zero_z3403309356797280102nteger )
         => ( ( sgn_sgn_Code_integer @ A )
            = ( uminus1351360451143612070nteger @ ( sgn_sgn_Code_integer @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6824_sgn__not__eq__imp,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
       != ( sgn_sgn_rat @ A ) )
     => ( ( ( sgn_sgn_rat @ A )
         != zero_zero_rat )
       => ( ( ( sgn_sgn_rat @ B )
           != zero_zero_rat )
         => ( ( sgn_sgn_rat @ A )
            = ( uminus_uminus_rat @ ( sgn_sgn_rat @ B ) ) ) ) ) ) ).

% sgn_not_eq_imp
thf(fact_6825_sgn__minus__1,axiom,
    ( ( sgn_sgn_real @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sgn_minus_1
thf(fact_6826_sgn__minus__1,axiom,
    ( ( sgn_sgn_int @ ( uminus_uminus_int @ one_one_int ) )
    = ( uminus_uminus_int @ one_one_int ) ) ).

% sgn_minus_1
thf(fact_6827_sgn__minus__1,axiom,
    ( ( sgn_sgn_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% sgn_minus_1
thf(fact_6828_sgn__minus__1,axiom,
    ( ( sgn_sgn_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
    = ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ).

% sgn_minus_1
thf(fact_6829_sgn__minus__1,axiom,
    ( ( sgn_sgn_rat @ ( uminus_uminus_rat @ one_one_rat ) )
    = ( uminus_uminus_rat @ one_one_rat ) ) ).

% sgn_minus_1
thf(fact_6830_AND__upper2_H,axiom,
    ! [Y: int,Z: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ Z ) ) ) ).

% AND_upper2'
thf(fact_6831_AND__upper1_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_eq_int @ Y @ Z )
       => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1'
thf(fact_6832_AND__upper2,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ Y ) ) ).

% AND_upper2
thf(fact_6833_AND__upper1,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ord_less_eq_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ X4 ) ) ).

% AND_upper1
thf(fact_6834_AND__lower,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ord_less_eq_int @ zero_zero_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) ) ) ).

% AND_lower
thf(fact_6835_not__diff__distrib,axiom,
    ! [A: int,B: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A @ B ) )
      = ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).

% not_diff_distrib
thf(fact_6836_not__add__distrib,axiom,
    ! [A: int,B: int] :
      ( ( bit_ri7919022796975470100ot_int @ ( plus_plus_int @ A @ B ) )
      = ( minus_minus_int @ ( bit_ri7919022796975470100ot_int @ A ) @ B ) ) ).

% not_add_distrib
thf(fact_6837_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_Code_integer
    = ( ^ [K3: code_integer] : ( times_3573771949741848930nteger @ K3 @ ( sgn_sgn_Code_integer @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_6838_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_rat
    = ( ^ [K3: rat] : ( times_times_rat @ K3 @ ( sgn_sgn_rat @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_6839_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_real
    = ( ^ [K3: real] : ( times_times_real @ K3 @ ( sgn_sgn_real @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_6840_linordered__idom__class_Oabs__sgn,axiom,
    ( abs_abs_int
    = ( ^ [K3: int] : ( times_times_int @ K3 @ ( sgn_sgn_int @ K3 ) ) ) ) ).

% linordered_idom_class.abs_sgn
thf(fact_6841_abs__mult__sgn,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( abs_abs_Code_integer @ A ) @ ( sgn_sgn_Code_integer @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6842_abs__mult__sgn,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( abs_abs_rat @ A ) @ ( sgn_sgn_rat @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6843_abs__mult__sgn,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( abs_abs_complex @ A ) @ ( sgn_sgn_complex @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6844_abs__mult__sgn,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( abs_abs_real @ A ) @ ( sgn_sgn_real @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6845_abs__mult__sgn,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( abs_abs_int @ A ) @ ( sgn_sgn_int @ A ) )
      = A ) ).

% abs_mult_sgn
thf(fact_6846_sgn__mult__abs,axiom,
    ! [A: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ A ) @ ( abs_abs_Code_integer @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6847_sgn__mult__abs,axiom,
    ! [A: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ A ) @ ( abs_abs_rat @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6848_sgn__mult__abs,axiom,
    ! [A: complex] :
      ( ( times_times_complex @ ( sgn_sgn_complex @ A ) @ ( abs_abs_complex @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6849_sgn__mult__abs,axiom,
    ! [A: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( abs_abs_real @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6850_sgn__mult__abs,axiom,
    ! [A: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ A ) @ ( abs_abs_int @ A ) )
      = A ) ).

% sgn_mult_abs
thf(fact_6851_mult__sgn__abs,axiom,
    ! [X4: code_integer] :
      ( ( times_3573771949741848930nteger @ ( sgn_sgn_Code_integer @ X4 ) @ ( abs_abs_Code_integer @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_6852_mult__sgn__abs,axiom,
    ! [X4: rat] :
      ( ( times_times_rat @ ( sgn_sgn_rat @ X4 ) @ ( abs_abs_rat @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_6853_mult__sgn__abs,axiom,
    ! [X4: real] :
      ( ( times_times_real @ ( sgn_sgn_real @ X4 ) @ ( abs_abs_real @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_6854_mult__sgn__abs,axiom,
    ! [X4: int] :
      ( ( times_times_int @ ( sgn_sgn_int @ X4 ) @ ( abs_abs_int @ X4 ) )
      = X4 ) ).

% mult_sgn_abs
thf(fact_6855_same__sgn__abs__add,axiom,
    ! [B: code_integer,A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ B )
        = ( sgn_sgn_Code_integer @ A ) )
     => ( ( abs_abs_Code_integer @ ( plus_p5714425477246183910nteger @ A @ B ) )
        = ( plus_p5714425477246183910nteger @ ( abs_abs_Code_integer @ A ) @ ( abs_abs_Code_integer @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_6856_same__sgn__abs__add,axiom,
    ! [B: real,A: real] :
      ( ( ( sgn_sgn_real @ B )
        = ( sgn_sgn_real @ A ) )
     => ( ( abs_abs_real @ ( plus_plus_real @ A @ B ) )
        = ( plus_plus_real @ ( abs_abs_real @ A ) @ ( abs_abs_real @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_6857_same__sgn__abs__add,axiom,
    ! [B: rat,A: rat] :
      ( ( ( sgn_sgn_rat @ B )
        = ( sgn_sgn_rat @ A ) )
     => ( ( abs_abs_rat @ ( plus_plus_rat @ A @ B ) )
        = ( plus_plus_rat @ ( abs_abs_rat @ A ) @ ( abs_abs_rat @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_6858_same__sgn__abs__add,axiom,
    ! [B: int,A: int] :
      ( ( ( sgn_sgn_int @ B )
        = ( sgn_sgn_int @ A ) )
     => ( ( abs_abs_int @ ( plus_plus_int @ A @ B ) )
        = ( plus_plus_int @ ( abs_abs_int @ A ) @ ( abs_abs_int @ B ) ) ) ) ).

% same_sgn_abs_add
thf(fact_6859_exp__add__commuting,axiom,
    ! [X4: complex,Y: complex] :
      ( ( ( times_times_complex @ X4 @ Y )
        = ( times_times_complex @ Y @ X4 ) )
     => ( ( exp_complex @ ( plus_plus_complex @ X4 @ Y ) )
        = ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_6860_exp__add__commuting,axiom,
    ! [X4: real,Y: real] :
      ( ( ( times_times_real @ X4 @ Y )
        = ( times_times_real @ Y @ X4 ) )
     => ( ( exp_real @ ( plus_plus_real @ X4 @ Y ) )
        = ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ Y ) ) ) ) ).

% exp_add_commuting
thf(fact_6861_mult__exp__exp,axiom,
    ! [X4: complex,Y: complex] :
      ( ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y ) )
      = ( exp_complex @ ( plus_plus_complex @ X4 @ Y ) ) ) ).

% mult_exp_exp
thf(fact_6862_mult__exp__exp,axiom,
    ! [X4: real,Y: real] :
      ( ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ Y ) )
      = ( exp_real @ ( plus_plus_real @ X4 @ Y ) ) ) ).

% mult_exp_exp
thf(fact_6863_exp__diff,axiom,
    ! [X4: real,Y: real] :
      ( ( exp_real @ ( minus_minus_real @ X4 @ Y ) )
      = ( divide_divide_real @ ( exp_real @ X4 ) @ ( exp_real @ Y ) ) ) ).

% exp_diff
thf(fact_6864_exp__diff,axiom,
    ! [X4: complex,Y: complex] :
      ( ( exp_complex @ ( minus_minus_complex @ X4 @ Y ) )
      = ( divide1717551699836669952omplex @ ( exp_complex @ X4 ) @ ( exp_complex @ Y ) ) ) ).

% exp_diff
thf(fact_6865_and__not__numerals_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% and_not_numerals(6)
thf(fact_6866_and__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% and_not_numerals(9)
thf(fact_6867_mask__nonnegative__int,axiom,
    ! [N2: nat] : ( ord_less_eq_int @ zero_zero_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% mask_nonnegative_int
thf(fact_6868_not__mask__negative__int,axiom,
    ! [N2: nat] :
      ~ ( ord_less_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ zero_zero_int ) ).

% not_mask_negative_int
thf(fact_6869_minus__exp__eq__not__mask,axiom,
    ! [N2: nat] :
      ( ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) )
      = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N2 ) ) ) ).

% minus_exp_eq_not_mask
thf(fact_6870_minus__exp__eq__not__mask,axiom,
    ! [N2: nat] :
      ( ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ).

% minus_exp_eq_not_mask
thf(fact_6871_exp__gt__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ one_one_real @ ( exp_real @ X4 ) ) ) ).

% exp_gt_one
thf(fact_6872_sgn__1__pos,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = one_one_Code_integer )
      = ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ A ) ) ).

% sgn_1_pos
thf(fact_6873_sgn__1__pos,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = one_one_real )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% sgn_1_pos
thf(fact_6874_sgn__1__pos,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = one_one_rat )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% sgn_1_pos
thf(fact_6875_sgn__1__pos,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = one_one_int )
      = ( ord_less_int @ zero_zero_int @ A ) ) ).

% sgn_1_pos
thf(fact_6876_exp__ge__add__one__self,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ).

% exp_ge_add_one_self
thf(fact_6877_abs__sgn__eq,axiom,
    ! [A: code_integer] :
      ( ( ( A = zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
          = zero_z3403309356797280102nteger ) )
      & ( ( A != zero_z3403309356797280102nteger )
       => ( ( abs_abs_Code_integer @ ( sgn_sgn_Code_integer @ A ) )
          = one_one_Code_integer ) ) ) ).

% abs_sgn_eq
thf(fact_6878_abs__sgn__eq,axiom,
    ! [A: real] :
      ( ( ( A = zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = zero_zero_real ) )
      & ( ( A != zero_zero_real )
       => ( ( abs_abs_real @ ( sgn_sgn_real @ A ) )
          = one_one_real ) ) ) ).

% abs_sgn_eq
thf(fact_6879_abs__sgn__eq,axiom,
    ! [A: rat] :
      ( ( ( A = zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
          = zero_zero_rat ) )
      & ( ( A != zero_zero_rat )
       => ( ( abs_abs_rat @ ( sgn_sgn_rat @ A ) )
          = one_one_rat ) ) ) ).

% abs_sgn_eq
thf(fact_6880_abs__sgn__eq,axiom,
    ! [A: int] :
      ( ( ( A = zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = zero_zero_int ) )
      & ( ( A != zero_zero_int )
       => ( ( abs_abs_int @ ( sgn_sgn_int @ A ) )
          = one_one_int ) ) ) ).

% abs_sgn_eq
thf(fact_6881_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_6882_AND__upper1_H_H,axiom,
    ! [Y: int,Z: int,Ya: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ Y @ Ya ) @ Z ) ) ) ).

% AND_upper1''
thf(fact_6883_AND__upper2_H_H,axiom,
    ! [Y: int,Z: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( ord_less_int @ Y @ Z )
       => ( ord_less_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ Z ) ) ) ).

% AND_upper2''
thf(fact_6884_minus__eq__not__plus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A3: code_integer] : ( plus_p5714425477246183910nteger @ ( bit_ri7632146776885996613nteger @ A3 ) @ one_one_Code_integer ) ) ) ).

% minus_eq_not_plus_1
thf(fact_6885_minus__eq__not__plus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A3: int] : ( plus_plus_int @ ( bit_ri7919022796975470100ot_int @ A3 ) @ one_one_int ) ) ) ).

% minus_eq_not_plus_1
thf(fact_6886_not__eq__complement,axiom,
    ( bit_ri7632146776885996613nteger
    = ( ^ [A3: code_integer] : ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ A3 ) @ one_one_Code_integer ) ) ) ).

% not_eq_complement
thf(fact_6887_not__eq__complement,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [A3: int] : ( minus_minus_int @ ( uminus_uminus_int @ A3 ) @ one_one_int ) ) ) ).

% not_eq_complement
thf(fact_6888_minus__eq__not__minus__1,axiom,
    ( uminus1351360451143612070nteger
    = ( ^ [A3: code_integer] : ( bit_ri7632146776885996613nteger @ ( minus_8373710615458151222nteger @ A3 @ one_one_Code_integer ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_6889_minus__eq__not__minus__1,axiom,
    ( uminus_uminus_int
    = ( ^ [A3: int] : ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ A3 @ one_one_int ) ) ) ) ).

% minus_eq_not_minus_1
thf(fact_6890_div__sgn__abs__cancel,axiom,
    ! [V: int,K: int,L: int] :
      ( ( V != zero_zero_int )
     => ( ( divide_divide_int @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ K ) ) @ ( times_times_int @ ( sgn_sgn_int @ V ) @ ( abs_abs_int @ L ) ) )
        = ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ).

% div_sgn_abs_cancel
thf(fact_6891_not__int__def,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( minus_minus_int @ ( uminus_uminus_int @ K3 ) @ one_one_int ) ) ) ).

% not_int_def
thf(fact_6892_exp__minus__inverse,axiom,
    ! [X4: real] :
      ( ( times_times_real @ ( exp_real @ X4 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) )
      = one_one_real ) ).

% exp_minus_inverse
thf(fact_6893_exp__minus__inverse,axiom,
    ! [X4: complex] :
      ( ( times_times_complex @ ( exp_complex @ X4 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X4 ) ) )
      = one_one_complex ) ).

% exp_minus_inverse
thf(fact_6894_exp__of__nat2__mult,axiom,
    ! [X4: complex,N2: nat] :
      ( ( exp_complex @ ( times_times_complex @ X4 @ ( semiri8010041392384452111omplex @ N2 ) ) )
      = ( power_power_complex @ ( exp_complex @ X4 ) @ N2 ) ) ).

% exp_of_nat2_mult
thf(fact_6895_exp__of__nat2__mult,axiom,
    ! [X4: real,N2: nat] :
      ( ( exp_real @ ( times_times_real @ X4 @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = ( power_power_real @ ( exp_real @ X4 ) @ N2 ) ) ).

% exp_of_nat2_mult
thf(fact_6896_exp__of__nat__mult,axiom,
    ! [N2: nat,X4: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ X4 ) )
      = ( power_power_complex @ ( exp_complex @ X4 ) @ N2 ) ) ).

% exp_of_nat_mult
thf(fact_6897_exp__of__nat__mult,axiom,
    ! [N2: nat,X4: real] :
      ( ( exp_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ X4 ) )
      = ( power_power_real @ ( exp_real @ X4 ) @ N2 ) ) ).

% exp_of_nat_mult
thf(fact_6898_div__dvd__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( dvd_dvd_int @ L @ K )
     => ( ( divide_divide_int @ K @ L )
        = ( times_times_int @ ( times_times_int @ ( sgn_sgn_int @ K ) @ ( sgn_sgn_int @ L ) ) @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_dvd_sgn_abs
thf(fact_6899_sgn__mod,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ~ ( dvd_dvd_int @ L @ K )
       => ( ( sgn_sgn_int @ ( modulo_modulo_int @ K @ L ) )
          = ( sgn_sgn_int @ L ) ) ) ) ).

% sgn_mod
thf(fact_6900_and__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% and_not_numerals(8)
thf(fact_6901_minus__numeral__inc__eq,axiom,
    ! [N2: num] :
      ( ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( inc @ N2 ) ) )
      = ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ N2 ) ) ) ).

% minus_numeral_inc_eq
thf(fact_6902_minus__numeral__inc__eq,axiom,
    ! [N2: num] :
      ( ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ N2 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% minus_numeral_inc_eq
thf(fact_6903_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_6904_even__and__iff,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se3949692690581998587nteger @ A @ B ) )
      = ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_6905_even__and__iff,axiom,
    ! [A: int,B: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_6906_even__and__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ B ) ) ) ).

% even_and_iff
thf(fact_6907_sgn__1__neg,axiom,
    ! [A: real] :
      ( ( ( sgn_sgn_real @ A )
        = ( uminus_uminus_real @ one_one_real ) )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% sgn_1_neg
thf(fact_6908_sgn__1__neg,axiom,
    ! [A: int] :
      ( ( ( sgn_sgn_int @ A )
        = ( uminus_uminus_int @ one_one_int ) )
      = ( ord_less_int @ A @ zero_zero_int ) ) ).

% sgn_1_neg
thf(fact_6909_sgn__1__neg,axiom,
    ! [A: code_integer] :
      ( ( ( sgn_sgn_Code_integer @ A )
        = ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( ord_le6747313008572928689nteger @ A @ zero_z3403309356797280102nteger ) ) ).

% sgn_1_neg
thf(fact_6910_sgn__1__neg,axiom,
    ! [A: rat] :
      ( ( ( sgn_sgn_rat @ A )
        = ( uminus_uminus_rat @ one_one_rat ) )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% sgn_1_neg
thf(fact_6911_sgn__if,axiom,
    ( sgn_sgn_real
    = ( ^ [X: real] : ( if_real @ ( X = zero_zero_real ) @ zero_zero_real @ ( if_real @ ( ord_less_real @ zero_zero_real @ X ) @ one_one_real @ ( uminus_uminus_real @ one_one_real ) ) ) ) ) ).

% sgn_if
thf(fact_6912_sgn__if,axiom,
    ( sgn_sgn_int
    = ( ^ [X: int] : ( if_int @ ( X = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ X ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% sgn_if
thf(fact_6913_sgn__if,axiom,
    ( sgn_sgn_Code_integer
    = ( ^ [X: code_integer] : ( if_Code_integer @ ( X = zero_z3403309356797280102nteger ) @ zero_z3403309356797280102nteger @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ X ) @ one_one_Code_integer @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) ) ) ).

% sgn_if
thf(fact_6914_sgn__if,axiom,
    ( sgn_sgn_rat
    = ( ^ [X: rat] : ( if_rat @ ( X = zero_zero_rat ) @ zero_zero_rat @ ( if_rat @ ( ord_less_rat @ zero_zero_rat @ X ) @ one_one_rat @ ( uminus_uminus_rat @ one_one_rat ) ) ) ) ) ).

% sgn_if
thf(fact_6915_not__int__div__2,axiom,
    ! [K: int] :
      ( ( divide_divide_int @ ( bit_ri7919022796975470100ot_int @ K ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% not_int_div_2
thf(fact_6916_even__not__iff__int,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_not_iff_int
thf(fact_6917_exp__ge__add__one__self__aux,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( exp_real @ X4 ) ) ) ).

% exp_ge_add_one_self_aux
thf(fact_6918_even__and__iff__int,axiom,
    ! [K: int,L: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ L ) ) ) ).

% even_and_iff_int
thf(fact_6919_lemma__exp__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ Y )
     => ? [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
          & ( ord_less_eq_real @ X5 @ ( minus_minus_real @ Y @ one_one_real ) )
          & ( ( exp_real @ X5 )
            = Y ) ) ) ).

% lemma_exp_total
thf(fact_6920_ln__ge__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ Y @ ( ln_ln_real @ X4 ) )
        = ( ord_less_eq_real @ ( exp_real @ Y ) @ X4 ) ) ) ).

% ln_ge_iff
thf(fact_6921_zsgn__def,axiom,
    ( sgn_sgn_int
    = ( ^ [I3: int] : ( if_int @ ( I3 = zero_zero_int ) @ zero_zero_int @ ( if_int @ ( ord_less_int @ zero_zero_int @ I3 ) @ one_one_int @ ( uminus_uminus_int @ one_one_int ) ) ) ) ) ).

% zsgn_def
thf(fact_6922_ln__x__over__x__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( exp_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ord_less_eq_real @ ( divide_divide_real @ ( ln_ln_real @ Y ) @ Y ) @ ( divide_divide_real @ ( ln_ln_real @ X4 ) @ X4 ) ) ) ) ).

% ln_x_over_x_mono
thf(fact_6923_not__numeral__Bit0__eq,axiom,
    ! [N2: num] :
      ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit1 @ N2 ) ) ) ) ).

% not_numeral_Bit0_eq
thf(fact_6924_not__numeral__Bit0__eq,axiom,
    ! [N2: num] :
      ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).

% not_numeral_Bit0_eq
thf(fact_6925_norm__sgn,axiom,
    ! [X4: real] :
      ( ( ( X4 = zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X4 ) )
          = zero_zero_real ) )
      & ( ( X4 != zero_zero_real )
       => ( ( real_V7735802525324610683m_real @ ( sgn_sgn_real @ X4 ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_6926_norm__sgn,axiom,
    ! [X4: complex] :
      ( ( ( X4 = zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X4 ) )
          = zero_zero_real ) )
      & ( ( X4 != zero_zero_complex )
       => ( ( real_V1022390504157884413omplex @ ( sgn_sgn_complex @ X4 ) )
          = one_one_real ) ) ) ).

% norm_sgn
thf(fact_6927_bit__minus__int__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( uminus_uminus_int @ K ) @ N2 )
      = ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ ( minus_minus_int @ K @ one_one_int ) ) @ N2 ) ) ).

% bit_minus_int_iff
thf(fact_6928_not__numeral__BitM__eq,axiom,
    ! [N2: num] :
      ( ( bit_ri7632146776885996613nteger @ ( numera6620942414471956472nteger @ ( bitM @ N2 ) ) )
      = ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ N2 ) ) ) ) ).

% not_numeral_BitM_eq
thf(fact_6929_not__numeral__BitM__eq,axiom,
    ! [N2: num] :
      ( ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bitM @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).

% not_numeral_BitM_eq
thf(fact_6930_one__and__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ one_one_Code_integer @ A )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6931_one__and__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ one_one_int @ A )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6932_one__and__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ one_one_nat @ A )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% one_and_eq
thf(fact_6933_and__one__eq,axiom,
    ! [A: code_integer] :
      ( ( bit_se3949692690581998587nteger @ A @ one_one_Code_integer )
      = ( modulo364778990260209775nteger @ A @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6934_and__one__eq,axiom,
    ! [A: int] :
      ( ( bit_se725231765392027082nd_int @ A @ one_one_int )
      = ( modulo_modulo_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6935_and__one__eq,axiom,
    ! [A: nat] :
      ( ( bit_se727722235901077358nd_nat @ A @ one_one_nat )
      = ( modulo_modulo_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% and_one_eq
thf(fact_6936_exp__le,axiom,
    ord_less_eq_real @ ( exp_real @ one_one_real ) @ ( numeral_numeral_real @ ( bit1 @ one ) ) ).

% exp_le
thf(fact_6937_take__bit__eq__mask__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( ( bit_se2923211474154528505it_int @ N2 @ ( plus_plus_int @ K @ one_one_int ) )
        = zero_zero_int ) ) ).

% take_bit_eq_mask_iff
thf(fact_6938_exp__divide__power__eq,axiom,
    ! [N2: nat,X4: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_complex @ ( exp_complex @ ( divide1717551699836669952omplex @ X4 @ ( semiri8010041392384452111omplex @ N2 ) ) ) @ N2 )
        = ( exp_complex @ X4 ) ) ) ).

% exp_divide_power_eq
thf(fact_6939_exp__divide__power__eq,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( power_power_real @ ( exp_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 )
        = ( exp_real @ X4 ) ) ) ).

% exp_divide_power_eq
thf(fact_6940_eucl__rel__int__remainderI,axiom,
    ! [R3: int,L: int,K: int,Q3: int] :
      ( ( ( sgn_sgn_int @ R3 )
        = ( sgn_sgn_int @ L ) )
     => ( ( ord_less_int @ ( abs_abs_int @ R3 ) @ ( abs_abs_int @ L ) )
       => ( ( K
            = ( plus_plus_int @ ( times_times_int @ Q3 @ L ) @ R3 ) )
         => ( eucl_rel_int @ K @ L @ ( product_Pair_int_int @ Q3 @ R3 ) ) ) ) ) ).

% eucl_rel_int_remainderI
thf(fact_6941_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_6942_tanh__altdef,axiom,
    ( tanh_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).

% tanh_altdef
thf(fact_6943_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_6944_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_6945_exp__half__le2,axiom,
    ord_less_eq_real @ ( exp_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% exp_half_le2
thf(fact_6946_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_6947_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_6948_bit__not__iff__eq,axiom,
    ! [A: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_ri7919022796975470100ot_int @ A ) @ N2 )
      = ( ( ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
         != zero_zero_int )
        & ~ ( bit_se1146084159140164899it_int @ A @ N2 ) ) ) ).

% bit_not_iff_eq
thf(fact_6949_exp__double,axiom,
    ! [Z: complex] :
      ( ( exp_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_complex @ ( exp_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_6950_exp__double,axiom,
    ! [Z: real] :
      ( ( exp_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ Z ) )
      = ( power_power_real @ ( exp_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% exp_double
thf(fact_6951_eucl__rel__int_Osimps,axiom,
    ( eucl_rel_int
    = ( ^ [A1: int,A22: int,A32: product_prod_int_int] :
          ( ? [K3: int] :
              ( ( A1 = K3 )
              & ( A22 = zero_zero_int )
              & ( A32
                = ( product_Pair_int_int @ zero_zero_int @ K3 ) ) )
          | ? [L2: int,K3: int,Q5: int] :
              ( ( A1 = K3 )
              & ( A22 = L2 )
              & ( A32
                = ( 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 )
              & ( A32
                = ( 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_6952_eucl__rel__int_Ocases,axiom,
    ! [A12: int,A23: int,A33: product_prod_int_int] :
      ( ( eucl_rel_int @ A12 @ A23 @ A33 )
     => ( ( ( A23 = zero_zero_int )
         => ( A33
           != ( product_Pair_int_int @ zero_zero_int @ A12 ) ) )
       => ( ! [Q2: int] :
              ( ( A33
                = ( product_Pair_int_int @ Q2 @ zero_zero_int ) )
             => ( ( A23 != zero_zero_int )
               => ( A12
                 != ( times_times_int @ Q2 @ A23 ) ) ) )
         => ~ ! [R2: int,Q2: int] :
                ( ( A33
                  = ( product_Pair_int_int @ Q2 @ R2 ) )
               => ( ( ( sgn_sgn_int @ R2 )
                    = ( sgn_sgn_int @ A23 ) )
                 => ( ( ord_less_int @ ( abs_abs_int @ R2 ) @ ( abs_abs_int @ A23 ) )
                   => ( A12
                     != ( plus_plus_int @ ( times_times_int @ Q2 @ A23 ) @ R2 ) ) ) ) ) ) ) ) ).

% eucl_rel_int.cases
thf(fact_6953_div__noneq__sgn__abs,axiom,
    ! [L: int,K: int] :
      ( ( L != zero_zero_int )
     => ( ( ( sgn_sgn_int @ K )
         != ( sgn_sgn_int @ L ) )
       => ( ( divide_divide_int @ K @ L )
          = ( minus_minus_int @ ( uminus_uminus_int @ ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) ) )
            @ ( zero_n2684676970156552555ol_int
              @ ~ ( dvd_dvd_int @ L @ K ) ) ) ) ) ) ).

% div_noneq_sgn_abs
thf(fact_6954_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_6955_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_6956_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_6957_mask__half__int,axiom,
    ! [N2: nat] :
      ( ( divide_divide_int @ ( bit_se2000444600071755411sk_int @ N2 ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
      = ( bit_se2000444600071755411sk_int @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ).

% mask_half_int
thf(fact_6958_mask__nat__def,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ).

% mask_nat_def
thf(fact_6959_mask__int__def,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ).

% mask_int_def
thf(fact_6960_exp__bound__half,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_6961_exp__bound__half,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% exp_bound_half
thf(fact_6962_mask__eq__exp__minus__1,axiom,
    ( bit_se2002935070580805687sk_nat
    = ( ^ [N: nat] : ( minus_minus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_6963_mask__eq__exp__minus__1,axiom,
    ( bit_se2000444600071755411sk_int
    = ( ^ [N: nat] : ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) @ one_one_int ) ) ) ).

% mask_eq_exp_minus_1
thf(fact_6964_exp__bound,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% exp_bound
thf(fact_6965_not__int__rec,axiom,
    ( bit_ri7919022796975470100ot_int
    = ( ^ [K3: int] : ( plus_plus_int @ ( zero_n2684676970156552555ol_int @ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K3 ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_ri7919022796975470100ot_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% not_int_rec
thf(fact_6966_and__int__rec,axiom,
    ( bit_se725231765392027082nd_int
    = ( ^ [K3: int,L2: int] :
          ( 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_rec
thf(fact_6967_real__exp__bound__lemma,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ ( exp_real @ X4 ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) ) ) ) ).

% real_exp_bound_lemma
thf(fact_6968_exp__ge__one__plus__x__over__n__power__n,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ X4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ord_less_eq_real @ ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ X4 ) ) ) ) ).

% exp_ge_one_plus_x_over_n_power_n
thf(fact_6969_exp__ge__one__minus__x__over__n__power__n,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_eq_real @ X4 @ ( 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 @ X4 @ ( semiri5074537144036343181t_real @ N2 ) ) ) @ N2 ) @ ( exp_real @ ( uminus_uminus_real @ X4 ) ) ) ) ) ).

% exp_ge_one_minus_x_over_n_power_n
thf(fact_6970_exp__bound__lemma,axiom,
    ! [Z: real] :
      ( ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( exp_real @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V7735802525324610683m_real @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_6971_exp__bound__lemma,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( exp_complex @ Z ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ) ).

% exp_bound_lemma
thf(fact_6972_take__bit__eq__mask__iff__exp__dvd,axiom,
    ! [N2: nat,K: int] :
      ( ( ( bit_se2923211474154528505it_int @ N2 @ K )
        = ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ ( plus_plus_int @ K @ one_one_int ) ) ) ).

% take_bit_eq_mask_iff_exp_dvd
thf(fact_6973_exp__lower__Taylor__quadratic,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X4 ) @ ( divide_divide_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( exp_real @ X4 ) ) ) ).

% exp_lower_Taylor_quadratic
thf(fact_6974_log__base__10__eq1,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
        = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( ln_ln_real @ X4 ) ) ) ) ).

% log_base_10_eq1
thf(fact_6975_modulo__int__def,axiom,
    ( modulo_modulo_int
    = ( ^ [K3: int,L2: int] :
          ( if_int @ ( L2 = zero_zero_int ) @ K3
          @ ( if_int
            @ ( ( sgn_sgn_int @ K3 )
              = ( sgn_sgn_int @ L2 ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 ) @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) )
            @ ( times_times_int @ ( sgn_sgn_int @ L2 )
              @ ( minus_minus_int
                @ ( times_times_int @ ( abs_abs_int @ L2 )
                  @ ( zero_n2684676970156552555ol_int
                    @ ~ ( dvd_dvd_int @ L2 @ K3 ) ) )
                @ ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K3 ) ) @ ( nat2 @ ( abs_abs_int @ L2 ) ) ) ) ) ) ) ) ) ) ).

% modulo_int_def
thf(fact_6976_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_6977_arctan__half,axiom,
    ( arctan
    = ( ^ [X: real] : ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ X @ ( plus_plus_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% arctan_half
thf(fact_6978_log__base__10__eq2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ X4 )
        = ( times_times_real @ ( log @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ ( bit0 @ one ) ) ) ) @ ( exp_real @ one_one_real ) ) @ ( ln_ln_real @ X4 ) ) ) ) ).

% log_base_10_eq2
thf(fact_6979_machin,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% machin
thf(fact_6980_zero__le__sgn__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sgn_sgn_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% zero_le_sgn_iff
thf(fact_6981_sgn__le__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sgn_sgn_real @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% sgn_le_0_iff
thf(fact_6982_real__sqrt__less__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( sqrt @ X4 ) @ ( sqrt @ Y ) )
      = ( ord_less_real @ X4 @ Y ) ) ).

% real_sqrt_less_iff
thf(fact_6983_real__sqrt__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% real_sqrt_le_iff
thf(fact_6984_real__sqrt__one,axiom,
    ( ( sqrt @ one_one_real )
    = one_one_real ) ).

% real_sqrt_one
thf(fact_6985_real__sqrt__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( sqrt @ X4 )
        = one_one_real )
      = ( X4 = one_one_real ) ) ).

% real_sqrt_eq_1_iff
thf(fact_6986_real__sqrt__lt__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( sqrt @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% real_sqrt_lt_0_iff
thf(fact_6987_real__sqrt__gt__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_gt_0_iff
thf(fact_6988_nat__numeral,axiom,
    ! [K: num] :
      ( ( nat2 @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_numeral
thf(fact_6989_real__sqrt__le__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% real_sqrt_le_0_iff
thf(fact_6990_real__sqrt__ge__0__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ).

% real_sqrt_ge_0_iff
thf(fact_6991_real__sqrt__lt__1__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( sqrt @ X4 ) @ one_one_real )
      = ( ord_less_real @ X4 @ one_one_real ) ) ).

% real_sqrt_lt_1_iff
thf(fact_6992_real__sqrt__gt__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_real @ one_one_real @ Y ) ) ).

% real_sqrt_gt_1_iff
thf(fact_6993_real__sqrt__le__1__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ one_one_real )
      = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).

% real_sqrt_le_1_iff
thf(fact_6994_real__sqrt__ge__1__iff,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( sqrt @ Y ) )
      = ( ord_less_eq_real @ one_one_real @ Y ) ) ).

% real_sqrt_ge_1_iff
thf(fact_6995_log__one,axiom,
    ! [A: real] :
      ( ( log @ A @ one_one_real )
      = zero_zero_real ) ).

% log_one
thf(fact_6996_real__sqrt__four,axiom,
    ( ( sqrt @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) )
    = ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% real_sqrt_four
thf(fact_6997_nat__1,axiom,
    ( ( nat2 @ one_one_int )
    = ( suc @ zero_zero_nat ) ) ).

% nat_1
thf(fact_6998_nat__0__iff,axiom,
    ! [I2: int] :
      ( ( ( nat2 @ I2 )
        = zero_zero_nat )
      = ( ord_less_eq_int @ I2 @ zero_zero_int ) ) ).

% nat_0_iff
thf(fact_6999_nat__le__0,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ Z )
        = zero_zero_nat ) ) ).

% nat_le_0
thf(fact_7000_zless__nat__conj,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
      = ( ( ord_less_int @ zero_zero_int @ Z )
        & ( ord_less_int @ W @ Z ) ) ) ).

% zless_nat_conj
thf(fact_7001_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_7002_log__less__cancel__iff,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) )
            = ( ord_less_real @ X4 @ Y ) ) ) ) ) ).

% log_less_cancel_iff
thf(fact_7003_log__less__one__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( log @ A @ X4 ) @ one_one_real )
          = ( ord_less_real @ X4 @ A ) ) ) ) ).

% log_less_one_cancel_iff
thf(fact_7004_one__less__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ one_one_real @ ( log @ A @ X4 ) )
          = ( ord_less_real @ A @ X4 ) ) ) ) ).

% one_less_log_cancel_iff
thf(fact_7005_log__less__zero__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( log @ A @ X4 ) @ zero_zero_real )
          = ( ord_less_real @ X4 @ one_one_real ) ) ) ) ).

% log_less_zero_cancel_iff
thf(fact_7006_zero__less__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ zero_zero_real @ ( log @ A @ X4 ) )
          = ( ord_less_real @ one_one_real @ X4 ) ) ) ) ).

% zero_less_log_cancel_iff
thf(fact_7007_int__nat__eq,axiom,
    ! [Z: int] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = Z ) )
      & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
       => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
          = zero_zero_int ) ) ) ).

% int_nat_eq
thf(fact_7008_zero__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ zero_zero_nat @ ( nat2 @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% zero_less_nat_eq
thf(fact_7009_zero__le__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( log @ A @ X4 ) )
          = ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ) ).

% zero_le_log_cancel_iff
thf(fact_7010_log__le__zero__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ zero_zero_real )
          = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ) ).

% log_le_zero_cancel_iff
thf(fact_7011_one__le__log__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ one_one_real @ ( log @ A @ X4 ) )
          = ( ord_less_eq_real @ A @ X4 ) ) ) ) ).

% one_le_log_cancel_iff
thf(fact_7012_log__le__one__cancel__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ one_one_real )
          = ( ord_less_eq_real @ X4 @ A ) ) ) ) ).

% log_le_one_cancel_iff
thf(fact_7013_log__le__cancel__iff,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) )
            = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ) ).

% log_le_cancel_iff
thf(fact_7014_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_7015_and__nat__numerals_I3_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = zero_zero_nat ) ).

% and_nat_numerals(3)
thf(fact_7016_and__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = zero_zero_nat ) ).

% and_nat_numerals(1)
thf(fact_7017_nat__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( nat2 @ Y )
        = ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% nat_eq_numeral_power_cancel_iff
thf(fact_7018_numeral__power__eq__nat__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 )
        = ( nat2 @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_nat_cancel_iff
thf(fact_7019_one__less__nat__eq,axiom,
    ! [Z: int] :
      ( ( ord_less_nat @ ( suc @ zero_zero_nat ) @ ( nat2 @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% one_less_nat_eq
thf(fact_7020_real__sqrt__abs,axiom,
    ! [X4: real] :
      ( ( sqrt @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X4 ) ) ).

% real_sqrt_abs
thf(fact_7021_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_7022_and__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se727722235901077358nd_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = one_one_nat ) ).

% and_nat_numerals(2)
thf(fact_7023_and__nat__numerals_I4_J,axiom,
    ! [X4: num] :
      ( ( bit_se727722235901077358nd_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = one_one_nat ) ).

% and_nat_numerals(4)
thf(fact_7024_real__sqrt__pow2,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X4 ) ) ).

% real_sqrt_pow2
thf(fact_7025_real__sqrt__pow2__iff,axiom,
    ! [X4: real] :
      ( ( ( power_power_real @ ( sqrt @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X4 )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% real_sqrt_pow2_iff
thf(fact_7026_real__sqrt__sum__squares__mult__squared__eq,axiom,
    ! [X4: real,Y: real,Xa: real,Ya: real] :
      ( ( power_power_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_squared_eq
thf(fact_7027_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_7028_nat__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% nat_less_numeral_power_cancel_iff
thf(fact_7029_numeral__power__less__nat__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ A ) ) ).

% numeral_power_less_nat_cancel_iff
thf(fact_7030_numeral__power__le__nat__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_eq_nat @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) @ ( nat2 @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ A ) ) ).

% numeral_power_le_nat_cancel_iff
thf(fact_7031_nat__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ A ) @ ( power_power_nat @ ( numeral_numeral_nat @ X4 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% nat_le_numeral_power_cancel_iff
thf(fact_7032_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_7033_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_7034_nat__mask__eq,axiom,
    ! [N2: nat] :
      ( ( nat2 @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( bit_se2002935070580805687sk_nat @ N2 ) ) ).

% nat_mask_eq
thf(fact_7035_and__nat__def,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se725231765392027082nd_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% and_nat_def
thf(fact_7036_real__sqrt__less__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ( ord_less_real @ ( sqrt @ X4 ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_less_mono
thf(fact_7037_real__sqrt__le__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ord_less_eq_real @ ( sqrt @ X4 ) @ ( sqrt @ Y ) ) ) ).

% real_sqrt_le_mono
thf(fact_7038_real__sqrt__power,axiom,
    ! [X4: real,K: nat] :
      ( ( sqrt @ ( power_power_real @ X4 @ K ) )
      = ( power_power_real @ ( sqrt @ X4 ) @ K ) ) ).

% real_sqrt_power
thf(fact_7039_real__sqrt__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_gt_zero
thf(fact_7040_real__sqrt__eq__zero__cancel,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( sqrt @ X4 )
          = zero_zero_real )
       => ( X4 = zero_zero_real ) ) ) ).

% real_sqrt_eq_zero_cancel
thf(fact_7041_real__sqrt__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_ge_zero
thf(fact_7042_real__sqrt__ge__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ord_less_eq_real @ one_one_real @ ( sqrt @ X4 ) ) ) ).

% real_sqrt_ge_one
thf(fact_7043_nat__numeral__as__int,axiom,
    ( numeral_numeral_nat
    = ( ^ [I3: num] : ( nat2 @ ( numeral_numeral_int @ I3 ) ) ) ) ).

% nat_numeral_as_int
thf(fact_7044_nat__mono,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ X4 @ Y )
     => ( ord_less_eq_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ).

% nat_mono
thf(fact_7045_ex__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ? [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ? [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
          & ( P4 @ ( nat2 @ X ) ) ) ) ) ).

% ex_nat
thf(fact_7046_all__nat,axiom,
    ( ( ^ [P3: nat > $o] :
        ! [X6: nat] : ( P3 @ X6 ) )
    = ( ^ [P4: nat > $o] :
        ! [X: int] :
          ( ( ord_less_eq_int @ zero_zero_int @ X )
         => ( P4 @ ( nat2 @ X ) ) ) ) ) ).

% all_nat
thf(fact_7047_eq__nat__nat__iff,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( ( nat2 @ Z )
            = ( nat2 @ Z6 ) )
          = ( Z = Z6 ) ) ) ) ).

% eq_nat_nat_iff
thf(fact_7048_nat__one__as__int,axiom,
    ( one_one_nat
    = ( nat2 @ one_one_int ) ) ).

% nat_one_as_int
thf(fact_7049_pi__not__less__zero,axiom,
    ~ ( ord_less_real @ pi @ zero_zero_real ) ).

% pi_not_less_zero
thf(fact_7050_pi__gt__zero,axiom,
    ord_less_real @ zero_zero_real @ pi ).

% pi_gt_zero
thf(fact_7051_pi__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ pi ).

% pi_ge_zero
thf(fact_7052_unset__bit__nat__def,axiom,
    ( bit_se4205575877204974255it_nat
    = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se4203085406695923979it_int @ M6 @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% unset_bit_nat_def
thf(fact_7053_real__div__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( divide_divide_real @ X4 @ ( sqrt @ X4 ) )
        = ( sqrt @ X4 ) ) ) ).

% real_div_sqrt
thf(fact_7054_sqrt__add__le__add__sqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ X4 @ Y ) ) @ ( plus_plus_real @ ( sqrt @ X4 ) @ ( sqrt @ Y ) ) ) ) ) ).

% sqrt_add_le_add_sqrt
thf(fact_7055_le__real__sqrt__sumsq,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( times_times_real @ X4 @ X4 ) @ ( times_times_real @ Y @ Y ) ) ) ) ).

% le_real_sqrt_sumsq
thf(fact_7056_nat__mono__iff,axiom,
    ! [Z: int,W: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_mono_iff
thf(fact_7057_zless__nat__eq__int__zless,axiom,
    ! [M: nat,Z: int] :
      ( ( ord_less_nat @ M @ ( nat2 @ Z ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ M ) @ Z ) ) ).

% zless_nat_eq_int_zless
thf(fact_7058_nat__le__iff,axiom,
    ! [X4: int,N2: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ X4 ) @ N2 )
      = ( ord_less_eq_int @ X4 @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% nat_le_iff
thf(fact_7059_nat__0__le,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = Z ) ) ).

% nat_0_le
thf(fact_7060_int__eq__iff,axiom,
    ! [M: nat,Z: int] :
      ( ( ( semiri1314217659103216013at_int @ M )
        = Z )
      = ( ( M
          = ( nat2 @ Z ) )
        & ( ord_less_eq_int @ zero_zero_int @ Z ) ) ) ).

% int_eq_iff
thf(fact_7061_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_7062_log__ln,axiom,
    ( ln_ln_real
    = ( log @ ( exp_real @ one_one_real ) ) ) ).

% log_ln
thf(fact_7063_xor__nat__def,axiom,
    ( bit_se6528837805403552850or_nat
    = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se6526347334894502574or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% xor_nat_def
thf(fact_7064_sqrt2__less__2,axiom,
    ord_less_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% sqrt2_less_2
thf(fact_7065_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_7066_log__base__change,axiom,
    ! [A: real,B: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ B @ X4 )
          = ( divide_divide_real @ ( log @ A @ X4 ) @ ( log @ A @ B ) ) ) ) ) ).

% log_base_change
thf(fact_7067_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_7068_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_7069_nat__less__eq__zless,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_int @ W @ Z ) ) ) ).

% nat_less_eq_zless
thf(fact_7070_nat__le__eq__zle,axiom,
    ! [W: int,Z: int] :
      ( ( ( ord_less_int @ zero_zero_int @ W )
        | ( ord_less_eq_int @ zero_zero_int @ Z ) )
     => ( ( ord_less_eq_nat @ ( nat2 @ W ) @ ( nat2 @ Z ) )
        = ( ord_less_eq_int @ W @ Z ) ) ) ).

% nat_le_eq_zle
thf(fact_7071_nat__eq__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ( nat2 @ W )
        = M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff
thf(fact_7072_nat__eq__iff2,axiom,
    ! [M: nat,W: int] :
      ( ( M
        = ( nat2 @ W ) )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ W )
         => ( W
            = ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ W )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_eq_iff2
thf(fact_7073_split__nat,axiom,
    ! [P: nat > $o,I2: int] :
      ( ( P @ ( nat2 @ I2 ) )
      = ( ! [N: nat] :
            ( ( I2
              = ( semiri1314217659103216013at_int @ N ) )
           => ( P @ N ) )
        & ( ( ord_less_int @ I2 @ zero_zero_int )
         => ( P @ zero_zero_nat ) ) ) ) ).

% split_nat
thf(fact_7074_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_7075_nat__add__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
       => ( ( nat2 @ ( plus_plus_int @ Z @ Z6 ) )
          = ( plus_plus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_add_distrib
thf(fact_7076_nat__mult__distrib,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ).

% nat_mult_distrib
thf(fact_7077_Suc__as__int,axiom,
    ( suc
    = ( ^ [A3: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ one_one_int ) ) ) ) ).

% Suc_as_int
thf(fact_7078_nat__diff__distrib_H,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( minus_minus_int @ X4 @ Y ) )
          = ( minus_minus_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_diff_distrib'
thf(fact_7079_nat__diff__distrib,axiom,
    ! [Z6: int,Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z6 )
     => ( ( ord_less_eq_int @ Z6 @ Z )
       => ( ( nat2 @ ( minus_minus_int @ Z @ Z6 ) )
          = ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) ) ) ) ) ).

% nat_diff_distrib
thf(fact_7080_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_7081_nat__div__distrib,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( nat2 @ ( divide_divide_int @ X4 @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib
thf(fact_7082_nat__div__distrib_H,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Y )
     => ( ( nat2 @ ( divide_divide_int @ X4 @ Y ) )
        = ( divide_divide_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ) ).

% nat_div_distrib'
thf(fact_7083_nat__power__eq,axiom,
    ! [Z: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( nat2 @ ( power_power_int @ Z @ N2 ) )
        = ( power_power_nat @ ( nat2 @ Z ) @ N2 ) ) ) ).

% nat_power_eq
thf(fact_7084_nat__mod__distrib,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ( nat2 @ ( modulo_modulo_int @ X4 @ Y ) )
          = ( modulo_modulo_nat @ ( nat2 @ X4 ) @ ( nat2 @ Y ) ) ) ) ) ).

% nat_mod_distrib
thf(fact_7085_pi__less__4,axiom,
    ord_less_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ).

% pi_less_4
thf(fact_7086_pi__ge__two,axiom,
    ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ).

% pi_ge_two
thf(fact_7087_div__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( divide_divide_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% div_abs_eq_div_nat
thf(fact_7088_pi__half__neq__two,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_neq_two
thf(fact_7089_mod__abs__eq__div__nat,axiom,
    ! [K: int,L: int] :
      ( ( modulo_modulo_int @ ( abs_abs_int @ K ) @ ( abs_abs_int @ L ) )
      = ( semiri1314217659103216013at_int @ ( modulo_modulo_nat @ ( nat2 @ ( abs_abs_int @ K ) ) @ ( nat2 @ ( abs_abs_int @ L ) ) ) ) ) ).

% mod_abs_eq_div_nat
thf(fact_7090_take__bit__nat__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) )
        = ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ) ).

% take_bit_nat_eq
thf(fact_7091_nat__take__bit__eq,axiom,
    ! [K: int,N2: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) )
        = ( bit_se2925701944663578781it_nat @ N2 @ ( nat2 @ K ) ) ) ) ).

% nat_take_bit_eq
thf(fact_7092_arctan__inverse,axiom,
    ! [X4: real] :
      ( ( X4 != zero_zero_real )
     => ( ( arctan @ ( divide_divide_real @ one_one_real @ X4 ) )
        = ( minus_minus_real @ ( divide_divide_real @ ( times_times_real @ ( sgn_sgn_real @ X4 ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( arctan @ X4 ) ) ) ) ).

% arctan_inverse
thf(fact_7093_bit__nat__iff,axiom,
    ! [K: int,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( nat2 @ K ) @ N2 )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( bit_se1146084159140164899it_int @ K @ N2 ) ) ) ).

% bit_nat_iff
thf(fact_7094_real__less__rsqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_real @ X4 @ ( sqrt @ Y ) ) ) ).

% real_less_rsqrt
thf(fact_7095_sqrt__le__D,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y )
     => ( ord_less_eq_real @ X4 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sqrt_le_D
thf(fact_7096_real__le__rsqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y )
     => ( ord_less_eq_real @ X4 @ ( sqrt @ Y ) ) ) ).

% real_le_rsqrt
thf(fact_7097_nat__2,axiom,
    ( ( nat2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) )
    = ( suc @ ( suc @ zero_zero_nat ) ) ) ).

% nat_2
thf(fact_7098_log__mult,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( times_times_real @ X4 @ Y ) )
              = ( plus_plus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_mult
thf(fact_7099_sgn__power__injE,axiom,
    ! [A: real,N2: nat,X4: real,B: real] :
      ( ( ( times_times_real @ ( sgn_sgn_real @ A ) @ ( power_power_real @ ( abs_abs_real @ A ) @ N2 ) )
        = X4 )
     => ( ( X4
          = ( 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_7100_log__divide,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( ord_less_real @ zero_zero_real @ Y )
           => ( ( log @ A @ ( divide_divide_real @ X4 @ Y ) )
              = ( minus_minus_real @ ( log @ A @ X4 ) @ ( log @ A @ Y ) ) ) ) ) ) ) ).

% log_divide
thf(fact_7101_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_7102_log__base__pow,axiom,
    ! [A: real,N2: nat,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( log @ ( power_power_real @ A @ N2 ) @ X4 )
        = ( divide_divide_real @ ( log @ A @ X4 ) @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ).

% log_base_pow
thf(fact_7103_log__nat__power,axiom,
    ! [X4: real,B: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( log @ B @ ( power_power_real @ X4 @ N2 ) )
        = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X4 ) ) ) ) ).

% log_nat_power
thf(fact_7104_Suc__nat__eq__nat__zadd1,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( suc @ ( nat2 @ Z ) )
        = ( nat2 @ ( plus_plus_int @ one_one_int @ Z ) ) ) ) ).

% Suc_nat_eq_nat_zadd1
thf(fact_7105_nat__less__iff,axiom,
    ! [W: int,M: nat] :
      ( ( ord_less_eq_int @ zero_zero_int @ W )
     => ( ( ord_less_nat @ ( nat2 @ W ) @ M )
        = ( ord_less_int @ W @ ( semiri1314217659103216013at_int @ M ) ) ) ) ).

% nat_less_iff
thf(fact_7106_nat__mult__distrib__neg,axiom,
    ! [Z: int,Z6: int] :
      ( ( ord_less_eq_int @ Z @ zero_zero_int )
     => ( ( nat2 @ ( times_times_int @ Z @ Z6 ) )
        = ( times_times_nat @ ( nat2 @ ( uminus_uminus_int @ Z ) ) @ ( nat2 @ ( uminus_uminus_int @ Z6 ) ) ) ) ) ).

% nat_mult_distrib_neg
thf(fact_7107_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_7108_pi__half__neq__zero,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% pi_half_neq_zero
thf(fact_7109_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_7110_pi__half__le__two,axiom,
    ord_less_eq_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ).

% pi_half_le_two
thf(fact_7111_real__le__lsqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ X4 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sqrt @ X4 ) @ Y ) ) ) ) ).

% real_le_lsqrt
thf(fact_7112_real__sqrt__unique,axiom,
    ! [Y: real,X4: real] :
      ( ( ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( sqrt @ X4 )
          = Y ) ) ) ).

% real_sqrt_unique
thf(fact_7113_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_7114_real__sqrt__sum__squares__eq__cancel,axiom,
    ! [X4: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = X4 )
     => ( Y = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel
thf(fact_7115_real__sqrt__sum__squares__eq__cancel2,axiom,
    ! [X4: real,Y: real] :
      ( ( ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        = Y )
     => ( X4 = zero_zero_real ) ) ).

% real_sqrt_sum_squares_eq_cancel2
thf(fact_7116_real__sqrt__sum__squares__ge1,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge1
thf(fact_7117_real__sqrt__sum__squares__ge2,axiom,
    ! [Y: real,X4: real] : ( ord_less_eq_real @ Y @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_sum_squares_ge2
thf(fact_7118_real__sqrt__sum__squares__triangle__ineq,axiom,
    ! [A: real,C: real,B: real,D: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( plus_plus_real @ A @ C ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( plus_plus_real @ B @ D ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ C @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ D @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_triangle_ineq
thf(fact_7119_sqrt__ge__absD,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ Y ) )
     => ( ord_less_eq_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ Y ) ) ).

% sqrt_ge_absD
thf(fact_7120_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_7121_log2__of__power__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( M
        = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) )
     => ( ( semiri5074537144036343181t_real @ N2 )
        = ( log @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) ) ) ).

% log2_of_power_eq
thf(fact_7122_log__eq__div__ln__mult__log,axiom,
    ! [A: real,B: real,X4: 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 @ X4 )
             => ( ( log @ A @ X4 )
                = ( times_times_real @ ( divide_divide_real @ ( ln_ln_real @ B ) @ ( ln_ln_real @ A ) ) @ ( log @ B @ X4 ) ) ) ) ) ) ) ) ).

% log_eq_div_ln_mult_log
thf(fact_7123_nat__dvd__iff,axiom,
    ! [Z: int,M: nat] :
      ( ( dvd_dvd_nat @ ( nat2 @ Z ) @ M )
      = ( ( ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( dvd_dvd_int @ Z @ ( semiri1314217659103216013at_int @ M ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ Z )
         => ( M = zero_zero_nat ) ) ) ) ).

% nat_dvd_iff
thf(fact_7124_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_7125_pi__half__ge__zero,axiom,
    ord_less_eq_real @ zero_zero_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ).

% pi_half_ge_zero
thf(fact_7126_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_7127_arctan__ubound,axiom,
    ! [Y: real] : ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arctan_ubound
thf(fact_7128_arctan__one,axiom,
    ( ( arctan @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% arctan_one
thf(fact_7129_real__less__lsqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ X4 @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sqrt @ X4 ) @ Y ) ) ) ) ).

% real_less_lsqrt
thf(fact_7130_sqrt__sum__squares__le__sum,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ X4 @ Y ) ) ) ) ).

% sqrt_sum_squares_le_sum
thf(fact_7131_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_7132_sqrt__sum__squares__le__sum__abs,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( plus_plus_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ Y ) ) ) ).

% sqrt_sum_squares_le_sum_abs
thf(fact_7133_real__sqrt__ge__abs2,axiom,
    ! [Y: real,X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs2
thf(fact_7134_real__sqrt__ge__abs1,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% real_sqrt_ge_abs1
thf(fact_7135_ln__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ln_ln_real @ ( sqrt @ X4 ) )
        = ( divide_divide_real @ ( ln_ln_real @ X4 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% ln_sqrt
thf(fact_7136_sqrt__even__pow2,axiom,
    ! [N2: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( sqrt @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% sqrt_even_pow2
thf(fact_7137_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_7138_arctan__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arctan_bounded
thf(fact_7139_arctan__lbound,axiom,
    ! [Y: real] : ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) ) ).

% arctan_lbound
thf(fact_7140_and__nat__unfold,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N: nat] :
          ( if_nat
          @ ( ( M6 = zero_zero_nat )
            | ( N = zero_zero_nat ) )
          @ zero_zero_nat
          @ ( plus_plus_nat @ ( times_times_nat @ ( modulo_modulo_nat @ M6 @ ( 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 @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% and_nat_unfold
thf(fact_7141_arsinh__real__aux,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ).

% arsinh_real_aux
thf(fact_7142_real__sqrt__sum__squares__mult__ge__zero,axiom,
    ! [X4: real,Y: real,Xa: real,Ya: real] : ( ord_less_eq_real @ zero_zero_real @ ( sqrt @ ( times_times_real @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( plus_plus_real @ ( power_power_real @ Xa @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Ya @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_sum_squares_mult_ge_zero
thf(fact_7143_real__sqrt__power__even,axiom,
    ! [N2: nat,X4: real] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( power_power_real @ ( sqrt @ X4 ) @ N2 )
          = ( power_power_real @ X4 @ ( divide_divide_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% real_sqrt_power_even
thf(fact_7144_arith__geo__mean__sqrt,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ord_less_eq_real @ ( sqrt @ ( times_times_real @ X4 @ Y ) ) @ ( divide_divide_real @ ( plus_plus_real @ X4 @ Y ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arith_geo_mean_sqrt
thf(fact_7145_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_7146_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_7147_and__nat__rec,axiom,
    ( bit_se727722235901077358nd_nat
    = ( ^ [M6: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se727722235901077358nd_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% and_nat_rec
thf(fact_7148_even__nat__iff,axiom,
    ! [K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ K )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( nat2 @ K ) )
        = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ) ).

% even_nat_iff
thf(fact_7149_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_7150_cos__x__y__le__one,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( divide_divide_real @ X4 @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ one_one_real ) ).

% cos_x_y_le_one
thf(fact_7151_real__sqrt__sum__squares__less,axiom,
    ! [X4: real,U: real,Y: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
     => ( ( ord_less_real @ ( abs_abs_real @ Y ) @ ( divide_divide_real @ U @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ).

% real_sqrt_sum_squares_less
thf(fact_7152_arcosh__real__def,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( arcosh_real @ X4 )
        = ( ln_ln_real @ ( plus_plus_real @ X4 @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arcosh_real_def
thf(fact_7153_sqrt__sum__squares__half__less,axiom,
    ! [X4: real,U: real,Y: real] :
      ( ( ord_less_real @ X4 @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ U @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
         => ( ( ord_less_eq_real @ zero_zero_real @ Y )
           => ( ord_less_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ U ) ) ) ) ) ).

% sqrt_sum_squares_half_less
thf(fact_7154_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_7155_machin__Euler,axiom,
    ( ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ ( bit0 @ one ) ) ) @ ( arctan @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ one ) ) ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( arctan @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit1 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ).

% machin_Euler
thf(fact_7156_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_7157_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_7158_arsinh__real__def,axiom,
    ( arsinh_real
    = ( ^ [X: real] : ( ln_ln_real @ ( plus_plus_real @ X @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) ) ) ) ).

% arsinh_real_def
thf(fact_7159_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_7160_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_7161_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_7162_cos__zero,axiom,
    ( ( cos_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cos_zero
thf(fact_7163_cos__zero,axiom,
    ( ( cos_real @ zero_zero_real )
    = one_one_real ) ).

% cos_zero
thf(fact_7164_ceiling__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% ceiling_numeral
thf(fact_7165_ceiling__one,axiom,
    ( ( archim2889992004027027881ng_rat @ one_one_rat )
    = one_one_int ) ).

% ceiling_one
thf(fact_7166_ceiling__one,axiom,
    ( ( archim7802044766580827645g_real @ one_one_real )
    = one_one_int ) ).

% ceiling_one
thf(fact_7167_cos__pi,axiom,
    ( ( cos_real @ pi )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_pi
thf(fact_7168_sin__cos__squared__add3,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ X4 ) ) @ ( times_times_complex @ ( sin_complex @ X4 ) @ ( sin_complex @ X4 ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add3
thf(fact_7169_sin__cos__squared__add3,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ X4 ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ X4 ) ) )
      = one_one_real ) ).

% sin_cos_squared_add3
thf(fact_7170_ceiling__le__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% ceiling_le_zero
thf(fact_7171_ceiling__le__zero,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X4 @ zero_zero_rat ) ) ).

% ceiling_le_zero
thf(fact_7172_zero__less__ceiling,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ zero_zero_rat @ X4 ) ) ).

% zero_less_ceiling
thf(fact_7173_zero__less__ceiling,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% zero_less_ceiling
thf(fact_7174_ceiling__le__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X4 @ ( numeral_numeral_real @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_7175_ceiling__le__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X4 @ ( numeral_numeral_rat @ V ) ) ) ).

% ceiling_le_numeral
thf(fact_7176_ceiling__less__one,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% ceiling_less_one
thf(fact_7177_ceiling__less__one,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int )
      = ( ord_less_eq_rat @ X4 @ zero_zero_rat ) ) ).

% ceiling_less_one
thf(fact_7178_one__le__ceiling,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ zero_zero_rat @ X4 ) ) ).

% one_le_ceiling
thf(fact_7179_one__le__ceiling,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% one_le_ceiling
thf(fact_7180_numeral__less__ceiling,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( numeral_numeral_rat @ V ) @ X4 ) ) ).

% numeral_less_ceiling
thf(fact_7181_numeral__less__ceiling,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( numeral_numeral_real @ V ) @ X4 ) ) ).

% numeral_less_ceiling
thf(fact_7182_ceiling__le__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int )
      = ( ord_less_eq_real @ X4 @ one_one_real ) ) ).

% ceiling_le_one
thf(fact_7183_ceiling__le__one,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int )
      = ( ord_less_eq_rat @ X4 @ one_one_rat ) ) ).

% ceiling_le_one
thf(fact_7184_one__less__ceiling,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ one_one_rat @ X4 ) ) ).

% one_less_ceiling
thf(fact_7185_one__less__ceiling,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ one_one_int @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ one_one_real @ X4 ) ) ).

% one_less_ceiling
thf(fact_7186_ceiling__add__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_7187_ceiling__add__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_add_numeral
thf(fact_7188_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_7189_ceiling__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim2889992004027027881ng_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_neg_numeral
thf(fact_7190_ceiling__add__one,axiom,
    ! [X4: rat] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_7191_ceiling__add__one,axiom,
    ! [X4: real] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ one_one_real ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int ) ) ).

% ceiling_add_one
thf(fact_7192_ceiling__diff__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_7193_ceiling__diff__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% ceiling_diff_numeral
thf(fact_7194_ceiling__diff__one,axiom,
    ! [X4: rat] :
      ( ( archim2889992004027027881ng_rat @ ( minus_minus_rat @ X4 @ one_one_rat ) )
      = ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_7195_ceiling__diff__one,axiom,
    ! [X4: real] :
      ( ( archim7802044766580827645g_real @ ( minus_minus_real @ X4 @ one_one_real ) )
      = ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ one_one_int ) ) ).

% ceiling_diff_one
thf(fact_7196_ceiling__numeral__power,axiom,
    ! [X4: num,N2: nat] :
      ( ( archim7802044766580827645g_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ).

% ceiling_numeral_power
thf(fact_7197_nat__ceiling__le__eq,axiom,
    ! [X4: real,A: nat] :
      ( ( ord_less_eq_nat @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) @ A )
      = ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ A ) ) ) ).

% nat_ceiling_le_eq
thf(fact_7198_ceiling__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ zero_zero_int )
      = ( ord_less_eq_real @ X4 @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% ceiling_less_zero
thf(fact_7199_ceiling__less__zero,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ zero_zero_int )
      = ( ord_less_eq_rat @ X4 @ ( uminus_uminus_rat @ one_one_rat ) ) ) ).

% ceiling_less_zero
thf(fact_7200_zero__le__ceiling,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 ) ) ).

% zero_le_ceiling
thf(fact_7201_zero__le__ceiling,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ one_one_rat ) @ X4 ) ) ).

% zero_le_ceiling
thf(fact_7202_ceiling__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_divide_eq_div_numeral
thf(fact_7203_ceiling__less__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% ceiling_less_numeral
thf(fact_7204_ceiling__less__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% ceiling_less_numeral
thf(fact_7205_numeral__le__ceiling,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X4 ) ) ).

% numeral_le_ceiling
thf(fact_7206_numeral__le__ceiling,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X4 ) ) ).

% numeral_le_ceiling
thf(fact_7207_ceiling__le__neg__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7208_ceiling__le__neg__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% ceiling_le_neg_numeral
thf(fact_7209_neg__numeral__less__ceiling,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X4 ) ) ).

% neg_numeral_less_ceiling
thf(fact_7210_neg__numeral__less__ceiling,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X4 ) ) ).

% neg_numeral_less_ceiling
thf(fact_7211_cos__pi__half,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = zero_zero_real ) ).

% cos_pi_half
thf(fact_7212_sin__two__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = zero_zero_real ) ).

% sin_two_pi
thf(fact_7213_sin__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_pi_half
thf(fact_7214_cos__two__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_real ) ).

% cos_two_pi
thf(fact_7215_cos__periodic,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cos_real @ X4 ) ) ).

% cos_periodic
thf(fact_7216_sin__periodic,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( sin_real @ X4 ) ) ).

% sin_periodic
thf(fact_7217_cos__2pi__minus,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X4 ) )
      = ( cos_real @ X4 ) ) ).

% cos_2pi_minus
thf(fact_7218_cos__npi,axiom,
    ! [N2: nat] :
      ( ( cos_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ pi ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% cos_npi
thf(fact_7219_cos__npi2,axiom,
    ! [N2: nat] :
      ( ( cos_real @ ( times_times_real @ pi @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) ) ).

% cos_npi2
thf(fact_7220_ceiling__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim7802044766580827645g_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ) ).

% ceiling_minus_divide_eq_div_numeral
thf(fact_7221_sin__cos__squared__add2,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add2
thf(fact_7222_sin__cos__squared__add2,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add2
thf(fact_7223_sin__cos__squared__add,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% sin_cos_squared_add
thf(fact_7224_sin__cos__squared__add,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% sin_cos_squared_add
thf(fact_7225_sin__2npi,axiom,
    ! [N2: nat] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
      = zero_zero_real ) ).

% sin_2npi
thf(fact_7226_cos__2npi,axiom,
    ! [N2: nat] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ N2 ) ) @ pi ) )
      = one_one_real ) ).

% cos_2npi
thf(fact_7227_sin__2pi__minus,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ X4 ) )
      = ( uminus_uminus_real @ ( sin_real @ X4 ) ) ) ).

% sin_2pi_minus
thf(fact_7228_ceiling__less__neg__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7229_ceiling__less__neg__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% ceiling_less_neg_numeral
thf(fact_7230_neg__numeral__le__ceiling,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X4 ) ) ).

% neg_numeral_le_ceiling
thf(fact_7231_neg__numeral__le__ceiling,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X4 ) ) ).

% neg_numeral_le_ceiling
thf(fact_7232_cos__3over2__pi,axiom,
    ( ( cos_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = zero_zero_real ) ).

% cos_3over2_pi
thf(fact_7233_sin__3over2__pi,axiom,
    ( ( sin_real @ ( times_times_real @ ( divide_divide_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% sin_3over2_pi
thf(fact_7234_cos__one__sin__zero,axiom,
    ! [X4: complex] :
      ( ( ( cos_complex @ X4 )
        = one_one_complex )
     => ( ( sin_complex @ X4 )
        = zero_zero_complex ) ) ).

% cos_one_sin_zero
thf(fact_7235_cos__one__sin__zero,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = one_one_real )
     => ( ( sin_real @ X4 )
        = zero_zero_real ) ) ).

% cos_one_sin_zero
thf(fact_7236_sin__add,axiom,
    ! [X4: complex,Y: complex] :
      ( ( sin_complex @ ( plus_plus_complex @ X4 @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( sin_complex @ X4 ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( sin_complex @ Y ) ) ) ) ).

% sin_add
thf(fact_7237_sin__add,axiom,
    ! [X4: real,Y: real] :
      ( ( sin_real @ ( plus_plus_real @ X4 @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( sin_real @ Y ) ) ) ) ).

% sin_add
thf(fact_7238_cos__diff,axiom,
    ! [X4: complex,Y: complex] :
      ( ( cos_complex @ ( minus_minus_complex @ X4 @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X4 ) @ ( sin_complex @ Y ) ) ) ) ).

% cos_diff
thf(fact_7239_cos__diff,axiom,
    ! [X4: real,Y: real] :
      ( ( cos_real @ ( minus_minus_real @ X4 @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y ) ) ) ) ).

% cos_diff
thf(fact_7240_cos__add,axiom,
    ! [X4: complex,Y: complex] :
      ( ( cos_complex @ ( plus_plus_complex @ X4 @ Y ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y ) ) @ ( times_times_complex @ ( sin_complex @ X4 ) @ ( sin_complex @ Y ) ) ) ) ).

% cos_add
thf(fact_7241_cos__add,axiom,
    ! [X4: real,Y: real] :
      ( ( cos_real @ ( plus_plus_real @ X4 @ Y ) )
      = ( minus_minus_real @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y ) ) ) ) ).

% cos_add
thf(fact_7242_sin__zero__norm__cos__one,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
     => ( ( real_V7735802525324610683m_real @ ( cos_real @ X4 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_7243_sin__zero__norm__cos__one,axiom,
    ! [X4: complex] :
      ( ( ( sin_complex @ X4 )
        = zero_zero_complex )
     => ( ( real_V1022390504157884413omplex @ ( cos_complex @ X4 ) )
        = one_one_real ) ) ).

% sin_zero_norm_cos_one
thf(fact_7244_sin__zero__abs__cos__one,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
     => ( ( abs_abs_real @ ( cos_real @ X4 ) )
        = one_one_real ) ) ).

% sin_zero_abs_cos_one
thf(fact_7245_sin__double,axiom,
    ! [X4: complex] :
      ( ( sin_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ X4 ) ) @ ( cos_complex @ X4 ) ) ) ).

% sin_double
thf(fact_7246_sin__double,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ X4 ) ) @ ( cos_real @ X4 ) ) ) ).

% sin_double
thf(fact_7247_sincos__principal__value,axiom,
    ! [X4: real] :
    ? [Y3: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ Y3 )
      & ( ord_less_eq_real @ Y3 @ pi )
      & ( ( sin_real @ Y3 )
        = ( sin_real @ X4 ) )
      & ( ( cos_real @ Y3 )
        = ( cos_real @ X4 ) ) ) ).

% sincos_principal_value
thf(fact_7248_ceiling__mono,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ Y @ X4 )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ Y ) @ ( archim7802044766580827645g_real @ X4 ) ) ) ).

% ceiling_mono
thf(fact_7249_ceiling__mono,axiom,
    ! [Y: rat,X4: rat] :
      ( ( ord_less_eq_rat @ Y @ X4 )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ Y ) @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ).

% ceiling_mono
thf(fact_7250_sin__x__le__x,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( sin_real @ X4 ) @ X4 ) ) ).

% sin_x_le_x
thf(fact_7251_ceiling__less__cancel,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim2889992004027027881ng_rat @ Y ) )
     => ( ord_less_rat @ X4 @ Y ) ) ).

% ceiling_less_cancel
thf(fact_7252_ceiling__less__cancel,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim7802044766580827645g_real @ Y ) )
     => ( ord_less_real @ X4 @ Y ) ) ).

% ceiling_less_cancel
thf(fact_7253_sin__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( sin_real @ X4 ) @ one_one_real ) ).

% sin_le_one
thf(fact_7254_cos__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( cos_real @ X4 ) @ one_one_real ) ).

% cos_le_one
thf(fact_7255_abs__sin__x__le__abs__x,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ ( abs_abs_real @ X4 ) ) ).

% abs_sin_x_le_abs_x
thf(fact_7256_sin__cos__le1,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( plus_plus_real @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) ) ) @ one_one_real ) ).

% sin_cos_le1
thf(fact_7257_sin__squared__eq,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_7258_sin__squared__eq,axiom,
    ! [X4: real] :
      ( ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% sin_squared_eq
thf(fact_7259_cos__squared__eq,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_7260_cos__squared__eq,axiom,
    ! [X4: real] :
      ( ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_squared_eq
thf(fact_7261_of__nat__ceiling,axiom,
    ! [R3: real] : ( ord_less_eq_real @ R3 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ R3 ) ) ) ) ).

% of_nat_ceiling
thf(fact_7262_of__nat__ceiling,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ R3 @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim2889992004027027881ng_rat @ R3 ) ) ) ) ).

% of_nat_ceiling
thf(fact_7263_sin__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ pi )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_gt_zero
thf(fact_7264_sin__x__ge__neg__x,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( uminus_uminus_real @ X4 ) @ ( sin_real @ X4 ) ) ) ).

% sin_x_ge_neg_x
thf(fact_7265_sin__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_ge_zero
thf(fact_7266_ceiling__add__le,axiom,
    ! [X4: rat,Y: rat] : ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ Y ) ) @ ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim2889992004027027881ng_rat @ Y ) ) ) ).

% ceiling_add_le
thf(fact_7267_ceiling__add__le,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_int @ ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ Y ) ) @ ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim7802044766580827645g_real @ Y ) ) ) ).

% ceiling_add_le
thf(fact_7268_real__nat__ceiling__ge,axiom,
    ! [X4: real] : ( ord_less_eq_real @ X4 @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ).

% real_nat_ceiling_ge
thf(fact_7269_sin__ge__minus__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( sin_real @ X4 ) ) ).

% sin_ge_minus_one
thf(fact_7270_cos__inj__pi,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ( cos_real @ X4 )
                = ( cos_real @ Y ) )
             => ( X4 = Y ) ) ) ) ) ) ).

% cos_inj_pi
thf(fact_7271_cos__mono__le__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) )
              = ( ord_less_eq_real @ Y @ X4 ) ) ) ) ) ) ).

% cos_mono_le_eq
thf(fact_7272_cos__monotone__0__pi__le,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ pi )
         => ( ord_less_eq_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi_le
thf(fact_7273_cos__ge__minus__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ ( cos_real @ X4 ) ) ).

% cos_ge_minus_one
thf(fact_7274_abs__sin__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( sin_real @ X4 ) ) @ one_one_real ) ).

% abs_sin_le_one
thf(fact_7275_abs__cos__le__one,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( cos_real @ X4 ) ) @ one_one_real ) ).

% abs_cos_le_one
thf(fact_7276_sin__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_7277_sin__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_sin
thf(fact_7278_sin__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( sin_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_7279_sin__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( sin_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% sin_times_cos
thf(fact_7280_cos__times__sin,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( sin_real @ Z ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( sin_real @ ( plus_plus_real @ W @ Z ) ) @ ( sin_real @ ( minus_minus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_7281_cos__times__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( sin_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( sin_complex @ ( plus_plus_complex @ W @ Z ) ) @ ( sin_complex @ ( minus_minus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_sin
thf(fact_7282_sin__plus__sin,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_7283_sin__plus__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_plus_sin
thf(fact_7284_sin__diff__sin,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( sin_real @ W ) @ ( sin_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_7285_sin__diff__sin,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( sin_complex @ W ) @ ( sin_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_diff_sin
thf(fact_7286_cos__diff__cos,axiom,
    ! [W: real,Z: real] :
      ( ( minus_minus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sin_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( sin_real @ ( divide_divide_real @ ( minus_minus_real @ Z @ W ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_7287_cos__diff__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( minus_minus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( sin_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ Z @ W ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_diff_cos
thf(fact_7288_cos__double,axiom,
    ! [X4: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sin_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_7289_cos__double,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
      = ( minus_minus_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sin_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cos_double
thf(fact_7290_cos__double__sin,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( sin_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_7291_cos__double__sin,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ one_one_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( sin_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_double_sin
thf(fact_7292_cos__two__neq__zero,axiom,
    ( ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
   != zero_zero_real ) ).

% cos_two_neq_zero
thf(fact_7293_cos__monotone__0__pi,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ pi )
         => ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) ) ) ) ).

% cos_monotone_0_pi
thf(fact_7294_cos__mono__less__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( ord_less_eq_real @ zero_zero_real @ Y )
         => ( ( ord_less_eq_real @ Y @ pi )
           => ( ( ord_less_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) )
              = ( ord_less_real @ Y @ X4 ) ) ) ) ) ) ).

% cos_mono_less_eq
thf(fact_7295_sin__eq__0__pi,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X4 )
     => ( ( ord_less_real @ X4 @ pi )
       => ( ( ( sin_real @ X4 )
            = zero_zero_real )
         => ( X4 = zero_zero_real ) ) ) ) ).

% sin_eq_0_pi
thf(fact_7296_sin__zero__pi__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ pi )
     => ( ( ( sin_real @ X4 )
          = zero_zero_real )
        = ( X4 = zero_zero_real ) ) ) ).

% sin_zero_pi_iff
thf(fact_7297_cos__monotone__minus__pi__0_H,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
         => ( ord_less_eq_real @ ( cos_real @ Y ) @ ( cos_real @ X4 ) ) ) ) ) ).

% cos_monotone_minus_pi_0'
thf(fact_7298_sincos__total__pi,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
          = one_one_real )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ pi )
            & ( X4
              = ( cos_real @ T3 ) )
            & ( Y
              = ( sin_real @ T3 ) ) ) ) ) ).

% sincos_total_pi
thf(fact_7299_sin__cos__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sin_real @ X4 ) )
     => ( ( sin_real @ X4 )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_cos_sqrt
thf(fact_7300_sin__expansion__lemma,axiom,
    ! [X4: real,M: nat] :
      ( ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% sin_expansion_lemma
thf(fact_7301_cos__expansion__lemma,axiom,
    ! [X4: real,M: nat] :
      ( ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ M ) ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
      = ( uminus_uminus_real @ ( sin_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ M ) @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_expansion_lemma
thf(fact_7302_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_7303_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_7304_sin__gt__zero__02,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_gt_zero_02
thf(fact_7305_cos__two__less__zero,axiom,
    ord_less_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_less_zero
thf(fact_7306_cos__is__zero,axiom,
    ? [X5: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X5 )
      & ( ord_less_eq_real @ X5 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
      & ( ( cos_real @ X5 )
        = zero_zero_real )
      & ! [Y4: real] :
          ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
            & ( ord_less_eq_real @ Y4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ Y4 )
              = zero_zero_real ) )
         => ( Y4 = X5 ) ) ) ).

% cos_is_zero
thf(fact_7307_cos__two__le__zero,axiom,
    ord_less_eq_real @ ( cos_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ zero_zero_real ).

% cos_two_le_zero
thf(fact_7308_cos__monotone__minus__pi__0,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ Y )
     => ( ( ord_less_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
         => ( ord_less_real @ ( cos_real @ Y ) @ ( cos_real @ X4 ) ) ) ) ) ).

% cos_monotone_minus_pi_0
thf(fact_7309_cos__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X5: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X5 )
            & ( ord_less_eq_real @ X5 @ pi )
            & ( ( cos_real @ X5 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ zero_zero_real @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ pi )
                  & ( ( cos_real @ Y4 )
                    = Y ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% cos_total
thf(fact_7310_sincos__total__pi__half,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
            = one_one_real )
         => ? [T3: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( X4
                = ( cos_real @ T3 ) )
              & ( Y
                = ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_pi_half
thf(fact_7311_sincos__total__2pi__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
          & ( X4
            = ( cos_real @ T3 ) )
          & ( Y
            = ( sin_real @ T3 ) ) ) ) ).

% sincos_total_2pi_le
thf(fact_7312_sincos__total__2pi,axiom,
    ! [X4: real,Y: real] :
      ( ( ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( ( X4
                  = ( cos_real @ T3 ) )
               => ( Y
                 != ( sin_real @ T3 ) ) ) ) ) ) ).

% sincos_total_2pi
thf(fact_7313_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_7314_sin__45,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_45
thf(fact_7315_cos__45,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_45
thf(fact_7316_cos__times__cos,axiom,
    ! [W: real,Z: real] :
      ( ( times_times_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( cos_real @ ( minus_minus_real @ W @ Z ) ) @ ( cos_real @ ( plus_plus_real @ W @ Z ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_7317_cos__times__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( times_times_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( cos_complex @ ( minus_minus_complex @ W @ Z ) ) @ ( cos_complex @ ( plus_plus_complex @ W @ Z ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ).

% cos_times_cos
thf(fact_7318_cos__plus__cos,axiom,
    ! [W: real,Z: real] :
      ( ( plus_plus_real @ ( cos_real @ W ) @ ( cos_real @ Z ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( cos_real @ ( divide_divide_real @ ( plus_plus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ ( cos_real @ ( divide_divide_real @ ( minus_minus_real @ W @ Z ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_7319_cos__plus__cos,axiom,
    ! [W: complex,Z: complex] :
      ( ( plus_plus_complex @ ( cos_complex @ W ) @ ( cos_complex @ Z ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( plus_plus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) @ ( cos_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ W @ Z ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% cos_plus_cos
thf(fact_7320_sin__gt__zero2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( sin_real @ X4 ) ) ) ) ).

% sin_gt_zero2
thf(fact_7321_sin__lt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ pi @ X4 )
     => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).

% sin_lt_zero
thf(fact_7322_cos__double__less__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ord_less_real @ ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) @ one_one_real ) ) ) ).

% cos_double_less_one
thf(fact_7323_sin__30,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_30
thf(fact_7324_cos__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).

% cos_gt_zero
thf(fact_7325_sin__inj__pi,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ( sin_real @ X4 )
                = ( sin_real @ Y ) )
             => ( X4 = Y ) ) ) ) ) ) ).

% sin_inj_pi
thf(fact_7326_sin__mono__le__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( sin_real @ X4 ) @ ( sin_real @ Y ) )
              = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ) ) ).

% sin_mono_le_eq
thf(fact_7327_sin__monotone__2pi__le,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_eq_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( sin_real @ Y ) @ ( sin_real @ X4 ) ) ) ) ) ).

% sin_monotone_2pi_le
thf(fact_7328_cos__60,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_60
thf(fact_7329_sin__60,axiom,
    ( ( sin_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% sin_60
thf(fact_7330_cos__30,axiom,
    ( ( cos_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% cos_30
thf(fact_7331_cos__double__cos,axiom,
    ! [W: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( power_power_complex @ ( cos_complex @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_complex ) ) ).

% cos_double_cos
thf(fact_7332_cos__double__cos,axiom,
    ! [W: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ W ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( power_power_real @ ( cos_real @ W ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ one_one_real ) ) ).

% cos_double_cos
thf(fact_7333_cos__treble__cos,axiom,
    ! [X4: complex] :
      ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ X4 ) )
      = ( minus_minus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( cos_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit1 @ one ) ) @ ( cos_complex @ X4 ) ) ) ) ).

% cos_treble_cos
thf(fact_7334_cos__treble__cos,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ X4 ) )
      = ( minus_minus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( cos_real @ X4 ) @ ( numeral_numeral_nat @ ( bit1 @ one ) ) ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit1 @ one ) ) @ ( cos_real @ X4 ) ) ) ) ).

% cos_treble_cos
thf(fact_7335_sin__le__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ pi @ X4 )
     => ( ( ord_less_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
       => ( ord_less_eq_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).

% sin_le_zero
thf(fact_7336_sin__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ( ord_less_real @ ( sin_real @ X4 ) @ zero_zero_real ) ) ) ).

% sin_less_zero
thf(fact_7337_sin__mono__less__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( sin_real @ X4 ) @ ( sin_real @ Y ) )
              = ( ord_less_real @ X4 @ Y ) ) ) ) ) ) ).

% sin_mono_less_eq
thf(fact_7338_sin__monotone__2pi,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X4 )
       => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( sin_real @ Y ) @ ( sin_real @ X4 ) ) ) ) ) ).

% sin_monotone_2pi
thf(fact_7339_sin__total,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ? [X5: real] :
            ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
            & ( ord_less_eq_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( sin_real @ X5 )
              = Y )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
                  & ( ( sin_real @ Y4 )
                    = Y ) )
               => ( Y4 = X5 ) ) ) ) ) ).

% sin_total
thf(fact_7340_cos__gt__zero__pi,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).

% cos_gt_zero_pi
thf(fact_7341_cos__ge__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( cos_real @ X4 ) ) ) ) ).

% cos_ge_zero
thf(fact_7342_cos__one__2pi,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = one_one_real )
      = ( ? [X: nat] :
            ( X4
            = ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) )
        | ? [X: nat] :
            ( X4
            = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ) ).

% cos_one_2pi
thf(fact_7343_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_7344_sin__arctan,axiom,
    ! [X4: real] :
      ( ( sin_real @ ( arctan @ X4 ) )
      = ( divide_divide_real @ X4 @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arctan
thf(fact_7345_cos__arctan,axiom,
    ! [X4: real] :
      ( ( cos_real @ ( arctan @ X4 ) )
      = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% cos_arctan
thf(fact_7346_sin__zero__lemma,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( sin_real @ X4 )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_lemma
thf(fact_7347_sin__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
      = ( ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N: nat] :
            ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X4
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% sin_zero_iff
thf(fact_7348_cos__zero__lemma,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ( cos_real @ X4 )
          = zero_zero_real )
       => ? [N3: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N3 )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_lemma
thf(fact_7349_cos__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = zero_zero_real )
      = ( ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X4
              = ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) )
        | ? [N: nat] :
            ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N )
            & ( X4
              = ( uminus_uminus_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% cos_zero_iff
thf(fact_7350_tan__double,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
         != zero_zero_real )
       => ( ( tan_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
          = ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( tan_real @ X4 ) ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7351_tan__double,axiom,
    ! [X4: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
         != zero_zero_complex )
       => ( ( tan_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( tan_complex @ X4 ) ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% tan_double
thf(fact_7352_sin__tan,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( sin_real @ X4 )
        = ( divide_divide_real @ ( tan_real @ X4 ) @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_tan
thf(fact_7353_cos__tan,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( cos_real @ X4 )
        = ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_tan
thf(fact_7354_complex__unimodular__polar,axiom,
    ! [Z: complex] :
      ( ( ( real_V1022390504157884413omplex @ Z )
        = one_one_real )
     => ~ ! [T3: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ T3 )
           => ( ( ord_less_real @ T3 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
             => ( Z
               != ( complex2 @ ( cos_real @ T3 ) @ ( sin_real @ T3 ) ) ) ) ) ) ).

% complex_unimodular_polar
thf(fact_7355_ceiling__log__eq__powr__iff,axiom,
    ! [X4: real,B: real,K: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim7802044766580827645g_real @ ( log @ B @ X4 ) )
            = ( plus_plus_int @ ( semiri1314217659103216013at_int @ K ) @ one_one_int ) )
          = ( ( ord_less_real @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ K ) ) @ X4 )
            & ( ord_less_eq_real @ X4 @ ( powr_real @ B @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ K @ one_one_nat ) ) ) ) ) ) ) ) ).

% ceiling_log_eq_powr_iff
thf(fact_7356_powr__one__eq__one,axiom,
    ! [A: real] :
      ( ( powr_real @ one_one_real @ A )
      = one_one_real ) ).

% powr_one_eq_one
thf(fact_7357_powr__zero__eq__one,axiom,
    ! [X4: real] :
      ( ( ( X4 = zero_zero_real )
       => ( ( powr_real @ X4 @ zero_zero_real )
          = zero_zero_real ) )
      & ( ( X4 != zero_zero_real )
       => ( ( powr_real @ X4 @ zero_zero_real )
          = one_one_real ) ) ) ).

% powr_zero_eq_one
thf(fact_7358_powr__gt__zero,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( powr_real @ X4 @ A ) )
      = ( X4 != zero_zero_real ) ) ).

% powr_gt_zero
thf(fact_7359_powr__nonneg__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_eq_real @ ( powr_real @ A @ X4 ) @ zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% powr_nonneg_iff
thf(fact_7360_powr__less__cancel__iff,axiom,
    ! [X4: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B ) )
        = ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel_iff
thf(fact_7361_powr__eq__one__iff,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ one_one_real @ A )
     => ( ( ( powr_real @ A @ X4 )
          = one_one_real )
        = ( X4 = zero_zero_real ) ) ) ).

% powr_eq_one_iff
thf(fact_7362_powr__one__gt__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( powr_real @ X4 @ one_one_real )
        = X4 )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% powr_one_gt_zero_iff
thf(fact_7363_powr__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ one_one_real )
        = X4 ) ) ).

% powr_one
thf(fact_7364_powr__le__cancel__iff,axiom,
    ! [X4: real,A: real,B: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B ) )
        = ( ord_less_eq_real @ A @ B ) ) ) ).

% powr_le_cancel_iff
thf(fact_7365_numeral__powr__numeral__real,axiom,
    ! [M: num,N2: num] :
      ( ( powr_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_real @ N2 ) )
      = ( power_power_real @ ( numeral_numeral_real @ M ) @ ( numeral_numeral_nat @ N2 ) ) ) ).

% numeral_powr_numeral_real
thf(fact_7366_log__powr__cancel,axiom,
    ! [A: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( log @ A @ ( powr_real @ A @ Y ) )
          = Y ) ) ) ).

% log_powr_cancel
thf(fact_7367_powr__log__cancel,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( powr_real @ A @ ( log @ A @ X4 ) )
            = X4 ) ) ) ) ).

% powr_log_cancel
thf(fact_7368_tan__periodic__n,axiom,
    ! [X4: real,N2: num] :
      ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ N2 ) @ pi ) ) )
      = ( tan_real @ X4 ) ) ).

% tan_periodic_n
thf(fact_7369_norm__cos__sin,axiom,
    ! [T2: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ ( cos_real @ T2 ) @ ( sin_real @ T2 ) ) )
      = one_one_real ) ).

% norm_cos_sin
thf(fact_7370_powr__numeral,axiom,
    ! [X4: real,N2: num] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( numeral_numeral_real @ N2 ) )
        = ( power_power_real @ X4 @ ( numeral_numeral_nat @ N2 ) ) ) ) ).

% powr_numeral
thf(fact_7371_tan__periodic,axiom,
    ! [X4: real] :
      ( ( tan_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( tan_real @ X4 ) ) ).

% tan_periodic
thf(fact_7372_square__powr__half,axiom,
    ! [X4: real] :
      ( ( powr_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      = ( abs_abs_real @ X4 ) ) ).

% square_powr_half
thf(fact_7373_powr__non__neg,axiom,
    ! [A: real,X4: real] :
      ~ ( ord_less_real @ ( powr_real @ A @ X4 ) @ zero_zero_real ) ).

% powr_non_neg
thf(fact_7374_powr__less__mono2__neg,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ X4 @ Y )
         => ( ord_less_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).

% powr_less_mono2_neg
thf(fact_7375_powr__ge__pzero,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_real @ zero_zero_real @ ( powr_real @ X4 @ Y ) ) ).

% powr_ge_pzero
thf(fact_7376_powr__mono2,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ Y )
         => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mono2
thf(fact_7377_powr__less__cancel,axiom,
    ! [X4: real,A: real,B: real] :
      ( ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B ) )
     => ( ( ord_less_real @ one_one_real @ X4 )
       => ( ord_less_real @ A @ B ) ) ) ).

% powr_less_cancel
thf(fact_7378_powr__less__mono,axiom,
    ! [A: real,B: real,X4: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ one_one_real @ X4 )
       => ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B ) ) ) ) ).

% powr_less_mono
thf(fact_7379_powr__mono,axiom,
    ! [A: real,B: real,X4: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ one_one_real @ X4 )
       => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B ) ) ) ) ).

% powr_mono
thf(fact_7380_one__complex_Ocode,axiom,
    ( one_one_complex
    = ( complex2 @ one_one_real @ zero_zero_real ) ) ).

% one_complex.code
thf(fact_7381_Complex__eq__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = one_one_complex )
      = ( ( A = one_one_real )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_1
thf(fact_7382_Complex__eq__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( numera6690914467698888265omplex @ W ) )
      = ( ( A
          = ( numeral_numeral_real @ W ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_numeral
thf(fact_7383_powr__mono2_H,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ Y )
         => ( ord_less_eq_real @ ( powr_real @ Y @ A ) @ ( powr_real @ X4 @ A ) ) ) ) ) ).

% powr_mono2'
thf(fact_7384_powr__less__mono2,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ X4 @ Y )
         => ( ord_less_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_less_mono2
thf(fact_7385_powr__inj,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ( powr_real @ A @ X4 )
            = ( powr_real @ A @ Y ) )
          = ( X4 = Y ) ) ) ) ).

% powr_inj
thf(fact_7386_gr__one__powr,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ Y )
       => ( ord_less_real @ one_one_real @ ( powr_real @ X4 @ Y ) ) ) ) ).

% gr_one_powr
thf(fact_7387_ge__one__powr__ge__zero,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ one_one_real @ ( powr_real @ X4 @ A ) ) ) ) ).

% ge_one_powr_ge_zero
thf(fact_7388_powr__mono__both,axiom,
    ! [A: real,B: real,X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ B )
       => ( ( ord_less_eq_real @ one_one_real @ X4 )
         => ( ( ord_less_eq_real @ X4 @ Y )
           => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ B ) ) ) ) ) ) ).

% powr_mono_both
thf(fact_7389_powr__le1,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ one_one_real )
         => ( ord_less_eq_real @ ( powr_real @ X4 @ A ) @ one_one_real ) ) ) ) ).

% powr_le1
thf(fact_7390_powr__divide,axiom,
    ! [X4: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( divide_divide_real @ X4 @ Y ) @ A )
          = ( divide_divide_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_divide
thf(fact_7391_powr__mult,axiom,
    ! [X4: real,Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( powr_real @ ( times_times_real @ X4 @ Y ) @ A )
          = ( times_times_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ Y @ A ) ) ) ) ) ).

% powr_mult
thf(fact_7392_Complex__eq__neg__1,axiom,
    ! [A: real,B: real] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ one_one_complex ) )
      = ( ( A
          = ( uminus_uminus_real @ one_one_real ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_1
thf(fact_7393_Complex__eq__neg__numeral,axiom,
    ! [A: real,B: real,W: num] :
      ( ( ( complex2 @ A @ B )
        = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( ( A
          = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
        & ( B = zero_zero_real ) ) ) ).

% Complex_eq_neg_numeral
thf(fact_7394_powr__add,axiom,
    ! [X4: real,A: real,B: real] :
      ( ( powr_real @ X4 @ ( plus_plus_real @ A @ B ) )
      = ( times_times_real @ ( powr_real @ X4 @ A ) @ ( powr_real @ X4 @ B ) ) ) ).

% powr_add
thf(fact_7395_powr__diff,axiom,
    ! [W: real,Z1: real,Z22: real] :
      ( ( powr_real @ W @ ( minus_minus_real @ Z1 @ Z22 ) )
      = ( divide_divide_real @ ( powr_real @ W @ Z1 ) @ ( powr_real @ W @ Z22 ) ) ) ).

% powr_diff
thf(fact_7396_tan__def,axiom,
    ( tan_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sin_real @ X ) @ ( cos_real @ X ) ) ) ) ).

% tan_def
thf(fact_7397_tan__def,axiom,
    ( tan_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sin_complex @ X ) @ ( cos_complex @ X ) ) ) ) ).

% tan_def
thf(fact_7398_powr__realpow,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( semiri5074537144036343181t_real @ N2 ) )
        = ( power_power_real @ X4 @ N2 ) ) ) ).

% powr_realpow
thf(fact_7399_less__log__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ Y @ ( log @ B @ X4 ) )
          = ( ord_less_real @ ( powr_real @ B @ Y ) @ X4 ) ) ) ) ).

% less_log_iff
thf(fact_7400_log__less__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( log @ B @ X4 ) @ Y )
          = ( ord_less_real @ X4 @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_less_iff
thf(fact_7401_less__powr__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ X4 @ ( powr_real @ B @ Y ) )
          = ( ord_less_real @ ( log @ B @ X4 ) @ Y ) ) ) ) ).

% less_powr_iff
thf(fact_7402_powr__less__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_real @ ( powr_real @ B @ Y ) @ X4 )
          = ( ord_less_real @ Y @ ( log @ B @ X4 ) ) ) ) ) ).

% powr_less_iff
thf(fact_7403_powr__minus__divide,axiom,
    ! [X4: real,A: real] :
      ( ( powr_real @ X4 @ ( uminus_uminus_real @ A ) )
      = ( divide_divide_real @ one_one_real @ ( powr_real @ X4 @ A ) ) ) ).

% powr_minus_divide
thf(fact_7404_powr__neg__one,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( uminus_uminus_real @ one_one_real ) )
        = ( divide_divide_real @ one_one_real @ X4 ) ) ) ).

% powr_neg_one
thf(fact_7405_powr__mult__base,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( times_times_real @ X4 @ ( powr_real @ X4 @ Y ) )
        = ( powr_real @ X4 @ ( plus_plus_real @ one_one_real @ Y ) ) ) ) ).

% powr_mult_base
thf(fact_7406_le__log__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ Y @ ( log @ B @ X4 ) )
          = ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X4 ) ) ) ) ).

% le_log_iff
thf(fact_7407_log__le__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( log @ B @ X4 ) @ Y )
          = ( ord_less_eq_real @ X4 @ ( powr_real @ B @ Y ) ) ) ) ) ).

% log_le_iff
thf(fact_7408_le__powr__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ X4 @ ( powr_real @ B @ Y ) )
          = ( ord_less_eq_real @ ( log @ B @ X4 ) @ Y ) ) ) ) ).

% le_powr_iff
thf(fact_7409_powr__le__iff,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ one_one_real @ B )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( ord_less_eq_real @ ( powr_real @ B @ Y ) @ X4 )
          = ( ord_less_eq_real @ Y @ ( log @ B @ X4 ) ) ) ) ) ).

% powr_le_iff
thf(fact_7410_ln__powr__bound,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_eq_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( ln_ln_real @ X4 ) @ ( divide_divide_real @ ( powr_real @ X4 @ A ) @ A ) ) ) ) ).

% ln_powr_bound
thf(fact_7411_ln__powr__bound2,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( powr_real @ ( ln_ln_real @ X4 ) @ A ) @ ( times_times_real @ ( powr_real @ A @ A ) @ X4 ) ) ) ) ).

% ln_powr_bound2
thf(fact_7412_tan__45,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
    = one_one_real ) ).

% tan_45
thf(fact_7413_log__add__eq__powr,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( plus_plus_real @ ( log @ B @ X4 ) @ Y )
            = ( log @ B @ ( times_times_real @ X4 @ ( powr_real @ B @ Y ) ) ) ) ) ) ) ).

% log_add_eq_powr
thf(fact_7414_add__log__eq__powr,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( plus_plus_real @ Y @ ( log @ B @ X4 ) )
            = ( log @ B @ ( times_times_real @ ( powr_real @ B @ Y ) @ X4 ) ) ) ) ) ) ).

% add_log_eq_powr
thf(fact_7415_minus__log__eq__powr,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( minus_minus_real @ Y @ ( log @ B @ X4 ) )
            = ( log @ B @ ( divide_divide_real @ ( powr_real @ B @ Y ) @ X4 ) ) ) ) ) ) ).

% minus_log_eq_powr
thf(fact_7416_tan__60,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) )
    = ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ).

% tan_60
thf(fact_7417_lemma__tan__total,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ zero_zero_real @ Y )
     => ? [X5: real] :
          ( ( ord_less_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ord_less_real @ Y @ ( tan_real @ X5 ) ) ) ) ).

% lemma_tan_total
thf(fact_7418_tan__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).

% tan_gt_zero
thf(fact_7419_tan__total,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y )
      & ! [Y4: real] :
          ( ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y4 )
            & ( ord_less_real @ Y4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
            & ( ( tan_real @ Y4 )
              = Y ) )
         => ( Y4 = X5 ) ) ) ).

% tan_total
thf(fact_7420_tan__monotone,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ X4 )
       => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X4 ) ) ) ) ) ).

% tan_monotone
thf(fact_7421_tan__monotone_H,axiom,
    ! [Y: real,X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
     => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
         => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ Y @ X4 )
              = ( ord_less_real @ ( tan_real @ Y ) @ ( tan_real @ X4 ) ) ) ) ) ) ) ).

% tan_monotone'
thf(fact_7422_tan__mono__lt__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) )
              = ( ord_less_real @ X4 @ Y ) ) ) ) ) ) ).

% tan_mono_lt_eq
thf(fact_7423_lemma__tan__total1,axiom,
    ! [Y: real] :
    ? [X5: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X5 )
      & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ X5 )
        = Y ) ) ).

% lemma_tan_total1
thf(fact_7424_tan__minus__45,axiom,
    ( ( tan_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% tan_minus_45
thf(fact_7425_tan__inverse,axiom,
    ! [Y: real] :
      ( ( divide_divide_real @ one_one_real @ ( tan_real @ Y ) )
      = ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y ) ) ) ).

% tan_inverse
thf(fact_7426_log__minus__eq__powr,axiom,
    ! [B: real,X4: real,Y: real] :
      ( ( ord_less_real @ zero_zero_real @ B )
     => ( ( B != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( minus_minus_real @ ( log @ B @ X4 ) @ Y )
            = ( log @ B @ ( times_times_real @ X4 @ ( powr_real @ B @ ( uminus_uminus_real @ Y ) ) ) ) ) ) ) ) ).

% log_minus_eq_powr
thf(fact_7427_complex__norm,axiom,
    ! [X4: real,Y: real] :
      ( ( real_V1022390504157884413omplex @ ( complex2 @ X4 @ Y ) )
      = ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_norm
thf(fact_7428_add__tan__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( plus_plus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) )
          = ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ X4 @ Y ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_7429_add__tan__eq,axiom,
    ! [X4: complex,Y: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( plus_plus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y ) )
          = ( divide1717551699836669952omplex @ ( sin_complex @ ( plus_plus_complex @ X4 @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% add_tan_eq
thf(fact_7430_powr__half__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        = ( sqrt @ X4 ) ) ) ).

% powr_half_sqrt
thf(fact_7431_powr__neg__numeral,axiom,
    ! [X4: real,N2: num] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( powr_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
        = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ N2 ) ) ) ) ) ).

% powr_neg_numeral
thf(fact_7432_tan__total__pos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ? [X5: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X5 )
          & ( ord_less_real @ X5 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( tan_real @ X5 )
            = Y ) ) ) ).

% tan_total_pos
thf(fact_7433_tan__pos__pi2__le,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_eq_real @ zero_zero_real @ ( tan_real @ X4 ) ) ) ) ).

% tan_pos_pi2_le
thf(fact_7434_tan__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ( ord_less_real @ ( tan_real @ X4 ) @ zero_zero_real ) ) ) ).

% tan_less_zero
thf(fact_7435_tan__mono__le__eq,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ Y )
         => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) )
              = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ) ) ).

% tan_mono_le_eq
thf(fact_7436_tan__mono__le,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ( ord_less_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
         => ( ord_less_eq_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) ) ) ) ) ).

% tan_mono_le
thf(fact_7437_tan__bound__pi2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
     => ( ord_less_real @ ( abs_abs_real @ ( tan_real @ X4 ) ) @ one_one_real ) ) ).

% tan_bound_pi2
thf(fact_7438_tan__30,axiom,
    ( ( tan_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit1 @ one ) ) ) ) )
    = ( divide_divide_real @ one_one_real @ ( sqrt @ ( numeral_numeral_real @ ( bit1 @ one ) ) ) ) ) ).

% tan_30
thf(fact_7439_arctan__unique,axiom,
    ! [X4: real,Y: 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 )
            = Y )
         => ( ( arctan @ Y )
            = X4 ) ) ) ) ).

% arctan_unique
thf(fact_7440_arctan__tan,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arctan @ ( tan_real @ X4 ) )
          = X4 ) ) ) ).

% arctan_tan
thf(fact_7441_arctan,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arctan @ Y ) )
      & ( ord_less_real @ ( arctan @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
      & ( ( tan_real @ ( arctan @ Y ) )
        = Y ) ) ).

% arctan
thf(fact_7442_lemma__tan__add1,axiom,
    ! [X4: real,Y: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) ) )
          = ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ X4 @ Y ) ) @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_7443_lemma__tan__add1,axiom,
    ! [X4: complex,Y: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y ) ) )
          = ( divide1717551699836669952omplex @ ( cos_complex @ ( plus_plus_complex @ X4 @ Y ) ) @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y ) ) ) ) ) ) ).

% lemma_tan_add1
thf(fact_7444_tan__diff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( minus_minus_real @ X4 @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( minus_minus_real @ X4 @ Y ) )
            = ( divide_divide_real @ ( minus_minus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_7445_tan__diff,axiom,
    ! [X4: complex,Y: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( minus_minus_complex @ X4 @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( minus_minus_complex @ X4 @ Y ) )
            = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_diff
thf(fact_7446_tan__add,axiom,
    ! [X4: real,Y: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( ( cos_real @ Y )
         != zero_zero_real )
       => ( ( ( cos_real @ ( plus_plus_real @ X4 @ Y ) )
           != zero_zero_real )
         => ( ( tan_real @ ( plus_plus_real @ X4 @ Y ) )
            = ( divide_divide_real @ ( plus_plus_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) ) @ ( minus_minus_real @ one_one_real @ ( times_times_real @ ( tan_real @ X4 ) @ ( tan_real @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_7447_tan__add,axiom,
    ! [X4: complex,Y: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cos_complex @ Y )
         != zero_zero_complex )
       => ( ( ( cos_complex @ ( plus_plus_complex @ X4 @ Y ) )
           != zero_zero_complex )
         => ( ( tan_complex @ ( plus_plus_complex @ X4 @ Y ) )
            = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y ) ) @ ( minus_minus_complex @ one_one_complex @ ( times_times_complex @ ( tan_complex @ X4 ) @ ( tan_complex @ Y ) ) ) ) ) ) ) ) ).

% tan_add
thf(fact_7448_tan__total__pi4,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ? [Z2: real] :
          ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) ) @ Z2 )
          & ( ord_less_real @ Z2 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ ( bit0 @ one ) ) ) ) )
          & ( ( tan_real @ Z2 )
            = X4 ) ) ) ).

% tan_total_pi4
thf(fact_7449_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_7450_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_7451_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_7452_cos__arcsin,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X4 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_arcsin
thf(fact_7453_sin__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( sin_real @ ( arccos @ Y ) )
        = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ Y @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% sin_arccos_abs
thf(fact_7454_sin__arccos,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X4 ) )
          = ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_arccos
thf(fact_7455_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_7456_of__real__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( real_V1803761363581548252l_real @ X4 )
        = one_one_real )
      = ( X4 = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_7457_of__real__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( real_V4546457046886955230omplex @ X4 )
        = one_one_complex )
      = ( X4 = one_one_real ) ) ).

% of_real_eq_1_iff
thf(fact_7458_of__real__1,axiom,
    ( ( real_V1803761363581548252l_real @ one_one_real )
    = one_one_real ) ).

% of_real_1
thf(fact_7459_of__real__1,axiom,
    ( ( real_V4546457046886955230omplex @ one_one_real )
    = one_one_complex ) ).

% of_real_1
thf(fact_7460_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( numeral_numeral_real @ W ) )
      = ( numeral_numeral_real @ W ) ) ).

% of_real_numeral
thf(fact_7461_of__real__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( numeral_numeral_real @ W ) )
      = ( numera6690914467698888265omplex @ W ) ) ).

% of_real_numeral
thf(fact_7462_of__real__divide,axiom,
    ! [X4: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X4 @ Y ) )
      = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_divide
thf(fact_7463_of__real__divide,axiom,
    ! [X4: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X4 @ Y ) )
      = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_divide
thf(fact_7464_of__real__add,axiom,
    ! [X4: real,Y: real] :
      ( ( real_V1803761363581548252l_real @ ( plus_plus_real @ X4 @ Y ) )
      = ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ).

% of_real_add
thf(fact_7465_of__real__add,axiom,
    ! [X4: real,Y: real] :
      ( ( real_V4546457046886955230omplex @ ( plus_plus_real @ X4 @ Y ) )
      = ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ).

% of_real_add
thf(fact_7466_of__real__power,axiom,
    ! [X4: real,N2: nat] :
      ( ( real_V1803761363581548252l_real @ ( power_power_real @ X4 @ N2 ) )
      = ( power_power_real @ ( real_V1803761363581548252l_real @ X4 ) @ N2 ) ) ).

% of_real_power
thf(fact_7467_of__real__power,axiom,
    ! [X4: real,N2: nat] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ X4 @ N2 ) )
      = ( power_power_complex @ ( real_V4546457046886955230omplex @ X4 ) @ N2 ) ) ).

% of_real_power
thf(fact_7468_arccos__1,axiom,
    ( ( arccos @ one_one_real )
    = zero_zero_real ) ).

% arccos_1
thf(fact_7469_arccos__minus__1,axiom,
    ( ( arccos @ ( uminus_uminus_real @ one_one_real ) )
    = pi ) ).

% arccos_minus_1
thf(fact_7470_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V1803761363581548252l_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_7471_of__real__neg__numeral,axiom,
    ! [W: num] :
      ( ( real_V4546457046886955230omplex @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% of_real_neg_numeral
thf(fact_7472_cos__of__real__pi,axiom,
    ( ( cos_real @ ( real_V1803761363581548252l_real @ pi ) )
    = ( uminus_uminus_real @ one_one_real ) ) ).

% cos_of_real_pi
thf(fact_7473_cos__of__real__pi,axiom,
    ( ( cos_complex @ ( real_V4546457046886955230omplex @ pi ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% cos_of_real_pi
thf(fact_7474_cos__arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( cos_real @ ( arccos @ Y ) )
          = Y ) ) ) ).

% cos_arccos
thf(fact_7475_sin__arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( sin_real @ ( arcsin @ Y ) )
          = Y ) ) ) ).

% sin_arcsin
thf(fact_7476_norm__of__real__add1,axiom,
    ! [X4: real] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ one_one_real ) )
      = ( abs_abs_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_7477_norm__of__real__add1,axiom,
    ! [X4: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ one_one_complex ) )
      = ( abs_abs_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).

% norm_of_real_add1
thf(fact_7478_norm__of__real__addn,axiom,
    ! [X4: real,B: num] :
      ( ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( numeral_numeral_real @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_7479_norm__of__real__addn,axiom,
    ! [X4: real,B: num] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ X4 ) @ ( numera6690914467698888265omplex @ B ) ) )
      = ( abs_abs_real @ ( plus_plus_real @ X4 @ ( numeral_numeral_real @ B ) ) ) ) ).

% norm_of_real_addn
thf(fact_7480_arccos__0,axiom,
    ( ( arccos @ zero_zero_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arccos_0
thf(fact_7481_arcsin__1,axiom,
    ( ( arcsin @ one_one_real )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% arcsin_1
thf(fact_7482_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_7483_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_7484_sin__of__real__pi__half,axiom,
    ( ( sin_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = one_one_real ) ).

% sin_of_real_pi_half
thf(fact_7485_sin__of__real__pi__half,axiom,
    ( ( sin_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
    = one_one_complex ) ).

% sin_of_real_pi_half
thf(fact_7486_arcsin__minus__1,axiom,
    ( ( arcsin @ ( uminus_uminus_real @ one_one_real ) )
    = ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% arcsin_minus_1
thf(fact_7487_nonzero__of__real__divide,axiom,
    ! [Y: real,X4: real] :
      ( ( Y != zero_zero_real )
     => ( ( real_V1803761363581548252l_real @ ( divide_divide_real @ X4 @ Y ) )
        = ( divide_divide_real @ ( real_V1803761363581548252l_real @ X4 ) @ ( real_V1803761363581548252l_real @ Y ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_7488_nonzero__of__real__divide,axiom,
    ! [Y: real,X4: real] :
      ( ( Y != zero_zero_real )
     => ( ( real_V4546457046886955230omplex @ ( divide_divide_real @ X4 @ Y ) )
        = ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ X4 ) @ ( real_V4546457046886955230omplex @ Y ) ) ) ) ).

% nonzero_of_real_divide
thf(fact_7489_arccos__le__arccos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arccos @ Y ) @ ( arccos @ X4 ) ) ) ) ) ).

% arccos_le_arccos
thf(fact_7490_arccos__le__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arccos @ X4 ) @ ( arccos @ Y ) )
          = ( ord_less_eq_real @ Y @ X4 ) ) ) ) ).

% arccos_le_mono
thf(fact_7491_arccos__eq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
        & ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real ) )
     => ( ( ( arccos @ X4 )
          = ( arccos @ Y ) )
        = ( X4 = Y ) ) ) ).

% arccos_eq_iff
thf(fact_7492_arcsin__minus,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( arcsin @ ( uminus_uminus_real @ X4 ) )
          = ( uminus_uminus_real @ ( arcsin @ X4 ) ) ) ) ) ).

% arcsin_minus
thf(fact_7493_arcsin__le__arcsin,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_le_arcsin
thf(fact_7494_arcsin__le__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_eq_real @ ( arcsin @ X4 ) @ ( arcsin @ Y ) )
          = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ).

% arcsin_le_mono
thf(fact_7495_arcsin__eq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ( arcsin @ X4 )
            = ( arcsin @ Y ) )
          = ( X4 = Y ) ) ) ) ).

% arcsin_eq_iff
thf(fact_7496_norm__less__p1,axiom,
    ! [X4: real] : ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ ( real_V7735802525324610683m_real @ ( plus_plus_real @ ( real_V1803761363581548252l_real @ ( real_V7735802525324610683m_real @ X4 ) ) @ one_one_real ) ) ) ).

% norm_less_p1
thf(fact_7497_norm__less__p1,axiom,
    ! [X4: complex] : ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( real_V1022390504157884413omplex @ X4 ) ) @ one_one_complex ) ) ) ).

% norm_less_p1
thf(fact_7498_arccos__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) ) ) ) ).

% arccos_lbound
thf(fact_7499_arccos__less__arccos,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arccos @ Y ) @ ( arccos @ X4 ) ) ) ) ) ).

% arccos_less_arccos
thf(fact_7500_arccos__less__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arccos @ X4 ) @ ( arccos @ Y ) )
          = ( ord_less_real @ Y @ X4 ) ) ) ) ).

% arccos_less_mono
thf(fact_7501_arccos__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ).

% arccos_ubound
thf(fact_7502_arccos__cos,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ pi )
       => ( ( arccos @ ( cos_real @ X4 ) )
          = X4 ) ) ) ).

% arccos_cos
thf(fact_7503_arcsin__less__arcsin,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ Y )
       => ( ( ord_less_eq_real @ Y @ one_one_real )
         => ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y ) ) ) ) ) ).

% arcsin_less_arcsin
thf(fact_7504_arcsin__less__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
       => ( ( ord_less_real @ ( arcsin @ X4 ) @ ( arcsin @ Y ) )
          = ( ord_less_real @ X4 @ Y ) ) ) ) ).

% arcsin_less_mono
thf(fact_7505_cos__arccos__abs,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Y ) @ one_one_real )
     => ( ( cos_real @ ( arccos @ Y ) )
        = Y ) ) ).

% cos_arccos_abs
thf(fact_7506_norm__of__real__diff,axiom,
    ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( real_V1803761363581548252l_real @ B ) @ ( real_V1803761363581548252l_real @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% norm_of_real_diff
thf(fact_7507_norm__of__real__diff,axiom,
    ! [B: real,A: real] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( real_V4546457046886955230omplex @ B ) @ ( real_V4546457046886955230omplex @ A ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ B @ A ) ) ) ).

% norm_of_real_diff
thf(fact_7508_arccos__cos__eq__abs,axiom,
    ! [Theta: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ Theta ) @ pi )
     => ( ( arccos @ ( cos_real @ Theta ) )
        = ( abs_abs_real @ Theta ) ) ) ).

% arccos_cos_eq_abs
thf(fact_7509_arccos__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_lt_bounded
thf(fact_7510_arccos__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi ) ) ) ) ).

% arccos_bounded
thf(fact_7511_sin__arccos__nonzero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ( sin_real @ ( arccos @ X4 ) )
         != zero_zero_real ) ) ) ).

% sin_arccos_nonzero
thf(fact_7512_arccos__cos2,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ pi ) @ X4 )
       => ( ( arccos @ ( cos_real @ X4 ) )
          = ( uminus_uminus_real @ X4 ) ) ) ) ).

% arccos_cos2
thf(fact_7513_arccos__minus,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
          = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ) ).

% arccos_minus
thf(fact_7514_cos__arcsin__nonzero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( ( cos_real @ ( arcsin @ X4 ) )
         != zero_zero_real ) ) ) ).

% cos_arcsin_nonzero
thf(fact_7515_arccos,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ zero_zero_real @ ( arccos @ Y ) )
          & ( ord_less_eq_real @ ( arccos @ Y ) @ pi )
          & ( ( cos_real @ ( arccos @ Y ) )
            = Y ) ) ) ) ).

% arccos
thf(fact_7516_arccos__minus__abs,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( arccos @ ( uminus_uminus_real @ X4 ) )
        = ( minus_minus_real @ pi @ ( arccos @ X4 ) ) ) ) ).

% arccos_minus_abs
thf(fact_7517_cos__sin__eq,axiom,
    ( cos_real
    = ( ^ [X: real] : ( sin_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).

% cos_sin_eq
thf(fact_7518_cos__sin__eq,axiom,
    ( cos_complex
    = ( ^ [X: complex] : ( sin_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).

% cos_sin_eq
thf(fact_7519_sin__cos__eq,axiom,
    ( sin_real
    = ( ^ [X: real] : ( cos_real @ ( minus_minus_real @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).

% sin_cos_eq
thf(fact_7520_sin__cos__eq,axiom,
    ( sin_complex
    = ( ^ [X: complex] : ( cos_complex @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ X ) ) ) ) ).

% sin_cos_eq
thf(fact_7521_minus__sin__cos__eq,axiom,
    ! [X4: real] :
      ( ( uminus_uminus_real @ ( sin_real @ X4 ) )
      = ( cos_real @ ( plus_plus_real @ X4 @ ( divide_divide_real @ ( real_V1803761363581548252l_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_7522_minus__sin__cos__eq,axiom,
    ! [X4: complex] :
      ( ( uminus1482373934393186551omplex @ ( sin_complex @ X4 ) )
      = ( cos_complex @ ( plus_plus_complex @ X4 @ ( divide1717551699836669952omplex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ) ).

% minus_sin_cos_eq
thf(fact_7523_arccos__le__pi2,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arccos @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arccos_le_pi2
thf(fact_7524_arcsin__lt__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_real @ Y @ one_one_real )
       => ( ( ord_less_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_lt_bounded
thf(fact_7525_arcsin__bounded,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ).

% arcsin_bounded
thf(fact_7526_arcsin__ubound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% arcsin_ubound
thf(fact_7527_arcsin__lbound,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) ) ) ) ).

% arcsin_lbound
thf(fact_7528_arcsin__sin,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ( arcsin @ ( sin_real @ X4 ) )
          = X4 ) ) ) ).

% arcsin_sin
thf(fact_7529_arcsin,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin
thf(fact_7530_arcsin__pi,axiom,
    ! [Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ Y )
     => ( ( ord_less_eq_real @ Y @ one_one_real )
       => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( arcsin @ Y ) )
          & ( ord_less_eq_real @ ( arcsin @ Y ) @ pi )
          & ( ( sin_real @ ( arcsin @ Y ) )
            = Y ) ) ) ) ).

% arcsin_pi
thf(fact_7531_arcsin__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ ( arcsin @ X4 ) @ Y )
              = ( ord_less_eq_real @ X4 @ ( sin_real @ Y ) ) ) ) ) ) ) ).

% arcsin_le_iff
thf(fact_7532_le__arcsin__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( ( ord_less_eq_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ Y )
         => ( ( ord_less_eq_real @ Y @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
           => ( ( ord_less_eq_real @ Y @ ( arcsin @ X4 ) )
              = ( ord_less_eq_real @ ( sin_real @ Y ) @ X4 ) ) ) ) ) ) ).

% le_arcsin_iff
thf(fact_7533_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_7534_cos__npi__int,axiom,
    ! [N2: int] :
      ( ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
          = one_one_real ) )
      & ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 )
       => ( ( cos_real @ ( times_times_real @ pi @ ( ring_1_of_int_real @ N2 ) ) )
          = ( uminus_uminus_real @ one_one_real ) ) ) ) ).

% cos_npi_int
thf(fact_7535_cot__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ( ord_less_real @ ( cot_real @ X4 ) @ zero_zero_real ) ) ) ).

% cot_less_zero
thf(fact_7536_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_7537_cot__periodic,axiom,
    ! [X4: real] :
      ( ( cot_real @ ( plus_plus_real @ X4 @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
      = ( cot_real @ X4 ) ) ).

% cot_periodic
thf(fact_7538_of__int__floor__cancel,axiom,
    ! [X4: real] :
      ( ( ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) )
        = X4 )
      = ( ? [N: int] :
            ( X4
            = ( ring_1_of_int_real @ N ) ) ) ) ).

% of_int_floor_cancel
thf(fact_7539_of__int__floor__cancel,axiom,
    ! [X4: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) )
        = X4 )
      = ( ? [N: int] :
            ( X4
            = ( ring_1_of_int_rat @ N ) ) ) ) ).

% of_int_floor_cancel
thf(fact_7540_of__int__ceiling__cancel,axiom,
    ! [X4: rat] :
      ( ( ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) )
        = X4 )
      = ( ? [N: int] :
            ( X4
            = ( ring_1_of_int_rat @ N ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_7541_of__int__ceiling__cancel,axiom,
    ! [X4: real] :
      ( ( ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) )
        = X4 )
      = ( ? [N: int] :
            ( X4
            = ( ring_1_of_int_real @ N ) ) ) ) ).

% of_int_ceiling_cancel
thf(fact_7542_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_7543_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_7544_of__int__le__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ W @ Z ) ) ).

% of_int_le_iff
thf(fact_7545_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_rat @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_rat @ K ) ) ).

% of_int_numeral
thf(fact_7546_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( numeral_numeral_int @ K ) )
      = ( numera6690914467698888265omplex @ K ) ) ).

% of_int_numeral
thf(fact_7547_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_real @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_real @ K ) ) ).

% of_int_numeral
thf(fact_7548_of__int__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( numeral_numeral_int @ K ) )
      = ( numeral_numeral_int @ K ) ) ).

% of_int_numeral
thf(fact_7549_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_rat @ Z )
        = ( numeral_numeral_rat @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_7550_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = ( numera6690914467698888265omplex @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_7551_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_real @ Z )
        = ( numeral_numeral_real @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_7552_of__int__eq__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ( ring_1_of_int_int @ Z )
        = ( numeral_numeral_int @ N2 ) )
      = ( Z
        = ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_eq_numeral_iff
thf(fact_7553_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_7554_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_7555_of__int__less__iff,axiom,
    ! [W: int,Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ W @ Z ) ) ).

% of_int_less_iff
thf(fact_7556_of__int__1,axiom,
    ( ( ring_17405671764205052669omplex @ one_one_int )
    = one_one_complex ) ).

% of_int_1
thf(fact_7557_of__int__1,axiom,
    ( ( ring_1_of_int_int @ one_one_int )
    = one_one_int ) ).

% of_int_1
thf(fact_7558_of__int__1,axiom,
    ( ( ring_1_of_int_real @ one_one_int )
    = one_one_real ) ).

% of_int_1
thf(fact_7559_of__int__1,axiom,
    ( ( ring_1_of_int_rat @ one_one_int )
    = one_one_rat ) ).

% of_int_1
thf(fact_7560_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_17405671764205052669omplex @ Z )
        = one_one_complex )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_7561_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_int @ Z )
        = one_one_int )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_7562_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_real @ Z )
        = one_one_real )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_7563_of__int__eq__1__iff,axiom,
    ! [Z: int] :
      ( ( ( ring_1_of_int_rat @ Z )
        = one_one_rat )
      = ( Z = one_one_int ) ) ).

% of_int_eq_1_iff
thf(fact_7564_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_int @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_int @ ( ring_1_of_int_int @ W ) @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_add
thf(fact_7565_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_real @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ W ) @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_add
thf(fact_7566_of__int__add,axiom,
    ! [W: int,Z: int] :
      ( ( ring_1_of_int_rat @ ( plus_plus_int @ W @ Z ) )
      = ( plus_plus_rat @ ( ring_1_of_int_rat @ W ) @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_add
thf(fact_7567_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( numeral_numeral_real @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_7568_floor__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( numeral_numeral_rat @ V ) )
      = ( numeral_numeral_int @ V ) ) ).

% floor_numeral
thf(fact_7569_floor__one,axiom,
    ( ( archim6058952711729229775r_real @ one_one_real )
    = one_one_int ) ).

% floor_one
thf(fact_7570_floor__one,axiom,
    ( ( archim3151403230148437115or_rat @ one_one_rat )
    = one_one_int ) ).

% floor_one
thf(fact_7571_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_rat @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_rat @ ( ring_1_of_int_rat @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_7572_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_real @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_real @ ( ring_1_of_int_real @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_7573_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_1_of_int_int @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_int @ ( ring_1_of_int_int @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_7574_of__int__power,axiom,
    ! [Z: int,N2: nat] :
      ( ( ring_17405671764205052669omplex @ ( power_power_int @ Z @ N2 ) )
      = ( power_power_complex @ ( ring_17405671764205052669omplex @ Z ) @ N2 ) ) ).

% of_int_power
thf(fact_7575_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W )
        = ( ring_1_of_int_rat @ X4 ) )
      = ( ( power_power_int @ B @ W )
        = X4 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_7576_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ( power_power_real @ ( ring_1_of_int_real @ B ) @ W )
        = ( ring_1_of_int_real @ X4 ) )
      = ( ( power_power_int @ B @ W )
        = X4 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_7577_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ( power_power_int @ ( ring_1_of_int_int @ B ) @ W )
        = ( ring_1_of_int_int @ X4 ) )
      = ( ( power_power_int @ B @ W )
        = X4 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_7578_of__int__eq__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W )
        = ( ring_17405671764205052669omplex @ X4 ) )
      = ( ( power_power_int @ B @ W )
        = X4 ) ) ).

% of_int_eq_of_int_power_cancel_iff
thf(fact_7579_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_rat @ X4 )
        = ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( X4
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_7580_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_real @ X4 )
        = ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( X4
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_7581_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ( ring_1_of_int_int @ X4 )
        = ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( X4
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_7582_of__int__power__eq__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ( ring_17405671764205052669omplex @ X4 )
        = ( power_power_complex @ ( ring_17405671764205052669omplex @ B ) @ W ) )
      = ( X4
        = ( power_power_int @ B @ W ) ) ) ).

% of_int_power_eq_of_int_cancel_iff
thf(fact_7583_ceiling__add__of__int,axiom,
    ! [X4: rat,Z: int] :
      ( ( archim2889992004027027881ng_rat @ ( plus_plus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) )
      = ( plus_plus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_7584_ceiling__add__of__int,axiom,
    ! [X4: real,Z: int] :
      ( ( archim7802044766580827645g_real @ ( plus_plus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) )
      = ( plus_plus_int @ ( archim7802044766580827645g_real @ X4 ) @ Z ) ) ).

% ceiling_add_of_int
thf(fact_7585_of__nat__nat__take__bit__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( semiri681578069525770553at_rat @ ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
      = ( ring_1_of_int_rat @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% of_nat_nat_take_bit_eq
thf(fact_7586_of__nat__nat__take__bit__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( semiri5074537144036343181t_real @ ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
      = ( ring_1_of_int_real @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% of_nat_nat_take_bit_eq
thf(fact_7587_of__nat__nat__take__bit__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( semiri1314217659103216013at_int @ ( nat2 @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) )
      = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% of_nat_nat_take_bit_eq
thf(fact_7588_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_7589_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_7590_of__int__0__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ zero_zero_int @ Z ) ) ).

% of_int_0_le_iff
thf(fact_7591_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_7592_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_7593_of__int__le__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_eq_int @ Z @ zero_zero_int ) ) ).

% of_int_le_0_iff
thf(fact_7594_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_7595_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_7596_of__int__0__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ zero_zero_int @ Z ) ) ).

% of_int_0_less_iff
thf(fact_7597_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ zero_zero_real )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_7598_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ zero_zero_rat )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_7599_of__int__less__0__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ zero_zero_int )
      = ( ord_less_int @ Z @ zero_zero_int ) ) ).

% of_int_less_0_iff
thf(fact_7600_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_7601_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_7602_of__int__numeral__le__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_le_iff
thf(fact_7603_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_7604_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_7605_of__int__le__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_eq_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_le_numeral_iff
thf(fact_7606_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ ( numeral_numeral_rat @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_7607_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ ( numeral_numeral_real @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_7608_of__int__less__numeral__iff,axiom,
    ! [Z: int,N2: num] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ ( numeral_numeral_int @ N2 ) )
      = ( ord_less_int @ Z @ ( numeral_numeral_int @ N2 ) ) ) ).

% of_int_less_numeral_iff
thf(fact_7609_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_rat @ ( numeral_numeral_rat @ N2 ) @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_7610_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_real @ ( numeral_numeral_real @ N2 ) @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_7611_of__int__numeral__less__iff,axiom,
    ! [N2: num,Z: int] :
      ( ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ ( numeral_numeral_int @ N2 ) @ Z ) ) ).

% of_int_numeral_less_iff
thf(fact_7612_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_7613_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_7614_of__int__le__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_eq_int @ Z @ one_one_int ) ) ).

% of_int_le_1_iff
thf(fact_7615_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_7616_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_7617_of__int__1__le__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_eq_int @ one_one_int @ Z ) ) ).

% of_int_1_le_iff
thf(fact_7618_zero__le__floor,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% zero_le_floor
thf(fact_7619_zero__le__floor,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ X4 ) ) ).

% zero_le_floor
thf(fact_7620_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ one_one_real @ ( ring_1_of_int_real @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_7621_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ one_one_rat @ ( ring_1_of_int_rat @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_7622_of__int__1__less__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ one_one_int @ ( ring_1_of_int_int @ Z ) )
      = ( ord_less_int @ one_one_int @ Z ) ) ).

% of_int_1_less_iff
thf(fact_7623_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ one_one_real )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_7624_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_7625_of__int__less__1__iff,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ Z ) @ one_one_int )
      = ( ord_less_int @ Z @ one_one_int ) ) ).

% of_int_less_1_iff
thf(fact_7626_floor__less__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ zero_zero_int )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% floor_less_zero
thf(fact_7627_floor__less__zero,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ zero_zero_int )
      = ( ord_less_rat @ X4 @ zero_zero_rat ) ) ).

% floor_less_zero
thf(fact_7628_numeral__le__floor,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ V ) @ X4 ) ) ).

% numeral_le_floor
thf(fact_7629_numeral__le__floor,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_eq_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ V ) @ X4 ) ) ).

% numeral_le_floor
thf(fact_7630_zero__less__floor,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ zero_zero_int @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ one_one_real @ X4 ) ) ).

% zero_less_floor
thf(fact_7631_zero__less__floor,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ zero_zero_int @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ).

% zero_less_floor
thf(fact_7632_floor__le__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ zero_zero_int )
      = ( ord_less_real @ X4 @ one_one_real ) ) ).

% floor_le_zero
thf(fact_7633_floor__le__zero,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ zero_zero_int )
      = ( ord_less_rat @ X4 @ one_one_rat ) ) ).

% floor_le_zero
thf(fact_7634_floor__less__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X4 @ ( numeral_numeral_real @ V ) ) ) ).

% floor_less_numeral
thf(fact_7635_floor__less__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X4 @ ( numeral_numeral_rat @ V ) ) ) ).

% floor_less_numeral
thf(fact_7636_one__le__floor,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ one_one_real @ X4 ) ) ).

% one_le_floor
thf(fact_7637_one__le__floor,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ one_one_int @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ).

% one_le_floor
thf(fact_7638_floor__less__one,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
      = ( ord_less_real @ X4 @ one_one_real ) ) ).

% floor_less_one
thf(fact_7639_floor__less__one,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
      = ( ord_less_rat @ X4 @ one_one_rat ) ) ).

% floor_less_one
thf(fact_7640_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_7641_floor__neg__numeral,axiom,
    ! [V: num] :
      ( ( archim3151403230148437115or_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) ) ).

% floor_neg_numeral
thf(fact_7642_floor__diff__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ ( numeral_numeral_real @ V ) ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_7643_floor__diff__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ ( numeral_numeral_rat @ V ) ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) ) ) ).

% floor_diff_numeral
thf(fact_7644_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X4 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_eq_int @ X4 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_7645_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X4 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_eq_int @ X4 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_7646_of__int__power__le__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ X4 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_eq_int @ X4 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_le_of_int_cancel_iff
thf(fact_7647_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X4 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X4 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_7648_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X4 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X4 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_7649_of__int__le__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X4 ) )
      = ( ord_less_eq_int @ ( power_power_int @ B @ W ) @ X4 ) ) ).

% of_int_le_of_int_power_cancel_iff
thf(fact_7650_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N2 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_7651_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_7652_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_7653_numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 )
        = Y ) ) ).

% numeral_power_eq_of_int_cancel_iff
thf(fact_7654_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_7655_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_7656_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_7657_of__int__eq__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_eq_numeral_power_cancel_iff
thf(fact_7658_floor__diff__one,axiom,
    ! [X4: real] :
      ( ( archim6058952711729229775r_real @ ( minus_minus_real @ X4 @ one_one_real ) )
      = ( minus_minus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_7659_floor__diff__one,axiom,
    ! [X4: rat] :
      ( ( archim3151403230148437115or_rat @ ( minus_minus_rat @ X4 @ one_one_rat ) )
      = ( minus_minus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ).

% floor_diff_one
thf(fact_7660_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ord_less_real @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) @ ( ring_1_of_int_real @ X4 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X4 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_7661_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) @ ( ring_1_of_int_rat @ X4 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X4 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_7662_of__int__less__of__int__power__cancel__iff,axiom,
    ! [B: int,W: nat,X4: int] :
      ( ( ord_less_int @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) @ ( ring_1_of_int_int @ X4 ) )
      = ( ord_less_int @ ( power_power_int @ B @ W ) @ X4 ) ) ).

% of_int_less_of_int_power_cancel_iff
thf(fact_7663_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ X4 ) @ ( power_power_real @ ( ring_1_of_int_real @ B ) @ W ) )
      = ( ord_less_int @ X4 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_7664_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ X4 ) @ ( power_power_rat @ ( ring_1_of_int_rat @ B ) @ W ) )
      = ( ord_less_int @ X4 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_7665_of__int__power__less__of__int__cancel__iff,axiom,
    ! [X4: int,B: int,W: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ X4 ) @ ( power_power_int @ ( ring_1_of_int_int @ B ) @ W ) )
      = ( ord_less_int @ X4 @ ( power_power_int @ B @ W ) ) ) ).

% of_int_power_less_of_int_cancel_iff
thf(fact_7666_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri681578069525770553at_rat @ ( nat2 @ Z ) )
        = ( ring_1_of_int_rat @ Z ) ) ) ).

% of_nat_nat
thf(fact_7667_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri5074537144036343181t_real @ ( nat2 @ Z ) )
        = ( ring_1_of_int_real @ Z ) ) ) ).

% of_nat_nat
thf(fact_7668_of__nat__nat,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ( semiri1314217659103216013at_int @ ( nat2 @ Z ) )
        = ( ring_1_of_int_int @ Z ) ) ) ).

% of_nat_nat
thf(fact_7669_floor__numeral__power,axiom,
    ! [X4: num,N2: nat] :
      ( ( archim6058952711729229775r_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ).

% floor_numeral_power
thf(fact_7670_floor__numeral__power,axiom,
    ! [X4: num,N2: nat] :
      ( ( archim3151403230148437115or_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N2 ) )
      = ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ).

% floor_numeral_power
thf(fact_7671_floor__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ ( numeral_numeral_int @ A ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_divide_eq_div_numeral
thf(fact_7672_numeral__less__floor,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) @ X4 ) ) ).

% numeral_less_floor
thf(fact_7673_numeral__less__floor,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_int @ ( numeral_numeral_int @ V ) @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) @ X4 ) ) ).

% numeral_less_floor
thf(fact_7674_floor__le__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_real @ X4 @ ( plus_plus_real @ ( numeral_numeral_real @ V ) @ one_one_real ) ) ) ).

% floor_le_numeral
thf(fact_7675_floor__le__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( numeral_numeral_int @ V ) )
      = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( numeral_numeral_rat @ V ) @ one_one_rat ) ) ) ).

% floor_le_numeral
thf(fact_7676_one__less__floor,axiom,
    ! [X4: real] :
      ( ( ord_less_int @ one_one_int @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) ) ).

% one_less_floor
thf(fact_7677_one__less__floor,axiom,
    ! [X4: rat] :
      ( ( ord_less_int @ one_one_int @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ X4 ) ) ).

% one_less_floor
thf(fact_7678_floor__le__one,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
      = ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_7679_floor__le__one,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
      = ( ord_less_rat @ X4 @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% floor_le_one
thf(fact_7680_neg__numeral__le__floor,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ X4 ) ) ).

% neg_numeral_le_floor
thf(fact_7681_neg__numeral__le__floor,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_eq_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ X4 ) ) ).

% neg_numeral_le_floor
thf(fact_7682_floor__less__neg__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X4 @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_7683_floor__less__neg__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X4 @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) ) ) ).

% floor_less_neg_numeral
thf(fact_7684_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_7685_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_7686_numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ A ) ) ).

% numeral_power_le_of_int_cancel_iff
thf(fact_7687_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_7688_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_7689_of__int__le__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_le_numeral_power_cancel_iff
thf(fact_7690_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_7691_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_7692_numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) @ A ) ) ).

% numeral_power_less_of_int_cancel_iff
thf(fact_7693_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( numeral_numeral_rat @ X4 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_7694_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( numeral_numeral_real @ X4 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_7695_of__int__less__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( numeral_numeral_int @ X4 ) @ N2 ) ) ) ).

% of_int_less_numeral_power_cancel_iff
thf(fact_7696_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( ring_1_of_int_real @ Y )
        = ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7697_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( ring_1_of_int_int @ Y )
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7698_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( ring_17405671764205052669omplex @ Y )
        = ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X4 ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7699_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( ring_18347121197199848620nteger @ Y )
        = ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7700_of__int__eq__neg__numeral__power__cancel__iff,axiom,
    ! [Y: int,X4: num,N2: nat] :
      ( ( ( ring_1_of_int_rat @ Y )
        = ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N2 ) )
      = ( Y
        = ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_eq_neg_numeral_power_cancel_iff
thf(fact_7701_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N2 )
        = ( ring_1_of_int_real @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7702_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 )
        = ( ring_1_of_int_int @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7703_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_power_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ X4 ) ) @ N2 )
        = ( ring_17405671764205052669omplex @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7704_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N2 )
        = ( ring_18347121197199848620nteger @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7705_neg__numeral__power__eq__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,Y: int] :
      ( ( ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N2 )
        = ( ring_1_of_int_rat @ Y ) )
      = ( ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 )
        = Y ) ) ).

% neg_numeral_power_eq_of_int_cancel_iff
thf(fact_7706_floor__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) )
      = ( divide_divide_int @ one_one_int @ ( numeral_numeral_int @ B ) ) ) ).

% floor_one_divide_eq_div_numeral
thf(fact_7707_floor__minus__divide__eq__div__numeral,axiom,
    ! [A: num,B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( numeral_numeral_real @ A ) @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ A ) ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_divide_eq_div_numeral
thf(fact_7708_sin__int__2pin,axiom,
    ! [N2: int] :
      ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
      = zero_zero_real ) ).

% sin_int_2pin
thf(fact_7709_cos__int__2pin,axiom,
    ! [N2: int] :
      ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ ( ring_1_of_int_real @ N2 ) ) )
      = one_one_real ) ).

% cos_int_2pin
thf(fact_7710_neg__numeral__less__floor,axiom,
    ! [V: num,X4: real] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) @ X4 ) ) ).

% neg_numeral_less_floor
thf(fact_7711_neg__numeral__less__floor,axiom,
    ! [V: num,X4: rat] :
      ( ( ord_less_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) @ X4 ) ) ).

% neg_numeral_less_floor
thf(fact_7712_floor__le__neg__numeral,axiom,
    ! [X4: real,V: num] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_real @ X4 @ ( plus_plus_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ V ) ) @ one_one_real ) ) ) ).

% floor_le_neg_numeral
thf(fact_7713_floor__le__neg__numeral,axiom,
    ! [X4: rat,V: num] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ V ) ) )
      = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ V ) ) @ one_one_rat ) ) ) ).

% floor_le_neg_numeral
thf(fact_7714_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7715_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_le3102999989581377725nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7716_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7717_of__int__le__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_eq_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) )
      = ( ord_less_eq_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_le_neg_numeral_power_cancel_iff
thf(fact_7718_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_eq_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7719_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_le3102999989581377725nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7720_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_eq_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7721_neg__numeral__power__le__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_eq_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_le_of_int_cancel_iff
thf(fact_7722_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_real @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N2 ) @ ( ring_1_of_int_real @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7723_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ ( ring_1_of_int_int @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7724_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_le6747313008572928689nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N2 ) @ ( ring_18347121197199848620nteger @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7725_neg__numeral__power__less__of__int__cancel__iff,axiom,
    ! [X4: num,N2: nat,A: int] :
      ( ( ord_less_rat @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N2 ) @ ( ring_1_of_int_rat @ A ) )
      = ( ord_less_int @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) @ A ) ) ).

% neg_numeral_power_less_of_int_cancel_iff
thf(fact_7726_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ A ) @ ( power_power_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ X4 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7727_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_int @ ( ring_1_of_int_int @ A ) @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7728_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_le6747313008572928689nteger @ ( ring_18347121197199848620nteger @ A ) @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ X4 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7729_of__int__less__neg__numeral__power__cancel__iff,axiom,
    ! [A: int,X4: num,N2: nat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ A ) @ ( power_power_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ X4 ) ) @ N2 ) )
      = ( ord_less_int @ A @ ( power_power_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ X4 ) ) @ N2 ) ) ) ).

% of_int_less_neg_numeral_power_cancel_iff
thf(fact_7730_floor__minus__one__divide__eq__div__numeral,axiom,
    ! [B: num] :
      ( ( archim6058952711729229775r_real @ ( uminus_uminus_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ B ) ) ) )
      = ( divide_divide_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ B ) ) ) ).

% floor_minus_one_divide_eq_div_numeral
thf(fact_7731_ex__le__of__int,axiom,
    ! [X4: real] :
    ? [Z2: int] : ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_le_of_int
thf(fact_7732_ex__le__of__int,axiom,
    ! [X4: rat] :
    ? [Z2: int] : ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_le_of_int
thf(fact_7733_ex__of__int__less,axiom,
    ! [X4: real] :
    ? [Z2: int] : ( ord_less_real @ ( ring_1_of_int_real @ Z2 ) @ X4 ) ).

% ex_of_int_less
thf(fact_7734_ex__of__int__less,axiom,
    ! [X4: rat] :
    ? [Z2: int] : ( ord_less_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 ) ).

% ex_of_int_less
thf(fact_7735_ex__less__of__int,axiom,
    ! [X4: real] :
    ? [Z2: int] : ( ord_less_real @ X4 @ ( ring_1_of_int_real @ Z2 ) ) ).

% ex_less_of_int
thf(fact_7736_ex__less__of__int,axiom,
    ! [X4: rat] :
    ? [Z2: int] : ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ Z2 ) ) ).

% ex_less_of_int
thf(fact_7737_int__add__floor,axiom,
    ! [Z: int,X4: real] :
      ( ( plus_plus_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ) ).

% int_add_floor
thf(fact_7738_int__add__floor,axiom,
    ! [Z: int,X4: rat] :
      ( ( plus_plus_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ) ).

% int_add_floor
thf(fact_7739_floor__add__int,axiom,
    ! [X4: real,Z: int] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ) ).

% floor_add_int
thf(fact_7740_floor__add__int,axiom,
    ! [X4: rat,Z: int] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ) ).

% floor_add_int
thf(fact_7741_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( ring_1_of_int_real @ K ) @ ( ring_1_of_int_real @ L ) ) )
      = ( divide_divide_int @ K @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_7742_floor__divide__of__int__eq,axiom,
    ! [K: int,L: int] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ K ) @ ( ring_1_of_int_rat @ L ) ) )
      = ( divide_divide_int @ K @ L ) ) ).

% floor_divide_of_int_eq
thf(fact_7743_of__int__floor__le,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) @ X4 ) ).

% of_int_floor_le
thf(fact_7744_of__int__floor__le,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) @ X4 ) ).

% of_int_floor_le
thf(fact_7745_floor__less__iff,axiom,
    ! [X4: real,Z: int] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
      = ( ord_less_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ).

% floor_less_iff
thf(fact_7746_floor__less__iff,axiom,
    ! [X4: rat,Z: int] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
      = ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ).

% floor_less_iff
thf(fact_7747_le__floor__iff,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_eq_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ).

% le_floor_iff
thf(fact_7748_le__floor__iff,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ).

% le_floor_iff
thf(fact_7749_floor__power,axiom,
    ! [X4: real,N2: nat] :
      ( ( X4
        = ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) )
     => ( ( archim6058952711729229775r_real @ ( power_power_real @ X4 @ N2 ) )
        = ( power_power_int @ ( archim6058952711729229775r_real @ X4 ) @ N2 ) ) ) ).

% floor_power
thf(fact_7750_floor__power,axiom,
    ! [X4: rat,N2: nat] :
      ( ( X4
        = ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) )
     => ( ( archim3151403230148437115or_rat @ ( power_power_rat @ X4 @ N2 ) )
        = ( power_power_int @ ( archim3151403230148437115or_rat @ X4 ) @ N2 ) ) ) ).

% floor_power
thf(fact_7751_real__of__int__floor__add__one__gt,axiom,
    ! [R3: real] : ( ord_less_real @ R3 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_gt
thf(fact_7752_floor__eq,axiom,
    ! [N2: int,X4: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X4 )
          = N2 ) ) ) ).

% floor_eq
thf(fact_7753_real__of__int__floor__add__one__ge,axiom,
    ! [R3: real] : ( ord_less_eq_real @ R3 @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) @ one_one_real ) ) ).

% real_of_int_floor_add_one_ge
thf(fact_7754_real__of__int__floor__gt__diff__one,axiom,
    ! [R3: real] : ( ord_less_real @ ( minus_minus_real @ R3 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) ) ).

% real_of_int_floor_gt_diff_one
thf(fact_7755_real__of__int__floor__ge__diff__one,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( minus_minus_real @ R3 @ one_one_real ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ R3 ) ) ) ).

% real_of_int_floor_ge_diff_one
thf(fact_7756_floor__split,axiom,
    ! [P: int > $o,T2: real] :
      ( ( P @ ( archim6058952711729229775r_real @ T2 ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ I3 ) @ T2 )
              & ( ord_less_real @ T2 @ ( plus_plus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_7757_floor__split,axiom,
    ! [P: int > $o,T2: rat] :
      ( ( P @ ( archim3151403230148437115or_rat @ T2 ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ I3 ) @ T2 )
              & ( ord_less_rat @ T2 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) ) )
           => ( P @ I3 ) ) ) ) ).

% floor_split
thf(fact_7758_floor__eq__iff,axiom,
    ! [X4: real,A: int] :
      ( ( ( archim6058952711729229775r_real @ X4 )
        = A )
      = ( ( ord_less_eq_real @ ( ring_1_of_int_real @ A ) @ X4 )
        & ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) ) ) ) ).

% floor_eq_iff
thf(fact_7759_floor__eq__iff,axiom,
    ! [X4: rat,A: int] :
      ( ( ( archim3151403230148437115or_rat @ X4 )
        = A )
      = ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ A ) @ X4 )
        & ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) ) ) ) ).

% floor_eq_iff
thf(fact_7760_floor__unique,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X4 )
          = Z ) ) ) ).

% floor_unique
thf(fact_7761_floor__unique,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z ) @ X4 )
     => ( ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) )
       => ( ( archim3151403230148437115or_rat @ X4 )
          = Z ) ) ) ).

% floor_unique
thf(fact_7762_less__floor__iff,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_int @ Z @ ( archim6058952711729229775r_real @ X4 ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 ) ) ).

% less_floor_iff
thf(fact_7763_less__floor__iff,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_int @ Z @ ( archim3151403230148437115or_rat @ X4 ) )
      = ( ord_less_eq_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 ) ) ).

% less_floor_iff
thf(fact_7764_floor__le__iff,axiom,
    ! [X4: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ Z )
      = ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% floor_le_iff
thf(fact_7765_floor__le__iff,axiom,
    ! [X4: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ Z )
      = ( ord_less_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% floor_le_iff
thf(fact_7766_floor__correct,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ X4 ) ) @ X4 )
      & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_7767_floor__correct,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ X4 ) ) @ X4 )
      & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int ) ) ) ) ).

% floor_correct
thf(fact_7768_floor__eq2,axiom,
    ! [N2: int,X4: real] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ N2 ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim6058952711729229775r_real @ X4 )
          = N2 ) ) ) ).

% floor_eq2
thf(fact_7769_floor__divide__real__eq__div,axiom,
    ! [B: int,A: real] :
      ( ( ord_less_eq_int @ zero_zero_int @ B )
     => ( ( archim6058952711729229775r_real @ ( divide_divide_real @ A @ ( ring_1_of_int_real @ B ) ) )
        = ( divide_divide_int @ ( archim6058952711729229775r_real @ A ) @ B ) ) ) ).

% floor_divide_real_eq_div
thf(fact_7770_floor__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ Y )
     => ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y ) ) ) ).

% floor_mono
thf(fact_7771_floor__mono,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ).

% floor_mono
thf(fact_7772_floor__less__cancel,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y ) )
     => ( ord_less_real @ X4 @ Y ) ) ).

% floor_less_cancel
thf(fact_7773_floor__less__cancel,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y ) )
     => ( ord_less_rat @ X4 @ Y ) ) ).

% floor_less_cancel
thf(fact_7774_floor__le__ceiling,axiom,
    ! [X4: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim7802044766580827645g_real @ X4 ) ) ).

% floor_le_ceiling
thf(fact_7775_floor__le__ceiling,axiom,
    ! [X4: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim2889992004027027881ng_rat @ X4 ) ) ).

% floor_le_ceiling
thf(fact_7776_le__of__int__ceiling,axiom,
    ! [X4: real] : ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) ) ).

% le_of_int_ceiling
thf(fact_7777_le__of__int__ceiling,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ).

% le_of_int_ceiling
thf(fact_7778_floor__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ ( times_times_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ Q3 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_7779_floor__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ Q3 ) @ P2 ) ) ).

% floor_divide_lower
thf(fact_7780_take__bit__of__int,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se2923211474154528505it_int @ N2 @ ( ring_1_of_int_int @ K ) )
      = ( ring_1_of_int_int @ ( bit_se2923211474154528505it_int @ N2 @ K ) ) ) ).

% take_bit_of_int
thf(fact_7781_of__int__and__eq,axiom,
    ! [K: int,L: int] :
      ( ( ring_1_of_int_int @ ( bit_se725231765392027082nd_int @ K @ L ) )
      = ( bit_se725231765392027082nd_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).

% of_int_and_eq
thf(fact_7782_of__int__not__eq,axiom,
    ! [K: int] :
      ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ K ) )
      = ( bit_ri7919022796975470100ot_int @ ( ring_1_of_int_int @ K ) ) ) ).

% of_int_not_eq
thf(fact_7783_of__int__xor__eq,axiom,
    ! [K: int,L: int] :
      ( ( ring_1_of_int_int @ ( bit_se6526347334894502574or_int @ K @ L ) )
      = ( bit_se6526347334894502574or_int @ ( ring_1_of_int_int @ K ) @ ( ring_1_of_int_int @ L ) ) ) ).

% of_int_xor_eq
thf(fact_7784_of__int__mask__eq,axiom,
    ! [N2: nat] :
      ( ( ring_1_of_int_int @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( bit_se2000444600071755411sk_int @ N2 ) ) ).

% of_int_mask_eq
thf(fact_7785_floor__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ P2 @ ( times_times_real @ ( plus_plus_real @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_7786_floor__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ P2 @ ( times_times_rat @ ( plus_plus_rat @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) ) ) ).

% floor_divide_upper
thf(fact_7787_le__floor__add,axiom,
    ! [X4: real,Y: real] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y ) ) @ ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y ) ) ) ).

% le_floor_add
thf(fact_7788_le__floor__add,axiom,
    ! [X4: rat,Y: rat] : ( ord_less_eq_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y ) ) @ ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y ) ) ) ).

% le_floor_add
thf(fact_7789_ceiling__le,axiom,
    ! [X4: real,A: int] :
      ( ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ A ) )
     => ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ A ) ) ).

% ceiling_le
thf(fact_7790_ceiling__le,axiom,
    ! [X4: rat,A: int] :
      ( ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ A ) )
     => ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ A ) ) ).

% ceiling_le
thf(fact_7791_ceiling__le__iff,axiom,
    ! [X4: real,Z: int] :
      ( ( ord_less_eq_int @ ( archim7802044766580827645g_real @ X4 ) @ Z )
      = ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_7792_ceiling__le__iff,axiom,
    ! [X4: rat,Z: int] :
      ( ( ord_less_eq_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z )
      = ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z ) ) ) ).

% ceiling_le_iff
thf(fact_7793_less__ceiling__iff,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_int @ Z @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( ring_1_of_int_rat @ Z ) @ X4 ) ) ).

% less_ceiling_iff
thf(fact_7794_less__ceiling__iff,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_int @ Z @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( ring_1_of_int_real @ Z ) @ X4 ) ) ).

% less_ceiling_iff
thf(fact_7795_real__of__int__div4,axiom,
    ! [N2: int,X4: int] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X4 ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X4 ) ) ) ).

% real_of_int_div4
thf(fact_7796_real__of__int__div,axiom,
    ! [D: int,N2: int] :
      ( ( dvd_dvd_int @ D @ N2 )
     => ( ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ D ) )
        = ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div
thf(fact_7797_of__nat__floor,axiom,
    ! [R3: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R3 )
     => ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( archim6058952711729229775r_real @ R3 ) ) ) @ R3 ) ) ).

% of_nat_floor
thf(fact_7798_of__nat__floor,axiom,
    ! [R3: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ R3 )
     => ( ord_less_eq_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( archim3151403230148437115or_rat @ R3 ) ) ) @ R3 ) ) ).

% of_nat_floor
thf(fact_7799_one__add__floor,axiom,
    ! [X4: real] :
      ( ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ one_one_int )
      = ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ one_one_real ) ) ) ).

% one_add_floor
thf(fact_7800_one__add__floor,axiom,
    ! [X4: rat] :
      ( ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ one_one_int )
      = ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) ) ) ).

% one_add_floor
thf(fact_7801_le__mult__nat__floor,axiom,
    ! [A: real,B: real] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ B ) ) ) @ ( nat2 @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ).

% le_mult_nat_floor
thf(fact_7802_le__mult__nat__floor,axiom,
    ! [A: rat,B: rat] : ( ord_less_eq_nat @ ( times_times_nat @ ( nat2 @ ( archim3151403230148437115or_rat @ A ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( nat2 @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ).

% le_mult_nat_floor
thf(fact_7803_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( archim6058952711729229775r_real @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ M ) @ ( semiri5074537144036343181t_real @ N2 ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_7804_floor__divide__of__nat__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( archim3151403230148437115or_rat @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ M ) @ ( semiri681578069525770553at_rat @ N2 ) ) )
      = ( semiri1314217659103216013at_int @ ( divide_divide_nat @ M @ N2 ) ) ) ).

% floor_divide_of_nat_eq
thf(fact_7805_nat__floor__neg,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
        = zero_zero_nat ) ) ).

% nat_floor_neg
thf(fact_7806_floor__log__eq__powr__iff,axiom,
    ! [X4: real,B: real,K: int] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ one_one_real @ B )
       => ( ( ( archim6058952711729229775r_real @ ( log @ B @ X4 ) )
            = K )
          = ( ( ord_less_eq_real @ ( powr_real @ B @ ( ring_1_of_int_real @ K ) ) @ X4 )
            & ( ord_less_real @ X4 @ ( powr_real @ B @ ( ring_1_of_int_real @ ( plus_plus_int @ K @ one_one_int ) ) ) ) ) ) ) ) ).

% floor_log_eq_powr_iff
thf(fact_7807_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_nonneg
thf(fact_7808_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_nonneg
thf(fact_7809_of__int__nonneg,axiom,
    ! [Z: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ Z )
     => ( ord_less_eq_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_nonneg
thf(fact_7810_of__int__leD,axiom,
    ! [N2: int,X4: code_integer] :
      ( ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X4 )
     => ( ( N2 = zero_zero_int )
        | ( ord_le3102999989581377725nteger @ one_one_Code_integer @ X4 ) ) ) ).

% of_int_leD
thf(fact_7811_of__int__leD,axiom,
    ! [N2: int,X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X4 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% of_int_leD
thf(fact_7812_of__int__leD,axiom,
    ! [N2: int,X4: rat] :
      ( ( ord_less_eq_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X4 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ) ).

% of_int_leD
thf(fact_7813_of__int__leD,axiom,
    ! [N2: int,X4: int] :
      ( ( ord_less_eq_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X4 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_eq_int @ one_one_int @ X4 ) ) ) ).

% of_int_leD
thf(fact_7814_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_real @ zero_zero_real @ ( ring_1_of_int_real @ Z ) ) ) ).

% of_int_pos
thf(fact_7815_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_rat @ zero_zero_rat @ ( ring_1_of_int_rat @ Z ) ) ) ).

% of_int_pos
thf(fact_7816_of__int__pos,axiom,
    ! [Z: int] :
      ( ( ord_less_int @ zero_zero_int @ Z )
     => ( ord_less_int @ zero_zero_int @ ( ring_1_of_int_int @ Z ) ) ) ).

% of_int_pos
thf(fact_7817_of__int__lessD,axiom,
    ! [N2: int,X4: code_integer] :
      ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( ring_18347121197199848620nteger @ N2 ) ) @ X4 )
     => ( ( N2 = zero_zero_int )
        | ( ord_le6747313008572928689nteger @ one_one_Code_integer @ X4 ) ) ) ).

% of_int_lessD
thf(fact_7818_of__int__lessD,axiom,
    ! [N2: int,X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ ( ring_1_of_int_real @ N2 ) ) @ X4 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% of_int_lessD
thf(fact_7819_of__int__lessD,axiom,
    ! [N2: int,X4: rat] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( ring_1_of_int_rat @ N2 ) ) @ X4 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_rat @ one_one_rat @ X4 ) ) ) ).

% of_int_lessD
thf(fact_7820_of__int__lessD,axiom,
    ! [N2: int,X4: int] :
      ( ( ord_less_int @ ( abs_abs_int @ ( ring_1_of_int_int @ N2 ) ) @ X4 )
     => ( ( N2 = zero_zero_int )
        | ( ord_less_int @ one_one_int @ X4 ) ) ) ).

% of_int_lessD
thf(fact_7821_floor__eq3,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
          = N2 ) ) ) ).

% floor_eq3
thf(fact_7822_le__nat__floor,axiom,
    ! [X4: nat,A: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ X4 ) @ A )
     => ( ord_less_eq_nat @ X4 @ ( nat2 @ ( archim6058952711729229775r_real @ A ) ) ) ) ).

% le_nat_floor
thf(fact_7823_floor__exists1,axiom,
    ! [X4: real] :
    ? [X5: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ X5 ) @ X4 )
      & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y4 ) @ X4 )
            & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X5 ) ) ) ).

% floor_exists1
thf(fact_7824_floor__exists1,axiom,
    ! [X4: rat] :
    ? [X5: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ X5 ) @ X4 )
      & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ X5 @ one_one_int ) ) )
      & ! [Y4: int] :
          ( ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y4 ) @ X4 )
            & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Y4 @ one_one_int ) ) ) )
         => ( Y4 = X5 ) ) ) ).

% floor_exists1
thf(fact_7825_floor__exists,axiom,
    ! [X4: real] :
    ? [Z2: int] :
      ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z2 ) @ X4 )
      & ( ord_less_real @ X4 @ ( ring_1_of_int_real @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_7826_floor__exists,axiom,
    ! [X4: rat] :
    ? [Z2: int] :
      ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z2 ) @ X4 )
      & ( ord_less_rat @ X4 @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z2 @ one_one_int ) ) ) ) ).

% floor_exists
thf(fact_7827_of__int__ceiling__le__add__one,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R3 ) ) @ ( plus_plus_real @ R3 @ one_one_real ) ) ).

% of_int_ceiling_le_add_one
thf(fact_7828_of__int__ceiling__le__add__one,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R3 ) ) @ ( plus_plus_rat @ R3 @ one_one_rat ) ) ).

% of_int_ceiling_le_add_one
thf(fact_7829_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_7830_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_7831_of__int__neg__numeral,axiom,
    ! [K: num] :
      ( ( ring_17405671764205052669omplex @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ K ) ) ) ).

% of_int_neg_numeral
thf(fact_7832_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_7833_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_7834_of__int__ceiling__diff__one__le,axiom,
    ! [R3: real] : ( ord_less_eq_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ R3 ) ) @ one_one_real ) @ R3 ) ).

% of_int_ceiling_diff_one_le
thf(fact_7835_of__int__ceiling__diff__one__le,axiom,
    ! [R3: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ R3 ) ) @ one_one_rat ) @ R3 ) ).

% of_int_ceiling_diff_one_le
thf(fact_7836_ceiling__diff__floor__le__1,axiom,
    ! [X4: real] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim7802044766580827645g_real @ X4 ) @ ( archim6058952711729229775r_real @ X4 ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_7837_ceiling__diff__floor__le__1,axiom,
    ! [X4: rat] : ( ord_less_eq_int @ ( minus_minus_int @ ( archim2889992004027027881ng_rat @ X4 ) @ ( archim3151403230148437115or_rat @ X4 ) ) @ one_one_int ) ).

% ceiling_diff_floor_le_1
thf(fact_7838_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X4: int] :
      ( ( ord_less_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( ring_1_of_int_rat @ X4 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X4 ) ) ).

% of_nat_less_of_int_iff
thf(fact_7839_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X4: int] :
      ( ( ord_less_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( ring_1_of_int_real @ X4 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X4 ) ) ).

% of_nat_less_of_int_iff
thf(fact_7840_of__nat__less__of__int__iff,axiom,
    ! [N2: nat,X4: int] :
      ( ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ ( ring_1_of_int_int @ X4 ) )
      = ( ord_less_int @ ( semiri1314217659103216013at_int @ N2 ) @ X4 ) ) ).

% of_nat_less_of_int_iff
thf(fact_7841_int__le__real__less,axiom,
    ( ord_less_eq_int
    = ( ^ [N: int,M6: int] : ( ord_less_real @ ( ring_1_of_int_real @ N ) @ ( plus_plus_real @ ( ring_1_of_int_real @ M6 ) @ one_one_real ) ) ) ) ).

% int_le_real_less
thf(fact_7842_int__less__real__le,axiom,
    ( ord_less_int
    = ( ^ [N: int,M6: int] : ( ord_less_eq_real @ ( plus_plus_real @ ( ring_1_of_int_real @ N ) @ one_one_real ) @ ( ring_1_of_int_real @ M6 ) ) ) ) ).

% int_less_real_le
thf(fact_7843_of__int__not__numeral,axiom,
    ! [K: num] :
      ( ( ring_1_of_int_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ K ) ) ) ).

% of_int_not_numeral
thf(fact_7844_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_7845_ceiling__divide__eq__div,axiom,
    ! [A: int,B: int] :
      ( ( archim7802044766580827645g_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) )
      = ( uminus_uminus_int @ ( divide_divide_int @ ( uminus_uminus_int @ A ) @ B ) ) ) ).

% ceiling_divide_eq_div
thf(fact_7846_real__of__int__div__aux,axiom,
    ! [X4: int,D: int] :
      ( ( divide_divide_real @ ( ring_1_of_int_real @ X4 ) @ ( ring_1_of_int_real @ D ) )
      = ( plus_plus_real @ ( ring_1_of_int_real @ ( divide_divide_int @ X4 @ D ) ) @ ( divide_divide_real @ ( ring_1_of_int_real @ ( modulo_modulo_int @ X4 @ D ) ) @ ( ring_1_of_int_real @ D ) ) ) ) ).

% real_of_int_div_aux
thf(fact_7847_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_7848_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_7849_floor__eq4,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_eq_real @ ( semiri5074537144036343181t_real @ N2 ) @ X4 )
     => ( ( ord_less_real @ X4 @ ( semiri5074537144036343181t_real @ ( suc @ N2 ) ) )
       => ( ( nat2 @ ( archim6058952711729229775r_real @ X4 ) )
          = N2 ) ) ) ).

% floor_eq4
thf(fact_7850_ceiling__split,axiom,
    ! [P: int > $o,T2: real] :
      ( ( P @ ( archim7802044766580827645g_real @ T2 ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ I3 ) @ one_one_real ) @ T2 )
              & ( ord_less_eq_real @ T2 @ ( ring_1_of_int_real @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_7851_ceiling__split,axiom,
    ! [P: int > $o,T2: rat] :
      ( ( P @ ( archim2889992004027027881ng_rat @ T2 ) )
      = ( ! [I3: int] :
            ( ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ I3 ) @ one_one_rat ) @ T2 )
              & ( ord_less_eq_rat @ T2 @ ( ring_1_of_int_rat @ I3 ) ) )
           => ( P @ I3 ) ) ) ) ).

% ceiling_split
thf(fact_7852_ceiling__eq__iff,axiom,
    ! [X4: real,A: int] :
      ( ( ( archim7802044766580827645g_real @ X4 )
        = A )
      = ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ A ) @ one_one_real ) @ X4 )
        & ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_7853_ceiling__eq__iff,axiom,
    ! [X4: rat,A: int] :
      ( ( ( archim2889992004027027881ng_rat @ X4 )
        = A )
      = ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ A ) @ one_one_rat ) @ X4 )
        & ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ A ) ) ) ) ).

% ceiling_eq_iff
thf(fact_7854_ceiling__unique,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ Z ) )
       => ( ( archim7802044766580827645g_real @ X4 )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_7855_ceiling__unique,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ Z ) )
       => ( ( archim2889992004027027881ng_rat @ X4 )
          = Z ) ) ) ).

% ceiling_unique
thf(fact_7856_ceiling__correct,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) @ one_one_real ) @ X4 )
      & ( ord_less_eq_real @ X4 @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ X4 ) ) ) ) ).

% ceiling_correct
thf(fact_7857_ceiling__correct,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) @ one_one_rat ) @ X4 )
      & ( ord_less_eq_rat @ X4 @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ X4 ) ) ) ) ).

% ceiling_correct
thf(fact_7858_cot__def,axiom,
    ( cot_real
    = ( ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) ) ) ) ).

% cot_def
thf(fact_7859_cot__def,axiom,
    ( cot_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( cos_complex @ X ) @ ( sin_complex @ X ) ) ) ) ).

% cot_def
thf(fact_7860_ceiling__less__iff,axiom,
    ! [X4: real,Z: int] :
      ( ( ord_less_int @ ( archim7802044766580827645g_real @ X4 ) @ Z )
      = ( ord_less_eq_real @ X4 @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) ) ) ).

% ceiling_less_iff
thf(fact_7861_ceiling__less__iff,axiom,
    ! [X4: rat,Z: int] :
      ( ( ord_less_int @ ( archim2889992004027027881ng_rat @ X4 ) @ Z )
      = ( ord_less_eq_rat @ X4 @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) ) ) ).

% ceiling_less_iff
thf(fact_7862_le__ceiling__iff,axiom,
    ! [Z: int,X4: rat] :
      ( ( ord_less_eq_int @ Z @ ( archim2889992004027027881ng_rat @ X4 ) )
      = ( ord_less_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ Z ) @ one_one_rat ) @ X4 ) ) ).

% le_ceiling_iff
thf(fact_7863_le__ceiling__iff,axiom,
    ! [Z: int,X4: real] :
      ( ( ord_less_eq_int @ Z @ ( archim7802044766580827645g_real @ X4 ) )
      = ( ord_less_real @ ( minus_minus_real @ ( ring_1_of_int_real @ Z ) @ one_one_real ) @ X4 ) ) ).

% le_ceiling_iff
thf(fact_7864_real__of__int__div2,axiom,
    ! [N2: int,X4: int] : ( ord_less_eq_real @ zero_zero_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X4 ) ) ) ) ).

% real_of_int_div2
thf(fact_7865_real__of__int__div3,axiom,
    ! [N2: int,X4: int] : ( ord_less_eq_real @ ( minus_minus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ N2 ) @ ( ring_1_of_int_real @ X4 ) ) @ ( ring_1_of_int_real @ ( divide_divide_int @ N2 @ X4 ) ) ) @ one_one_real ) ).

% real_of_int_div3
thf(fact_7866_ceiling__divide__upper,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_eq_real @ P2 @ ( times_times_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_7867_ceiling__divide__upper,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_eq_rat @ P2 @ ( times_times_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ Q3 ) ) ) ).

% ceiling_divide_upper
thf(fact_7868_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( ring_18347121197199848620nteger @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_7869_even__of__int__iff,axiom,
    ! [K: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( ring_1_of_int_int @ K ) )
      = ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ K ) ) ).

% even_of_int_iff
thf(fact_7870_of__int__of__nat,axiom,
    ( ring_17405671764205052669omplex
    = ( ^ [K3: int] : ( if_complex @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri8010041392384452111omplex @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7871_of__int__of__nat,axiom,
    ( ring_18347121197199848620nteger
    = ( ^ [K3: int] : ( if_Code_integer @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus1351360451143612070nteger @ ( semiri4939895301339042750nteger @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri4939895301339042750nteger @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7872_of__int__of__nat,axiom,
    ( ring_1_of_int_rat
    = ( ^ [K3: int] : ( if_rat @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri681578069525770553at_rat @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7873_of__int__of__nat,axiom,
    ( ring_1_of_int_real
    = ( ^ [K3: int] : ( if_real @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri5074537144036343181t_real @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7874_of__int__of__nat,axiom,
    ( ring_1_of_int_int
    = ( ^ [K3: int] : ( if_int @ ( ord_less_int @ K3 @ zero_zero_int ) @ ( uminus_uminus_int @ ( semiri1314217659103216013at_int @ ( nat2 @ ( uminus_uminus_int @ K3 ) ) ) ) @ ( semiri1314217659103216013at_int @ ( nat2 @ K3 ) ) ) ) ) ).

% of_int_of_nat
thf(fact_7875_ceiling__divide__lower,axiom,
    ! [Q3: rat,P2: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ Q3 )
     => ( ord_less_rat @ ( times_times_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( divide_divide_rat @ P2 @ Q3 ) ) ) @ one_one_rat ) @ Q3 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_7876_ceiling__divide__lower,axiom,
    ! [Q3: real,P2: real] :
      ( ( ord_less_real @ zero_zero_real @ Q3 )
     => ( ord_less_real @ ( times_times_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( divide_divide_real @ P2 @ Q3 ) ) ) @ one_one_real ) @ Q3 ) @ P2 ) ) ).

% ceiling_divide_lower
thf(fact_7877_ceiling__eq,axiom,
    ! [N2: int,X4: real] :
      ( ( ord_less_real @ ( ring_1_of_int_real @ N2 ) @ X4 )
     => ( ( ord_less_eq_real @ X4 @ ( plus_plus_real @ ( ring_1_of_int_real @ N2 ) @ one_one_real ) )
       => ( ( archim7802044766580827645g_real @ X4 )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_7878_ceiling__eq,axiom,
    ! [N2: int,X4: rat] :
      ( ( ord_less_rat @ ( ring_1_of_int_rat @ N2 ) @ X4 )
     => ( ( ord_less_eq_rat @ X4 @ ( plus_plus_rat @ ( ring_1_of_int_rat @ N2 ) @ one_one_rat ) )
       => ( ( archim2889992004027027881ng_rat @ X4 )
          = ( plus_plus_int @ N2 @ one_one_int ) ) ) ) ).

% ceiling_eq
thf(fact_7879_cos__one__2pi__int,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = one_one_real )
      = ( ? [X: int] :
            ( X4
            = ( times_times_real @ ( times_times_real @ ( ring_1_of_int_real @ X ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ pi ) ) ) ) ).

% cos_one_2pi_int
thf(fact_7880_arccos__cos__eq__abs__2pi,axiom,
    ! [Theta: real] :
      ~ ! [K2: int] :
          ( ( arccos @ ( cos_real @ Theta ) )
         != ( abs_abs_real @ ( minus_minus_real @ Theta @ ( times_times_real @ ( ring_1_of_int_real @ K2 ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) ) ) ) ).

% arccos_cos_eq_abs_2pi
thf(fact_7881_cot__gt__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
       => ( ord_less_real @ zero_zero_real @ ( cot_real @ X4 ) ) ) ) ).

% cot_gt_zero
thf(fact_7882_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_7883_tan__cot_H,axiom,
    ! [X4: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) )
      = ( cot_real @ X4 ) ) ).

% tan_cot'
thf(fact_7884_cos__zero__iff__int,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X4
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% cos_zero_iff_int
thf(fact_7885_sin__zero__iff__int,axiom,
    ! [X4: real] :
      ( ( ( sin_real @ X4 )
        = zero_zero_real )
      = ( ? [I3: int] :
            ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ I3 )
            & ( X4
              = ( times_times_real @ ( ring_1_of_int_real @ I3 ) @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% sin_zero_iff_int
thf(fact_7886_powr__int,axiom,
    ! [X4: real,I2: int] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ I2 )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I2 ) )
            = ( power_power_real @ X4 @ ( nat2 @ I2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ I2 )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ I2 ) )
            = ( divide_divide_real @ one_one_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ I2 ) ) ) ) ) ) ) ) ).

% powr_int
thf(fact_7887_round__unique,axiom,
    ! [X4: real,Y: int] :
      ( ( ord_less_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ Y ) )
     => ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Y ) @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( archim8280529875227126926d_real @ X4 )
          = Y ) ) ) ).

% round_unique
thf(fact_7888_round__unique,axiom,
    ! [X4: rat,Y: int] :
      ( ( ord_less_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ Y ) )
     => ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Y ) @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) )
       => ( ( archim7778729529865785530nd_rat @ X4 )
          = Y ) ) ) ).

% round_unique
thf(fact_7889_round__unique_H,axiom,
    ! [X4: rat,N2: int] :
      ( ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ ( ring_1_of_int_rat @ N2 ) ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
     => ( ( archim7778729529865785530nd_rat @ X4 )
        = N2 ) ) ).

% round_unique'
thf(fact_7890_round__unique_H,axiom,
    ! [X4: real,N2: int] :
      ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ ( ring_1_of_int_real @ N2 ) ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
     => ( ( archim8280529875227126926d_real @ X4 )
        = N2 ) ) ).

% round_unique'
thf(fact_7891_of__int__round__abs__le,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) @ X4 ) ) @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7892_of__int__round__abs__le,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) @ X4 ) ) @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ).

% of_int_round_abs_le
thf(fact_7893_of__int__round__gt,axiom,
    ! [X4: rat] : ( ord_less_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) ) ).

% of_int_round_gt
thf(fact_7894_of__int__round__gt,axiom,
    ! [X4: real] : ( ord_less_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) ) ).

% of_int_round_gt
thf(fact_7895_of__int__round__ge,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( minus_minus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) ) ).

% of_int_round_ge
thf(fact_7896_of__int__round__ge,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ ( minus_minus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) ) ).

% of_int_round_ge
thf(fact_7897_round__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( numeral_numeral_real @ N2 ) )
      = ( numeral_numeral_int @ N2 ) ) ).

% round_numeral
thf(fact_7898_round__1,axiom,
    ( ( archim8280529875227126926d_real @ one_one_real )
    = one_one_int ) ).

% round_1
thf(fact_7899_round__1,axiom,
    ( ( archim7778729529865785530nd_rat @ one_one_rat )
    = one_one_int ) ).

% round_1
thf(fact_7900_round__neg__numeral,axiom,
    ! [N2: num] :
      ( ( archim8280529875227126926d_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% round_neg_numeral
thf(fact_7901_round__neg__numeral,axiom,
    ! [N2: num] :
      ( ( archim7778729529865785530nd_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) ) ).

% round_neg_numeral
thf(fact_7902_round__mono,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_rat @ X4 @ Y )
     => ( ord_less_eq_int @ ( archim7778729529865785530nd_rat @ X4 ) @ ( archim7778729529865785530nd_rat @ Y ) ) ) ).

% round_mono
thf(fact_7903_floor__le__round,axiom,
    ! [X4: real] : ( ord_less_eq_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim8280529875227126926d_real @ X4 ) ) ).

% floor_le_round
thf(fact_7904_floor__le__round,axiom,
    ! [X4: rat] : ( ord_less_eq_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim7778729529865785530nd_rat @ X4 ) ) ).

% floor_le_round
thf(fact_7905_ceiling__ge__round,axiom,
    ! [X4: real] : ( ord_less_eq_int @ ( archim8280529875227126926d_real @ X4 ) @ ( archim7802044766580827645g_real @ X4 ) ) ).

% ceiling_ge_round
thf(fact_7906_round__diff__minimal,axiom,
    ! [Z: real,M: int] : ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ Z ) ) ) ) @ ( abs_abs_real @ ( minus_minus_real @ Z @ ( ring_1_of_int_real @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_7907_round__diff__minimal,axiom,
    ! [Z: rat,M: int] : ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ Z ) ) ) ) @ ( abs_abs_rat @ ( minus_minus_rat @ Z @ ( ring_1_of_int_rat @ M ) ) ) ) ).

% round_diff_minimal
thf(fact_7908_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_7909_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_7910_of__int__round__le,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim8280529875227126926d_real @ X4 ) ) @ ( plus_plus_real @ X4 @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7911_of__int__round__le,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7778729529865785530nd_rat @ X4 ) ) @ ( plus_plus_rat @ X4 @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) ) ).

% of_int_round_le
thf(fact_7912_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_7913_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_7914_powr__real__of__int,axiom,
    ! [X4: real,N2: int] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N2 ) )
            = ( power_power_real @ X4 @ ( nat2 @ N2 ) ) ) )
        & ( ~ ( ord_less_eq_int @ zero_zero_int @ N2 )
         => ( ( powr_real @ X4 @ ( ring_1_of_int_real @ N2 ) )
            = ( inverse_inverse_real @ ( power_power_real @ X4 @ ( nat2 @ ( uminus_uminus_int @ N2 ) ) ) ) ) ) ) ) ).

% powr_real_of_int
thf(fact_7915_cis__2pi,axiom,
    ( ( cis @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) )
    = one_one_complex ) ).

% cis_2pi
thf(fact_7916_i__even__power,axiom,
    ! [N2: nat] :
      ( ( power_power_complex @ imaginary_unit @ ( times_times_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) ) ).

% i_even_power
thf(fact_7917_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_7918_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_7919_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_7920_inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
      = A ) ).

% inverse_inverse_eq
thf(fact_7921_inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
      = A ) ).

% inverse_inverse_eq
thf(fact_7922_inverse__eq__iff__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
      = ( A = B ) ) ).

% inverse_eq_iff_eq
thf(fact_7923_inverse__eq__iff__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = ( invers8013647133539491842omplex @ B ) )
      = ( A = B ) ) ).

% inverse_eq_iff_eq
thf(fact_7924_inverse__nonzero__iff__nonzero,axiom,
    ! [A: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = zero_zero_rat )
      = ( A = zero_zero_rat ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_7925_inverse__nonzero__iff__nonzero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
      = ( A = zero_zero_real ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_7926_inverse__nonzero__iff__nonzero,axiom,
    ! [A: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = zero_zero_complex )
      = ( A = zero_zero_complex ) ) ).

% inverse_nonzero_iff_nonzero
thf(fact_7927_inverse__zero,axiom,
    ( ( inverse_inverse_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% inverse_zero
thf(fact_7928_inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% inverse_zero
thf(fact_7929_inverse__zero,axiom,
    ( ( invers8013647133539491842omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% inverse_zero
thf(fact_7930_inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
      = ( times_times_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) ) ) ).

% inverse_mult_distrib
thf(fact_7931_inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
      = ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) ) ) ).

% inverse_mult_distrib
thf(fact_7932_inverse__eq__1__iff,axiom,
    ! [X4: rat] :
      ( ( ( inverse_inverse_rat @ X4 )
        = one_one_rat )
      = ( X4 = one_one_rat ) ) ).

% inverse_eq_1_iff
thf(fact_7933_inverse__eq__1__iff,axiom,
    ! [X4: real] :
      ( ( ( inverse_inverse_real @ X4 )
        = one_one_real )
      = ( X4 = one_one_real ) ) ).

% inverse_eq_1_iff
thf(fact_7934_inverse__eq__1__iff,axiom,
    ! [X4: complex] :
      ( ( ( invers8013647133539491842omplex @ X4 )
        = one_one_complex )
      = ( X4 = one_one_complex ) ) ).

% inverse_eq_1_iff
thf(fact_7935_inverse__1,axiom,
    ( ( inverse_inverse_rat @ one_one_rat )
    = one_one_rat ) ).

% inverse_1
thf(fact_7936_inverse__1,axiom,
    ( ( inverse_inverse_real @ one_one_real )
    = one_one_real ) ).

% inverse_1
thf(fact_7937_inverse__1,axiom,
    ( ( invers8013647133539491842omplex @ one_one_complex )
    = one_one_complex ) ).

% inverse_1
thf(fact_7938_inverse__divide,axiom,
    ! [A: real,B: real] :
      ( ( inverse_inverse_real @ ( divide_divide_real @ A @ B ) )
      = ( divide_divide_real @ B @ A ) ) ).

% inverse_divide
thf(fact_7939_inverse__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( invers8013647133539491842omplex @ ( divide1717551699836669952omplex @ A @ B ) )
      = ( divide1717551699836669952omplex @ B @ A ) ) ).

% inverse_divide
thf(fact_7940_inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
      = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ).

% inverse_minus_eq
thf(fact_7941_inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
      = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ).

% inverse_minus_eq
thf(fact_7942_inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
      = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ).

% inverse_minus_eq
thf(fact_7943_abs__inverse,axiom,
    ! [A: rat] :
      ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
      = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ).

% abs_inverse
thf(fact_7944_abs__inverse,axiom,
    ! [A: real] :
      ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
      = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ).

% abs_inverse
thf(fact_7945_abs__inverse,axiom,
    ! [A: complex] :
      ( ( abs_abs_complex @ ( invers8013647133539491842omplex @ A ) )
      = ( invers8013647133539491842omplex @ ( abs_abs_complex @ A ) ) ) ).

% abs_inverse
thf(fact_7946_inverse__sgn,axiom,
    ! [A: rat] :
      ( ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) )
      = ( sgn_sgn_rat @ A ) ) ).

% inverse_sgn
thf(fact_7947_inverse__sgn,axiom,
    ! [A: real] :
      ( ( inverse_inverse_real @ ( sgn_sgn_real @ A ) )
      = ( sgn_sgn_real @ A ) ) ).

% inverse_sgn
thf(fact_7948_sgn__inverse,axiom,
    ! [A: rat] :
      ( ( sgn_sgn_rat @ ( inverse_inverse_rat @ A ) )
      = ( inverse_inverse_rat @ ( sgn_sgn_rat @ A ) ) ) ).

% sgn_inverse
thf(fact_7949_sgn__inverse,axiom,
    ! [A: real] :
      ( ( sgn_sgn_real @ ( inverse_inverse_real @ A ) )
      = ( inverse_inverse_real @ ( sgn_sgn_real @ A ) ) ) ).

% sgn_inverse
thf(fact_7950_sgn__inverse,axiom,
    ! [A: complex] :
      ( ( sgn_sgn_complex @ ( invers8013647133539491842omplex @ A ) )
      = ( invers8013647133539491842omplex @ ( sgn_sgn_complex @ A ) ) ) ).

% sgn_inverse
thf(fact_7951_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_eq_rat @ zero_zero_rat @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_7952_inverse__nonnegative__iff__nonnegative,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_eq_real @ zero_zero_real @ A ) ) ).

% inverse_nonnegative_iff_nonnegative
thf(fact_7953_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_eq_rat @ A @ zero_zero_rat ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_7954_inverse__nonpositive__iff__nonpositive,axiom,
    ! [A: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_eq_real @ A @ zero_zero_real ) ) ).

% inverse_nonpositive_iff_nonpositive
thf(fact_7955_inverse__positive__iff__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
      = ( ord_less_rat @ zero_zero_rat @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_7956_inverse__positive__iff__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
      = ( ord_less_real @ zero_zero_real @ A ) ) ).

% inverse_positive_iff_positive
thf(fact_7957_inverse__negative__iff__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
      = ( ord_less_rat @ A @ zero_zero_rat ) ) ).

% inverse_negative_iff_negative
thf(fact_7958_inverse__negative__iff__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
      = ( ord_less_real @ A @ zero_zero_real ) ) ).

% inverse_negative_iff_negative
thf(fact_7959_inverse__less__iff__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_rat @ B @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_7960_inverse__less__iff__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_real @ B @ A ) ) ) ) ).

% inverse_less_iff_less_neg
thf(fact_7961_inverse__less__iff__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_rat @ B @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_7962_inverse__less__iff__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_real @ B @ A ) ) ) ) ).

% inverse_less_iff_less
thf(fact_7963_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_complex @ zero_zero_complex @ ( suc @ K ) )
      = zero_zero_complex ) ).

% gbinomial_0(2)
thf(fact_7964_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_real @ zero_zero_real @ ( suc @ K ) )
      = zero_zero_real ) ).

% gbinomial_0(2)
thf(fact_7965_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_rat @ zero_zero_rat @ ( suc @ K ) )
      = zero_zero_rat ) ).

% gbinomial_0(2)
thf(fact_7966_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_nat @ zero_zero_nat @ ( suc @ K ) )
      = zero_zero_nat ) ).

% gbinomial_0(2)
thf(fact_7967_gbinomial__0_I2_J,axiom,
    ! [K: nat] :
      ( ( gbinomial_int @ zero_zero_int @ ( suc @ K ) )
      = zero_zero_int ) ).

% gbinomial_0(2)
thf(fact_7968_gbinomial__0_I1_J,axiom,
    ! [A: complex] :
      ( ( gbinomial_complex @ A @ zero_zero_nat )
      = one_one_complex ) ).

% gbinomial_0(1)
thf(fact_7969_gbinomial__0_I1_J,axiom,
    ! [A: real] :
      ( ( gbinomial_real @ A @ zero_zero_nat )
      = one_one_real ) ).

% gbinomial_0(1)
thf(fact_7970_gbinomial__0_I1_J,axiom,
    ! [A: rat] :
      ( ( gbinomial_rat @ A @ zero_zero_nat )
      = one_one_rat ) ).

% gbinomial_0(1)
thf(fact_7971_gbinomial__0_I1_J,axiom,
    ! [A: nat] :
      ( ( gbinomial_nat @ A @ zero_zero_nat )
      = one_one_nat ) ).

% gbinomial_0(1)
thf(fact_7972_gbinomial__0_I1_J,axiom,
    ! [A: int] :
      ( ( gbinomial_int @ A @ zero_zero_nat )
      = one_one_int ) ).

% gbinomial_0(1)
thf(fact_7973_norm__ii,axiom,
    ( ( real_V1022390504157884413omplex @ imaginary_unit )
    = one_one_real ) ).

% norm_ii
thf(fact_7974_norm__cis,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( cis @ A ) )
      = one_one_real ) ).

% norm_cis
thf(fact_7975_inverse__le__iff__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_7976_inverse__le__iff__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_eq_real @ B @ A ) ) ) ) ).

% inverse_le_iff_le_neg
thf(fact_7977_inverse__le__iff__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ zero_zero_rat @ B )
       => ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( ord_less_eq_rat @ B @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_7978_inverse__le__iff__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( ord_less_eq_real @ B @ A ) ) ) ) ).

% inverse_le_iff_le
thf(fact_7979_right__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ A @ ( inverse_inverse_rat @ A ) )
        = one_one_rat ) ) ).

% right_inverse
thf(fact_7980_right__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ A @ ( inverse_inverse_real @ A ) )
        = one_one_real ) ) ).

% right_inverse
thf(fact_7981_right__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ A @ ( invers8013647133539491842omplex @ A ) )
        = one_one_complex ) ) ).

% right_inverse
thf(fact_7982_left__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% left_inverse
thf(fact_7983_left__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% left_inverse
thf(fact_7984_left__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% left_inverse
thf(fact_7985_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_rat @ ( numeral_numeral_rat @ W ) )
      = ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_7986_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_real @ ( numeral_numeral_real @ W ) )
      = ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_7987_inverse__eq__divide__numeral,axiom,
    ! [W: num] :
      ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ W ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ W ) ) ) ).

% inverse_eq_divide_numeral
thf(fact_7988_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) )
      = ( divide_divide_rat @ one_one_rat @ ( uminus_uminus_rat @ ( numeral_numeral_rat @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_7989_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( inverse_inverse_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) )
      = ( divide_divide_real @ one_one_real @ ( uminus_uminus_real @ ( numeral_numeral_real @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_7990_inverse__eq__divide__neg__numeral,axiom,
    ! [W: num] :
      ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide1717551699836669952omplex @ one_one_complex @ ( uminus1482373934393186551omplex @ ( numera6690914467698888265omplex @ W ) ) ) ) ).

% inverse_eq_divide_neg_numeral
thf(fact_7991_exp__two__pi__i_H,axiom,
    ( ( exp_complex @ ( times_times_complex @ imaginary_unit @ ( times_times_complex @ ( real_V4546457046886955230omplex @ pi ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) )
    = one_one_complex ) ).

% exp_two_pi_i'
thf(fact_7992_exp__two__pi__i,axiom,
    ( ( exp_complex @ ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( real_V4546457046886955230omplex @ pi ) ) @ imaginary_unit ) )
    = one_one_complex ) ).

% exp_two_pi_i
thf(fact_7993_cis__pi__half,axiom,
    ( ( cis @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
    = imaginary_unit ) ).

% cis_pi_half
thf(fact_7994_power2__i,axiom,
    ( ( power_power_complex @ imaginary_unit @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
    = ( uminus1482373934393186551omplex @ one_one_complex ) ) ).

% power2_i
thf(fact_7995_cis__minus__pi__half,axiom,
    ( ( cis @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
    = ( uminus1482373934393186551omplex @ imaginary_unit ) ) ).

% cis_minus_pi_half
thf(fact_7996_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: real,X4: real] :
      ( ( ( times_times_real @ Y @ X4 )
        = ( times_times_real @ X4 @ Y ) )
     => ( ( times_times_real @ ( inverse_inverse_real @ Y ) @ X4 )
        = ( times_times_real @ X4 @ ( inverse_inverse_real @ Y ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_7997_mult__commute__imp__mult__inverse__commute,axiom,
    ! [Y: complex,X4: complex] :
      ( ( ( times_times_complex @ Y @ X4 )
        = ( times_times_complex @ X4 @ Y ) )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ Y ) @ X4 )
        = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ Y ) ) ) ) ).

% mult_commute_imp_mult_inverse_commute
thf(fact_7998_power__inverse,axiom,
    ! [A: real,N2: nat] :
      ( ( power_power_real @ ( inverse_inverse_real @ A ) @ N2 )
      = ( inverse_inverse_real @ ( power_power_real @ A @ N2 ) ) ) ).

% power_inverse
thf(fact_7999_power__inverse,axiom,
    ! [A: complex,N2: nat] :
      ( ( power_power_complex @ ( invers8013647133539491842omplex @ A ) @ N2 )
      = ( invers8013647133539491842omplex @ ( power_power_complex @ A @ N2 ) ) ) ).

% power_inverse
thf(fact_8000_inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
thf(fact_8001_inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = ( invers8013647133539491842omplex @ B ) )
     => ( A = B ) ) ).

% inverse_eq_imp_eq
thf(fact_8002_field__class_Ofield__inverse__zero,axiom,
    ( ( inverse_inverse_rat @ zero_zero_rat )
    = zero_zero_rat ) ).

% field_class.field_inverse_zero
thf(fact_8003_field__class_Ofield__inverse__zero,axiom,
    ( ( inverse_inverse_real @ zero_zero_real )
    = zero_zero_real ) ).

% field_class.field_inverse_zero
thf(fact_8004_field__class_Ofield__inverse__zero,axiom,
    ( ( invers8013647133539491842omplex @ zero_zero_complex )
    = zero_zero_complex ) ).

% field_class.field_inverse_zero
thf(fact_8005_inverse__zero__imp__zero,axiom,
    ! [A: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = zero_zero_rat )
     => ( A = zero_zero_rat ) ) ).

% inverse_zero_imp_zero
thf(fact_8006_inverse__zero__imp__zero,axiom,
    ! [A: real] :
      ( ( ( inverse_inverse_real @ A )
        = zero_zero_real )
     => ( A = zero_zero_real ) ) ).

% inverse_zero_imp_zero
thf(fact_8007_inverse__zero__imp__zero,axiom,
    ! [A: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = zero_zero_complex )
     => ( A = zero_zero_complex ) ) ).

% inverse_zero_imp_zero
thf(fact_8008_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: rat,B: rat] :
      ( ( ( inverse_inverse_rat @ A )
        = ( inverse_inverse_rat @ B ) )
     => ( ( A != zero_zero_rat )
       => ( ( B != zero_zero_rat )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_8009_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: real,B: real] :
      ( ( ( inverse_inverse_real @ A )
        = ( inverse_inverse_real @ B ) )
     => ( ( A != zero_zero_real )
       => ( ( B != zero_zero_real )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_8010_nonzero__inverse__eq__imp__eq,axiom,
    ! [A: complex,B: complex] :
      ( ( ( invers8013647133539491842omplex @ A )
        = ( invers8013647133539491842omplex @ B ) )
     => ( ( A != zero_zero_complex )
       => ( ( B != zero_zero_complex )
         => ( A = B ) ) ) ) ).

% nonzero_inverse_eq_imp_eq
thf(fact_8011_nonzero__inverse__inverse__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ ( inverse_inverse_rat @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_8012_nonzero__inverse__inverse__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( inverse_inverse_real @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_8013_nonzero__inverse__inverse__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ ( invers8013647133539491842omplex @ A ) )
        = A ) ) ).

% nonzero_inverse_inverse_eq
thf(fact_8014_nonzero__imp__inverse__nonzero,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ A )
       != zero_zero_rat ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_8015_nonzero__imp__inverse__nonzero,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
       != zero_zero_real ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_8016_nonzero__imp__inverse__nonzero,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ A )
       != zero_zero_complex ) ) ).

% nonzero_imp_inverse_nonzero
thf(fact_8017_norm__inverse__le__norm,axiom,
    ! [R3: real,X4: real] :
      ( ( ord_less_eq_real @ R3 @ ( real_V7735802525324610683m_real @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ R3 )
       => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( inverse_inverse_real @ X4 ) ) @ ( inverse_inverse_real @ R3 ) ) ) ) ).

% norm_inverse_le_norm
thf(fact_8018_norm__inverse__le__norm,axiom,
    ! [R3: real,X4: complex] :
      ( ( ord_less_eq_real @ R3 @ ( real_V1022390504157884413omplex @ X4 ) )
     => ( ( ord_less_real @ zero_zero_real @ R3 )
       => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( invers8013647133539491842omplex @ X4 ) ) @ ( inverse_inverse_real @ R3 ) ) ) ) ).

% norm_inverse_le_norm
thf(fact_8019_positive__imp__inverse__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_8020_positive__imp__inverse__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) ) ) ).

% positive_imp_inverse_positive
thf(fact_8021_negative__imp__inverse__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ A @ zero_zero_rat )
     => ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat ) ) ).

% negative_imp_inverse_negative
thf(fact_8022_negative__imp__inverse__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ A @ zero_zero_real )
     => ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real ) ) ).

% negative_imp_inverse_negative
thf(fact_8023_inverse__positive__imp__positive,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( inverse_inverse_rat @ A ) )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ zero_zero_rat @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_8024_inverse__positive__imp__positive,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ ( inverse_inverse_real @ A ) )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ zero_zero_real @ A ) ) ) ).

% inverse_positive_imp_positive
thf(fact_8025_inverse__negative__imp__negative,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ zero_zero_rat )
     => ( ( A != zero_zero_rat )
       => ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% inverse_negative_imp_negative
thf(fact_8026_inverse__negative__imp__negative,axiom,
    ! [A: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ zero_zero_real )
     => ( ( A != zero_zero_real )
       => ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% inverse_negative_imp_negative
thf(fact_8027_less__imp__inverse__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_8028_less__imp__inverse__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less_neg
thf(fact_8029_inverse__less__imp__less__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_rat @ B @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_8030_inverse__less__imp__less__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_real @ B @ A ) ) ) ).

% inverse_less_imp_less_neg
thf(fact_8031_less__imp__inverse__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_8032_less__imp__inverse__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% less_imp_inverse_less
thf(fact_8033_inverse__less__imp__less,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_rat @ B @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_8034_inverse__less__imp__less,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_real @ B @ A ) ) ) ).

% inverse_less_imp_less
thf(fact_8035_nonzero__inverse__mult__distrib,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( inverse_inverse_rat @ ( times_times_rat @ A @ B ) )
          = ( times_times_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_8036_nonzero__inverse__mult__distrib,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( inverse_inverse_real @ ( times_times_real @ A @ B ) )
          = ( times_times_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_8037_nonzero__inverse__mult__distrib,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( invers8013647133539491842omplex @ ( times_times_complex @ A @ B ) )
          = ( times_times_complex @ ( invers8013647133539491842omplex @ B ) @ ( invers8013647133539491842omplex @ A ) ) ) ) ) ).

% nonzero_inverse_mult_distrib
thf(fact_8038_nonzero__inverse__minus__eq,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ ( uminus_uminus_rat @ A ) )
        = ( uminus_uminus_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_8039_nonzero__inverse__minus__eq,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ ( uminus_uminus_real @ A ) )
        = ( uminus_uminus_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_8040_nonzero__inverse__minus__eq,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ ( uminus1482373934393186551omplex @ A ) )
        = ( uminus1482373934393186551omplex @ ( invers8013647133539491842omplex @ A ) ) ) ) ).

% nonzero_inverse_minus_eq
thf(fact_8041_inverse__unique,axiom,
    ! [A: rat,B: rat] :
      ( ( ( times_times_rat @ A @ B )
        = one_one_rat )
     => ( ( inverse_inverse_rat @ A )
        = B ) ) ).

% inverse_unique
thf(fact_8042_inverse__unique,axiom,
    ! [A: real,B: real] :
      ( ( ( times_times_real @ A @ B )
        = one_one_real )
     => ( ( inverse_inverse_real @ A )
        = B ) ) ).

% inverse_unique
thf(fact_8043_inverse__unique,axiom,
    ! [A: complex,B: complex] :
      ( ( ( times_times_complex @ A @ B )
        = one_one_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = B ) ) ).

% inverse_unique
thf(fact_8044_inverse__numeral__1,axiom,
    ( ( inverse_inverse_real @ ( numeral_numeral_real @ one ) )
    = ( numeral_numeral_real @ one ) ) ).

% inverse_numeral_1
thf(fact_8045_inverse__numeral__1,axiom,
    ( ( invers8013647133539491842omplex @ ( numera6690914467698888265omplex @ one ) )
    = ( numera6690914467698888265omplex @ one ) ) ).

% inverse_numeral_1
thf(fact_8046_divide__inverse__commute,axiom,
    ( divide_divide_real
    = ( ^ [A3: real,B2: real] : ( times_times_real @ ( inverse_inverse_real @ B2 ) @ A3 ) ) ) ).

% divide_inverse_commute
thf(fact_8047_divide__inverse__commute,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ ( invers8013647133539491842omplex @ B2 ) @ A3 ) ) ) ).

% divide_inverse_commute
thf(fact_8048_divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A3: real,B2: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B2 ) ) ) ) ).

% divide_inverse
thf(fact_8049_divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).

% divide_inverse
thf(fact_8050_field__class_Ofield__divide__inverse,axiom,
    ( divide_divide_real
    = ( ^ [A3: real,B2: real] : ( times_times_real @ A3 @ ( inverse_inverse_real @ B2 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_8051_field__class_Ofield__divide__inverse,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B2: complex] : ( times_times_complex @ A3 @ ( invers8013647133539491842omplex @ B2 ) ) ) ) ).

% field_class.field_divide_inverse
thf(fact_8052_inverse__eq__divide,axiom,
    ( inverse_inverse_rat
    = ( divide_divide_rat @ one_one_rat ) ) ).

% inverse_eq_divide
thf(fact_8053_inverse__eq__divide,axiom,
    ( inverse_inverse_real
    = ( divide_divide_real @ one_one_real ) ) ).

% inverse_eq_divide
thf(fact_8054_inverse__eq__divide,axiom,
    ( invers8013647133539491842omplex
    = ( divide1717551699836669952omplex @ one_one_complex ) ) ).

% inverse_eq_divide
thf(fact_8055_power__mult__inverse__distrib,axiom,
    ! [X4: real,M: nat] :
      ( ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( inverse_inverse_real @ X4 ) )
      = ( times_times_real @ ( inverse_inverse_real @ X4 ) @ ( power_power_real @ X4 @ M ) ) ) ).

% power_mult_inverse_distrib
thf(fact_8056_power__mult__inverse__distrib,axiom,
    ! [X4: complex,M: nat] :
      ( ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( invers8013647133539491842omplex @ X4 ) )
      = ( times_times_complex @ ( invers8013647133539491842omplex @ X4 ) @ ( power_power_complex @ X4 @ M ) ) ) ).

% power_mult_inverse_distrib
thf(fact_8057_power__mult__power__inverse__commute,axiom,
    ! [X4: real,M: nat,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ N2 ) )
      = ( times_times_real @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ N2 ) @ ( power_power_real @ X4 @ M ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_8058_power__mult__power__inverse__commute,axiom,
    ! [X4: complex,M: nat,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ N2 ) )
      = ( times_times_complex @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ N2 ) @ ( power_power_complex @ X4 @ M ) ) ) ).

% power_mult_power_inverse_commute
thf(fact_8059_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X4: real] :
      ( ( times_times_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) @ X4 )
      = ( times_times_real @ X4 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ Xa ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_8060_mult__inverse__of__nat__commute,axiom,
    ! [Xa: nat,X4: complex] :
      ( ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) @ X4 )
      = ( times_times_complex @ X4 @ ( invers8013647133539491842omplex @ ( semiri8010041392384452111omplex @ Xa ) ) ) ) ).

% mult_inverse_of_nat_commute
thf(fact_8061_nonzero__abs__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( abs_abs_rat @ ( inverse_inverse_rat @ A ) )
        = ( inverse_inverse_rat @ ( abs_abs_rat @ A ) ) ) ) ).

% nonzero_abs_inverse
thf(fact_8062_nonzero__abs__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( abs_abs_real @ ( inverse_inverse_real @ A ) )
        = ( inverse_inverse_real @ ( abs_abs_real @ A ) ) ) ) ).

% nonzero_abs_inverse
thf(fact_8063_divide__real__def,axiom,
    ( divide_divide_real
    = ( ^ [X: real,Y5: real] : ( times_times_real @ X @ ( inverse_inverse_real @ Y5 ) ) ) ) ).

% divide_real_def
thf(fact_8064_frac__ge__0,axiom,
    ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X4 ) ) ).

% frac_ge_0
thf(fact_8065_frac__ge__0,axiom,
    ! [X4: rat] : ( ord_less_eq_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X4 ) ) ).

% frac_ge_0
thf(fact_8066_frac__lt__1,axiom,
    ! [X4: real] : ( ord_less_real @ ( archim2898591450579166408c_real @ X4 ) @ one_one_real ) ).

% frac_lt_1
thf(fact_8067_frac__lt__1,axiom,
    ! [X4: rat] : ( ord_less_rat @ ( archimedean_frac_rat @ X4 ) @ one_one_rat ) ).

% frac_lt_1
thf(fact_8068_frac__1__eq,axiom,
    ! [X4: real] :
      ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ one_one_real ) )
      = ( archim2898591450579166408c_real @ X4 ) ) ).

% frac_1_eq
thf(fact_8069_frac__1__eq,axiom,
    ! [X4: rat] :
      ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ one_one_rat ) )
      = ( archimedean_frac_rat @ X4 ) ) ).

% frac_1_eq
thf(fact_8070_le__imp__inverse__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% le_imp_inverse_le_neg
thf(fact_8071_le__imp__inverse__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% le_imp_inverse_le_neg
thf(fact_8072_inverse__le__imp__le__neg,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ B @ zero_zero_rat )
       => ( ord_less_eq_rat @ B @ A ) ) ) ).

% inverse_le_imp_le_neg
thf(fact_8073_inverse__le__imp__le__neg,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ B @ zero_zero_real )
       => ( ord_less_eq_real @ B @ A ) ) ) ).

% inverse_le_imp_le_neg
thf(fact_8074_le__imp__inverse__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ A @ B )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ ( inverse_inverse_rat @ B ) @ ( inverse_inverse_rat @ A ) ) ) ) ).

% le_imp_inverse_le
thf(fact_8075_le__imp__inverse__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ ( inverse_inverse_real @ B ) @ ( inverse_inverse_real @ A ) ) ) ) ).

% le_imp_inverse_le
thf(fact_8076_inverse__le__imp__le,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
     => ( ( ord_less_rat @ zero_zero_rat @ A )
       => ( ord_less_eq_rat @ B @ A ) ) ) ).

% inverse_le_imp_le
thf(fact_8077_inverse__le__imp__le,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
     => ( ( ord_less_real @ zero_zero_real @ A )
       => ( ord_less_eq_real @ B @ A ) ) ) ).

% inverse_le_imp_le
thf(fact_8078_inverse__le__1__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ X4 ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
        | ( ord_less_eq_rat @ one_one_rat @ X4 ) ) ) ).

% inverse_le_1_iff
thf(fact_8079_inverse__le__1__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
      = ( ( ord_less_eq_real @ X4 @ zero_zero_real )
        | ( ord_less_eq_real @ one_one_real @ X4 ) ) ) ).

% inverse_le_1_iff
thf(fact_8080_one__less__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_rat @ A @ one_one_rat )
       => ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_less_inverse
thf(fact_8081_one__less__inverse,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_real @ A @ one_one_real )
       => ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% one_less_inverse
thf(fact_8082_one__less__inverse__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ one_one_rat @ ( inverse_inverse_rat @ X4 ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X4 )
        & ( ord_less_rat @ X4 @ one_one_rat ) ) ) ).

% one_less_inverse_iff
thf(fact_8083_one__less__inverse__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
      = ( ( ord_less_real @ zero_zero_real @ X4 )
        & ( ord_less_real @ X4 @ one_one_real ) ) ) ).

% one_less_inverse_iff
thf(fact_8084_field__class_Ofield__inverse,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( times_times_rat @ ( inverse_inverse_rat @ A ) @ A )
        = one_one_rat ) ) ).

% field_class.field_inverse
thf(fact_8085_field__class_Ofield__inverse,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( times_times_real @ ( inverse_inverse_real @ A ) @ A )
        = one_one_real ) ) ).

% field_class.field_inverse
thf(fact_8086_field__class_Ofield__inverse,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ A )
        = one_one_complex ) ) ).

% field_class.field_inverse
thf(fact_8087_division__ring__inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( plus_plus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_8088_division__ring__inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( plus_plus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_8089_division__ring__inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
          = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( plus_plus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).

% division_ring_inverse_add
thf(fact_8090_inverse__add,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( plus_plus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( plus_plus_rat @ A @ B ) @ ( inverse_inverse_rat @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% inverse_add
thf(fact_8091_inverse__add,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( plus_plus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( plus_plus_real @ A @ B ) @ ( inverse_inverse_real @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% inverse_add
thf(fact_8092_inverse__add,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( plus_plus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
          = ( times_times_complex @ ( times_times_complex @ ( plus_plus_complex @ A @ B ) @ ( invers8013647133539491842omplex @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).

% inverse_add
thf(fact_8093_division__ring__inverse__diff,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ B @ A ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_8094_division__ring__inverse__diff,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ B @ A ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_8095_division__ring__inverse__diff,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
          = ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ B @ A ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ).

% division_ring_inverse_diff
thf(fact_8096_nonzero__inverse__eq__divide,axiom,
    ! [A: rat] :
      ( ( A != zero_zero_rat )
     => ( ( inverse_inverse_rat @ A )
        = ( divide_divide_rat @ one_one_rat @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_8097_nonzero__inverse__eq__divide,axiom,
    ! [A: real] :
      ( ( A != zero_zero_real )
     => ( ( inverse_inverse_real @ A )
        = ( divide_divide_real @ one_one_real @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_8098_nonzero__inverse__eq__divide,axiom,
    ! [A: complex] :
      ( ( A != zero_zero_complex )
     => ( ( invers8013647133539491842omplex @ A )
        = ( divide1717551699836669952omplex @ one_one_complex @ A ) ) ) ).

% nonzero_inverse_eq_divide
thf(fact_8099_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_8100_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_8101_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_8102_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_8103_inverse__powr,axiom,
    ! [Y: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ Y )
     => ( ( powr_real @ ( inverse_inverse_real @ Y ) @ A )
        = ( inverse_inverse_real @ ( powr_real @ Y @ A ) ) ) ) ).

% inverse_powr
thf(fact_8104_Complex__eq__i,axiom,
    ! [X4: real,Y: real] :
      ( ( ( complex2 @ X4 @ Y )
        = imaginary_unit )
      = ( ( X4 = zero_zero_real )
        & ( Y = one_one_real ) ) ) ).

% Complex_eq_i
thf(fact_8105_imaginary__unit_Ocode,axiom,
    ( imaginary_unit
    = ( complex2 @ zero_zero_real @ one_one_real ) ) ).

% imaginary_unit.code
thf(fact_8106_inverse__le__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_eq_rat @ B @ A ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
         => ( ord_less_eq_rat @ A @ B ) ) ) ) ).

% inverse_le_iff
thf(fact_8107_inverse__le__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_eq_real @ B @ A ) )
        & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
         => ( ord_less_eq_real @ A @ B ) ) ) ) ).

% inverse_le_iff
thf(fact_8108_inverse__less__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
      = ( ( ( ord_less_rat @ zero_zero_rat @ ( times_times_rat @ A @ B ) )
         => ( ord_less_rat @ B @ A ) )
        & ( ( ord_less_eq_rat @ ( times_times_rat @ A @ B ) @ zero_zero_rat )
         => ( ord_less_rat @ A @ B ) ) ) ) ).

% inverse_less_iff
thf(fact_8109_inverse__less__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ ( times_times_real @ A @ B ) )
         => ( ord_less_real @ B @ A ) )
        & ( ( ord_less_eq_real @ ( times_times_real @ A @ B ) @ zero_zero_real )
         => ( ord_less_real @ A @ B ) ) ) ) ).

% inverse_less_iff
thf(fact_8110_one__le__inverse,axiom,
    ! [A: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ A )
     => ( ( ord_less_eq_rat @ A @ one_one_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ A ) ) ) ) ).

% one_le_inverse
thf(fact_8111_one__le__inverse,axiom,
    ! [A: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ A ) ) ) ) ).

% one_le_inverse
thf(fact_8112_inverse__less__1__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ ( inverse_inverse_rat @ X4 ) @ one_one_rat )
      = ( ( ord_less_eq_rat @ X4 @ zero_zero_rat )
        | ( ord_less_rat @ one_one_rat @ X4 ) ) ) ).

% inverse_less_1_iff
thf(fact_8113_inverse__less__1__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( inverse_inverse_real @ X4 ) @ one_one_real )
      = ( ( ord_less_eq_real @ X4 @ zero_zero_real )
        | ( ord_less_real @ one_one_real @ X4 ) ) ) ).

% inverse_less_1_iff
thf(fact_8114_one__le__inverse__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_eq_rat @ one_one_rat @ ( inverse_inverse_rat @ X4 ) )
      = ( ( ord_less_rat @ zero_zero_rat @ X4 )
        & ( ord_less_eq_rat @ X4 @ one_one_rat ) ) ) ).

% one_le_inverse_iff
thf(fact_8115_one__le__inverse__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ one_one_real @ ( inverse_inverse_real @ X4 ) )
      = ( ( ord_less_real @ zero_zero_real @ X4 )
        & ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).

% one_le_inverse_iff
thf(fact_8116_inverse__diff__inverse,axiom,
    ! [A: rat,B: rat] :
      ( ( A != zero_zero_rat )
     => ( ( B != zero_zero_rat )
       => ( ( minus_minus_rat @ ( inverse_inverse_rat @ A ) @ ( inverse_inverse_rat @ B ) )
          = ( uminus_uminus_rat @ ( times_times_rat @ ( times_times_rat @ ( inverse_inverse_rat @ A ) @ ( minus_minus_rat @ A @ B ) ) @ ( inverse_inverse_rat @ B ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_8117_inverse__diff__inverse,axiom,
    ! [A: real,B: real] :
      ( ( A != zero_zero_real )
     => ( ( B != zero_zero_real )
       => ( ( minus_minus_real @ ( inverse_inverse_real @ A ) @ ( inverse_inverse_real @ B ) )
          = ( uminus_uminus_real @ ( times_times_real @ ( times_times_real @ ( inverse_inverse_real @ A ) @ ( minus_minus_real @ A @ B ) ) @ ( inverse_inverse_real @ B ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_8118_inverse__diff__inverse,axiom,
    ! [A: complex,B: complex] :
      ( ( A != zero_zero_complex )
     => ( ( B != zero_zero_complex )
       => ( ( minus_minus_complex @ ( invers8013647133539491842omplex @ A ) @ ( invers8013647133539491842omplex @ B ) )
          = ( uminus1482373934393186551omplex @ ( times_times_complex @ ( times_times_complex @ ( invers8013647133539491842omplex @ A ) @ ( minus_minus_complex @ A @ B ) ) @ ( invers8013647133539491842omplex @ B ) ) ) ) ) ) ).

% inverse_diff_inverse
thf(fact_8119_reals__Archimedean,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ? [N3: nat] : ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ ( suc @ N3 ) ) ) @ X4 ) ) ).

% reals_Archimedean
thf(fact_8120_reals__Archimedean,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [N3: nat] : ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N3 ) ) ) @ X4 ) ) ).

% reals_Archimedean
thf(fact_8121_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_8122_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_8123_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_8124_gbinomial__absorb__comp,axiom,
    ! [A: rat,K: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ A @ ( semiri681578069525770553at_rat @ K ) ) @ ( gbinomial_rat @ A @ K ) )
      = ( times_times_rat @ A @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_8125_gbinomial__absorb__comp,axiom,
    ! [A: complex,K: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ A @ ( semiri8010041392384452111omplex @ K ) ) @ ( gbinomial_complex @ A @ K ) )
      = ( times_times_complex @ A @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_8126_gbinomial__absorb__comp,axiom,
    ! [A: real,K: nat] :
      ( ( times_times_real @ ( minus_minus_real @ A @ ( semiri5074537144036343181t_real @ K ) ) @ ( gbinomial_real @ A @ K ) )
      = ( times_times_real @ A @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ K ) ) ) ).

% gbinomial_absorb_comp
thf(fact_8127_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_8128_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_8129_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_8130_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_8131_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_8132_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_8133_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_8134_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_8135_forall__pos__mono__1,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D3: real,E: real] :
          ( ( ord_less_real @ D3 @ E )
         => ( ( P @ D3 )
           => ( 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_8136_forall__pos__mono,axiom,
    ! [P: real > $o,E2: real] :
      ( ! [D3: real,E: real] :
          ( ( ord_less_real @ D3 @ E )
         => ( ( P @ D3 )
           => ( 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_8137_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_8138_sqrt__divide__self__eq,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( divide_divide_real @ ( sqrt @ X4 ) @ X4 )
        = ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) ) ).

% sqrt_divide_self_eq
thf(fact_8139_ln__inverse,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ln_ln_real @ ( inverse_inverse_real @ X4 ) )
        = ( uminus_uminus_real @ ( ln_ln_real @ X4 ) ) ) ) ).

% ln_inverse
thf(fact_8140_ex__inverse__of__nat__less,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ X4 )
     => ? [N3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N3 )
          & ( ord_less_rat @ ( inverse_inverse_rat @ ( semiri681578069525770553at_rat @ N3 ) ) @ X4 ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_8141_ex__inverse__of__nat__less,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [N3: nat] :
          ( ( ord_less_nat @ zero_zero_nat @ N3 )
          & ( ord_less_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ N3 ) ) @ X4 ) ) ) ).

% ex_inverse_of_nat_less
thf(fact_8142_power__diff__conv__inverse,axiom,
    ! [X4: rat,M: nat,N2: nat] :
      ( ( X4 != zero_zero_rat )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( power_power_rat @ X4 @ ( minus_minus_nat @ N2 @ M ) )
          = ( times_times_rat @ ( power_power_rat @ X4 @ N2 ) @ ( power_power_rat @ ( inverse_inverse_rat @ X4 ) @ M ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_8143_power__diff__conv__inverse,axiom,
    ! [X4: real,M: nat,N2: nat] :
      ( ( X4 != zero_zero_real )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( power_power_real @ X4 @ ( minus_minus_nat @ N2 @ M ) )
          = ( times_times_real @ ( power_power_real @ X4 @ N2 ) @ ( power_power_real @ ( inverse_inverse_real @ X4 ) @ M ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_8144_power__diff__conv__inverse,axiom,
    ! [X4: complex,M: nat,N2: nat] :
      ( ( X4 != zero_zero_complex )
     => ( ( ord_less_eq_nat @ M @ N2 )
       => ( ( power_power_complex @ X4 @ ( minus_minus_nat @ N2 @ M ) )
          = ( times_times_complex @ ( power_power_complex @ X4 @ N2 ) @ ( power_power_complex @ ( invers8013647133539491842omplex @ X4 ) @ M ) ) ) ) ) ).

% power_diff_conv_inverse
thf(fact_8145_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_8146_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_8147_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_8148_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_8149_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_8150_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_8151_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_8152_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_8153_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_8154_log__inverse,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_real @ zero_zero_real @ A )
     => ( ( A != one_one_real )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( log @ A @ ( inverse_inverse_real @ X4 ) )
            = ( uminus_uminus_real @ ( log @ A @ X4 ) ) ) ) ) ) ).

% log_inverse
thf(fact_8155_frac__eq,axiom,
    ! [X4: real] :
      ( ( ( archim2898591450579166408c_real @ X4 )
        = X4 )
      = ( ( ord_less_eq_real @ zero_zero_real @ X4 )
        & ( ord_less_real @ X4 @ one_one_real ) ) ) ).

% frac_eq
thf(fact_8156_frac__eq,axiom,
    ! [X4: rat] :
      ( ( ( archimedean_frac_rat @ X4 )
        = X4 )
      = ( ( ord_less_eq_rat @ zero_zero_rat @ X4 )
        & ( ord_less_rat @ X4 @ one_one_rat ) ) ) ).

% frac_eq
thf(fact_8157_frac__add,axiom,
    ! [X4: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ Y ) )
          = ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim2898591450579166408c_real @ ( plus_plus_real @ X4 @ Y ) )
          = ( minus_minus_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real ) ) ) ) ).

% frac_add
thf(fact_8158_frac__add,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ Y ) )
          = ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archimedean_frac_rat @ ( plus_plus_rat @ X4 @ Y ) )
          = ( minus_minus_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat ) ) ) ) ).

% frac_add
thf(fact_8159_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_8160_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_8161_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_8162_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_8163_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_8164_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_8165_gbinomial__negated__upper,axiom,
    ( gbinomial_complex
    = ( ^ [A3: complex,K3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( minus_minus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A3 ) @ one_one_complex ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8166_gbinomial__negated__upper,axiom,
    ( gbinomial_rat
    = ( ^ [A3: rat,K3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( minus_minus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A3 ) @ one_one_rat ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8167_gbinomial__negated__upper,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( gbinomial_real @ ( minus_minus_real @ ( minus_minus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A3 ) @ one_one_real ) @ K3 ) ) ) ) ).

% gbinomial_negated_upper
thf(fact_8168_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ K ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ N2 ) @ ( gbinomial_complex @ ( minus_minus_complex @ ( uminus1482373934393186551omplex @ ( semiri8010041392384452111omplex @ K ) ) @ one_one_complex ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_8169_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ K ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ N2 ) @ ( gbinomial_rat @ ( minus_minus_rat @ ( uminus_uminus_rat @ ( semiri681578069525770553at_rat @ K ) ) @ one_one_rat ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_8170_gbinomial__index__swap,axiom,
    ! [K: nat,N2: nat] :
      ( ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ K ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N2 ) @ ( gbinomial_real @ ( minus_minus_real @ ( uminus_uminus_real @ ( semiri5074537144036343181t_real @ K ) ) @ one_one_real ) @ N2 ) ) ) ).

% gbinomial_index_swap
thf(fact_8171_exp__plus__inverse__exp,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) ) ).

% exp_plus_inverse_exp
thf(fact_8172_cmod__unit__one,axiom,
    ! [A: real] :
      ( ( real_V1022390504157884413omplex @ ( plus_plus_complex @ ( real_V4546457046886955230omplex @ ( cos_real @ A ) ) @ ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sin_real @ A ) ) ) ) )
      = one_one_real ) ).

% cmod_unit_one
thf(fact_8173_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_8174_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_8175_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_8176_plus__inverse__ge__2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) ) ) ).

% plus_inverse_ge_2
thf(fact_8177_real__inv__sqrt__pow2,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( power_power_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( inverse_inverse_real @ X4 ) ) ) ).

% real_inv_sqrt_pow2
thf(fact_8178_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_8179_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_8180_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_8181_gbinomial__pochhammer,axiom,
    ( gbinomial_complex
    = ( ^ [A3: complex,K3: nat] : ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) @ ( comm_s2602460028002588243omplex @ ( uminus1482373934393186551omplex @ A3 ) @ K3 ) ) @ ( semiri5044797733671781792omplex @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8182_gbinomial__pochhammer,axiom,
    ( gbinomial_rat
    = ( ^ [A3: rat,K3: nat] : ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) @ ( comm_s4028243227959126397er_rat @ ( uminus_uminus_rat @ A3 ) @ K3 ) ) @ ( semiri773545260158071498ct_rat @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8183_gbinomial__pochhammer,axiom,
    ( gbinomial_real
    = ( ^ [A3: real,K3: nat] : ( divide_divide_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( comm_s7457072308508201937r_real @ ( uminus_uminus_real @ A3 ) @ K3 ) ) @ ( semiri2265585572941072030t_real @ K3 ) ) ) ) ).

% gbinomial_pochhammer
thf(fact_8184_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_8185_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_8186_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_8187_tan__cot,axiom,
    ! [X4: real] :
      ( ( tan_real @ ( minus_minus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ X4 ) )
      = ( inverse_inverse_real @ ( tan_real @ X4 ) ) ) ).

% tan_cot
thf(fact_8188_floor__add,axiom,
    ! [X4: real,Y: real] :
      ( ( ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y ) )
          = ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y ) ) ) )
      & ( ~ ( ord_less_real @ ( plus_plus_real @ ( archim2898591450579166408c_real @ X4 ) @ ( archim2898591450579166408c_real @ Y ) ) @ one_one_real )
       => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_8189_floor__add,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y ) )
          = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y ) ) ) )
      & ( ~ ( ord_less_rat @ ( plus_plus_rat @ ( archimedean_frac_rat @ X4 ) @ ( archimedean_frac_rat @ Y ) ) @ one_one_rat )
       => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y ) )
          = ( plus_plus_int @ ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y ) ) @ one_one_int ) ) ) ) ).

% floor_add
thf(fact_8190_real__le__x__sinh,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ X4 @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_x_sinh
thf(fact_8191_real__le__abs__sinh,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ ( abs_abs_real @ ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ X4 ) @ ( inverse_inverse_real @ ( exp_real @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% real_le_abs_sinh
thf(fact_8192_tan__sec,axiom,
    ! [X4: real] :
      ( ( ( cos_real @ X4 )
       != zero_zero_real )
     => ( ( plus_plus_real @ one_one_real @ ( power_power_real @ ( tan_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( power_power_real @ ( inverse_inverse_real @ ( cos_real @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% tan_sec
thf(fact_8193_tan__sec,axiom,
    ! [X4: complex] :
      ( ( ( cos_complex @ X4 )
       != zero_zero_complex )
     => ( ( plus_plus_complex @ one_one_complex @ ( power_power_complex @ ( tan_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = ( power_power_complex @ ( invers8013647133539491842omplex @ ( cos_complex @ X4 ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% tan_sec
thf(fact_8194_Arg__minus__ii,axiom,
    ( ( arg @ ( uminus1482373934393186551omplex @ imaginary_unit ) )
    = ( divide_divide_real @ ( uminus_uminus_real @ pi ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_minus_ii
thf(fact_8195_csqrt__ii,axiom,
    ( ( csqrt @ imaginary_unit )
    = ( divide1717551699836669952omplex @ ( plus_plus_complex @ one_one_complex @ imaginary_unit ) @ ( real_V4546457046886955230omplex @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt_ii
thf(fact_8196_Arg__ii,axiom,
    ( ( arg @ imaginary_unit )
    = ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ).

% Arg_ii
thf(fact_8197_sinh__ln__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( sinh_real @ ( ln_ln_real @ X4 ) )
        = ( divide_divide_real @ ( minus_minus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_ln_real
thf(fact_8198_cis__multiple__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( cis @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = one_one_complex ) ) ).

% cis_multiple_2pi
thf(fact_8199_sinh__real__less__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y ) )
      = ( ord_less_real @ X4 @ Y ) ) ).

% sinh_real_less_iff
thf(fact_8200_sinh__real__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% sinh_real_le_iff
thf(fact_8201_sinh__real__neg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( sinh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_real @ X4 @ zero_zero_real ) ) ).

% sinh_real_neg_iff
thf(fact_8202_sinh__real__pos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( sinh_real @ X4 ) )
      = ( ord_less_real @ zero_zero_real @ X4 ) ) ).

% sinh_real_pos_iff
thf(fact_8203_sinh__real__nonneg__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( sinh_real @ X4 ) )
      = ( ord_less_eq_real @ zero_zero_real @ X4 ) ) ).

% sinh_real_nonneg_iff
thf(fact_8204_sinh__real__nonpos__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( sinh_real @ X4 ) @ zero_zero_real )
      = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ).

% sinh_real_nonpos_iff
thf(fact_8205_floor__add2,axiom,
    ! [X4: real,Y: real] :
      ( ( ( member_real @ X4 @ ring_1_Ints_real )
        | ( member_real @ Y @ ring_1_Ints_real ) )
     => ( ( archim6058952711729229775r_real @ ( plus_plus_real @ X4 @ Y ) )
        = ( plus_plus_int @ ( archim6058952711729229775r_real @ X4 ) @ ( archim6058952711729229775r_real @ Y ) ) ) ) ).

% floor_add2
thf(fact_8206_floor__add2,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ( member_rat @ X4 @ ring_1_Ints_rat )
        | ( member_rat @ Y @ ring_1_Ints_rat ) )
     => ( ( archim3151403230148437115or_rat @ ( plus_plus_rat @ X4 @ Y ) )
        = ( plus_plus_int @ ( archim3151403230148437115or_rat @ X4 ) @ ( archim3151403230148437115or_rat @ Y ) ) ) ) ).

% floor_add2
thf(fact_8207_frac__gt__0__iff,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ ( archim2898591450579166408c_real @ X4 ) )
      = ( ~ ( member_real @ X4 @ ring_1_Ints_real ) ) ) ).

% frac_gt_0_iff
thf(fact_8208_frac__gt__0__iff,axiom,
    ! [X4: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ ( archimedean_frac_rat @ X4 ) )
      = ( ~ ( member_rat @ X4 @ ring_1_Ints_rat ) ) ) ).

% frac_gt_0_iff
thf(fact_8209_power2__csqrt,axiom,
    ! [Z: complex] :
      ( ( power_power_complex @ ( csqrt @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = Z ) ).

% power2_csqrt
thf(fact_8210_Ints__power,axiom,
    ! [A: real,N2: nat] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( member_real @ ( power_power_real @ A @ N2 ) @ ring_1_Ints_real ) ) ).

% Ints_power
thf(fact_8211_Ints__power,axiom,
    ! [A: int,N2: nat] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( member_int @ ( power_power_int @ A @ N2 ) @ ring_1_Ints_int ) ) ).

% Ints_power
thf(fact_8212_Ints__power,axiom,
    ! [A: complex,N2: nat] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( member_complex @ ( power_power_complex @ A @ N2 ) @ ring_1_Ints_complex ) ) ).

% Ints_power
thf(fact_8213_Ints__numeral,axiom,
    ! [N2: num] : ( member_complex @ ( numera6690914467698888265omplex @ N2 ) @ ring_1_Ints_complex ) ).

% Ints_numeral
thf(fact_8214_Ints__numeral,axiom,
    ! [N2: num] : ( member_real @ ( numeral_numeral_real @ N2 ) @ ring_1_Ints_real ) ).

% Ints_numeral
thf(fact_8215_Ints__numeral,axiom,
    ! [N2: num] : ( member_int @ ( numeral_numeral_int @ N2 ) @ ring_1_Ints_int ) ).

% Ints_numeral
thf(fact_8216_Ints__1,axiom,
    member_complex @ one_one_complex @ ring_1_Ints_complex ).

% Ints_1
thf(fact_8217_Ints__1,axiom,
    member_rat @ one_one_rat @ ring_1_Ints_rat ).

% Ints_1
thf(fact_8218_Ints__1,axiom,
    member_int @ one_one_int @ ring_1_Ints_int ).

% Ints_1
thf(fact_8219_Ints__1,axiom,
    member_real @ one_one_real @ ring_1_Ints_real ).

% Ints_1
thf(fact_8220_Ints__add,axiom,
    ! [A: complex,B: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( member_complex @ B @ ring_1_Ints_complex )
       => ( member_complex @ ( plus_plus_complex @ A @ B ) @ ring_1_Ints_complex ) ) ) ).

% Ints_add
thf(fact_8221_Ints__add,axiom,
    ! [A: real,B: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( member_real @ B @ ring_1_Ints_real )
       => ( member_real @ ( plus_plus_real @ A @ B ) @ ring_1_Ints_real ) ) ) ).

% Ints_add
thf(fact_8222_Ints__add,axiom,
    ! [A: rat,B: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( member_rat @ B @ ring_1_Ints_rat )
       => ( member_rat @ ( plus_plus_rat @ A @ B ) @ ring_1_Ints_rat ) ) ) ).

% Ints_add
thf(fact_8223_Ints__add,axiom,
    ! [A: int,B: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( member_int @ B @ ring_1_Ints_int )
       => ( member_int @ ( plus_plus_int @ A @ B ) @ ring_1_Ints_int ) ) ) ).

% Ints_add
thf(fact_8224_Ints__double__eq__0__iff,axiom,
    ! [A: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( ( plus_plus_complex @ A @ A )
          = zero_zero_complex )
        = ( A = zero_zero_complex ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_8225_Ints__double__eq__0__iff,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ( plus_plus_real @ A @ A )
          = zero_zero_real )
        = ( A = zero_zero_real ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_8226_Ints__double__eq__0__iff,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ( plus_plus_rat @ A @ A )
          = zero_zero_rat )
        = ( A = zero_zero_rat ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_8227_Ints__double__eq__0__iff,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ( plus_plus_int @ A @ A )
          = zero_zero_int )
        = ( A = zero_zero_int ) ) ) ).

% Ints_double_eq_0_iff
thf(fact_8228_Ints__odd__nonzero,axiom,
    ! [A: complex] :
      ( ( member_complex @ A @ ring_1_Ints_complex )
     => ( ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ A ) @ A )
       != zero_zero_complex ) ) ).

% Ints_odd_nonzero
thf(fact_8229_Ints__odd__nonzero,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A )
       != zero_zero_real ) ) ).

% Ints_odd_nonzero
thf(fact_8230_Ints__odd__nonzero,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A )
       != zero_zero_rat ) ) ).

% Ints_odd_nonzero
thf(fact_8231_Ints__odd__nonzero,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A )
       != zero_zero_int ) ) ).

% Ints_odd_nonzero
thf(fact_8232_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_rat @ ( divide_divide_rat @ ( ring_1_of_int_rat @ A ) @ ( ring_1_of_int_rat @ B ) ) @ ring_1_Ints_rat ) ) ).

% of_int_divide_in_Ints
thf(fact_8233_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_int @ ( divide_divide_int @ ( ring_1_of_int_int @ A ) @ ( ring_1_of_int_int @ B ) ) @ ring_1_Ints_int ) ) ).

% of_int_divide_in_Ints
thf(fact_8234_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_real @ ( divide_divide_real @ ( ring_1_of_int_real @ A ) @ ( ring_1_of_int_real @ B ) ) @ ring_1_Ints_real ) ) ).

% of_int_divide_in_Ints
thf(fact_8235_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_complex @ ( divide1717551699836669952omplex @ ( ring_17405671764205052669omplex @ A ) @ ( ring_17405671764205052669omplex @ B ) ) @ ring_1_Ints_complex ) ) ).

% of_int_divide_in_Ints
thf(fact_8236_of__int__divide__in__Ints,axiom,
    ! [B: int,A: int] :
      ( ( dvd_dvd_int @ B @ A )
     => ( member_Code_integer @ ( divide6298287555418463151nteger @ ( ring_18347121197199848620nteger @ A ) @ ( ring_18347121197199848620nteger @ B ) ) @ ring_11222124179247155820nteger ) ) ).

% of_int_divide_in_Ints
thf(fact_8237_Ints__odd__less__0,axiom,
    ! [A: real] :
      ( ( member_real @ A @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( plus_plus_real @ ( plus_plus_real @ one_one_real @ A ) @ A ) @ zero_zero_real )
        = ( ord_less_real @ A @ zero_zero_real ) ) ) ).

% Ints_odd_less_0
thf(fact_8238_Ints__odd__less__0,axiom,
    ! [A: rat] :
      ( ( member_rat @ A @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( plus_plus_rat @ ( plus_plus_rat @ one_one_rat @ A ) @ A ) @ zero_zero_rat )
        = ( ord_less_rat @ A @ zero_zero_rat ) ) ) ).

% Ints_odd_less_0
thf(fact_8239_Ints__odd__less__0,axiom,
    ! [A: int] :
      ( ( member_int @ A @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( plus_plus_int @ ( plus_plus_int @ one_one_int @ A ) @ A ) @ zero_zero_int )
        = ( ord_less_int @ A @ zero_zero_int ) ) ) ).

% Ints_odd_less_0
thf(fact_8240_Ints__nonzero__abs__ge1,axiom,
    ! [X4: code_integer] :
      ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
     => ( ( X4 != zero_z3403309356797280102nteger )
       => ( ord_le3102999989581377725nteger @ one_one_Code_integer @ ( abs_abs_Code_integer @ X4 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_8241_Ints__nonzero__abs__ge1,axiom,
    ! [X4: real] :
      ( ( member_real @ X4 @ ring_1_Ints_real )
     => ( ( X4 != zero_zero_real )
       => ( ord_less_eq_real @ one_one_real @ ( abs_abs_real @ X4 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_8242_Ints__nonzero__abs__ge1,axiom,
    ! [X4: rat] :
      ( ( member_rat @ X4 @ ring_1_Ints_rat )
     => ( ( X4 != zero_zero_rat )
       => ( ord_less_eq_rat @ one_one_rat @ ( abs_abs_rat @ X4 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_8243_Ints__nonzero__abs__ge1,axiom,
    ! [X4: int] :
      ( ( member_int @ X4 @ ring_1_Ints_int )
     => ( ( X4 != zero_zero_int )
       => ( ord_less_eq_int @ one_one_int @ ( abs_abs_int @ X4 ) ) ) ) ).

% Ints_nonzero_abs_ge1
thf(fact_8244_Ints__nonzero__abs__less1,axiom,
    ! [X4: code_integer] :
      ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
     => ( ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ X4 ) @ one_one_Code_integer )
       => ( X4 = zero_z3403309356797280102nteger ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_8245_Ints__nonzero__abs__less1,axiom,
    ! [X4: real] :
      ( ( member_real @ X4 @ ring_1_Ints_real )
     => ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
       => ( X4 = zero_zero_real ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_8246_Ints__nonzero__abs__less1,axiom,
    ! [X4: rat] :
      ( ( member_rat @ X4 @ ring_1_Ints_rat )
     => ( ( ord_less_rat @ ( abs_abs_rat @ X4 ) @ one_one_rat )
       => ( X4 = zero_zero_rat ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_8247_Ints__nonzero__abs__less1,axiom,
    ! [X4: int] :
      ( ( member_int @ X4 @ ring_1_Ints_int )
     => ( ( ord_less_int @ ( abs_abs_int @ X4 ) @ one_one_int )
       => ( X4 = zero_zero_int ) ) ) ).

% Ints_nonzero_abs_less1
thf(fact_8248_Ints__eq__abs__less1,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( member_Code_integer @ X4 @ ring_11222124179247155820nteger )
     => ( ( member_Code_integer @ Y @ ring_11222124179247155820nteger )
       => ( ( X4 = Y )
          = ( ord_le6747313008572928689nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ X4 @ Y ) ) @ one_one_Code_integer ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_8249_Ints__eq__abs__less1,axiom,
    ! [X4: real,Y: real] :
      ( ( member_real @ X4 @ ring_1_Ints_real )
     => ( ( member_real @ Y @ ring_1_Ints_real )
       => ( ( X4 = Y )
          = ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y ) ) @ one_one_real ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_8250_Ints__eq__abs__less1,axiom,
    ! [X4: rat,Y: rat] :
      ( ( member_rat @ X4 @ ring_1_Ints_rat )
     => ( ( member_rat @ Y @ ring_1_Ints_rat )
       => ( ( X4 = Y )
          = ( ord_less_rat @ ( abs_abs_rat @ ( minus_minus_rat @ X4 @ Y ) ) @ one_one_rat ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_8251_Ints__eq__abs__less1,axiom,
    ! [X4: int,Y: int] :
      ( ( member_int @ X4 @ ring_1_Ints_int )
     => ( ( member_int @ Y @ ring_1_Ints_int )
       => ( ( X4 = Y )
          = ( ord_less_int @ ( abs_abs_int @ ( minus_minus_int @ X4 @ Y ) ) @ one_one_int ) ) ) ) ).

% Ints_eq_abs_less1
thf(fact_8252_of__real__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( real_V4546457046886955230omplex @ ( sqrt @ X4 ) )
        = ( csqrt @ ( real_V4546457046886955230omplex @ X4 ) ) ) ) ).

% of_real_sqrt
thf(fact_8253_frac__neg,axiom,
    ! [X4: real] :
      ( ( ( member_real @ X4 @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X4 ) )
          = zero_zero_real ) )
      & ( ~ ( member_real @ X4 @ ring_1_Ints_real )
       => ( ( archim2898591450579166408c_real @ ( uminus_uminus_real @ X4 ) )
          = ( minus_minus_real @ one_one_real @ ( archim2898591450579166408c_real @ X4 ) ) ) ) ) ).

% frac_neg
thf(fact_8254_frac__neg,axiom,
    ! [X4: rat] :
      ( ( ( member_rat @ X4 @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X4 ) )
          = zero_zero_rat ) )
      & ( ~ ( member_rat @ X4 @ ring_1_Ints_rat )
       => ( ( archimedean_frac_rat @ ( uminus_uminus_rat @ X4 ) )
          = ( minus_minus_rat @ one_one_rat @ ( archimedean_frac_rat @ X4 ) ) ) ) ) ).

% frac_neg
thf(fact_8255_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_real @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_8256_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_8257_le__mult__floor__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim6058952711729229775r_real @ A ) @ ( archim6058952711729229775r_real @ B ) ) ) @ ( ring_1_of_int_int @ ( archim6058952711729229775r_real @ ( times_times_real @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_8258_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_real @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_8259_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_rat @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_8260_le__mult__floor__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( times_times_int @ ( archim3151403230148437115or_rat @ A ) @ ( archim3151403230148437115or_rat @ B ) ) ) @ ( ring_1_of_int_int @ ( archim3151403230148437115or_rat @ ( times_times_rat @ A @ B ) ) ) ) ) ) ).

% le_mult_floor_Ints
thf(fact_8261_frac__unique__iff,axiom,
    ! [X4: real,A: real] :
      ( ( ( archim2898591450579166408c_real @ X4 )
        = A )
      = ( ( member_real @ ( minus_minus_real @ X4 @ A ) @ ring_1_Ints_real )
        & ( ord_less_eq_real @ zero_zero_real @ A )
        & ( ord_less_real @ A @ one_one_real ) ) ) ).

% frac_unique_iff
thf(fact_8262_frac__unique__iff,axiom,
    ! [X4: rat,A: rat] :
      ( ( ( archimedean_frac_rat @ X4 )
        = A )
      = ( ( member_rat @ ( minus_minus_rat @ X4 @ A ) @ ring_1_Ints_rat )
        & ( ord_less_eq_rat @ zero_zero_rat @ A )
        & ( ord_less_rat @ A @ one_one_rat ) ) ) ).

% frac_unique_iff
thf(fact_8263_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_8264_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_8265_mult__ceiling__le__Ints,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( member_real @ A @ ring_1_Ints_real )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim7802044766580827645g_real @ ( times_times_real @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim7802044766580827645g_real @ A ) @ ( archim7802044766580827645g_real @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_8266_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_real @ ( ring_1_of_int_real @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_real @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_8267_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_rat @ ( ring_1_of_int_rat @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_rat @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_8268_mult__ceiling__le__Ints,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_rat @ zero_zero_rat @ A )
     => ( ( member_rat @ A @ ring_1_Ints_rat )
       => ( ord_less_eq_int @ ( ring_1_of_int_int @ ( archim2889992004027027881ng_rat @ ( times_times_rat @ A @ B ) ) ) @ ( ring_1_of_int_int @ ( times_times_int @ ( archim2889992004027027881ng_rat @ A ) @ ( archim2889992004027027881ng_rat @ B ) ) ) ) ) ) ).

% mult_ceiling_le_Ints
thf(fact_8269_Arg__bounded,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
      & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ).

% Arg_bounded
thf(fact_8270_sin__integer__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( sin_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = zero_zero_real ) ) ).

% sin_integer_2pi
thf(fact_8271_cos__integer__2pi,axiom,
    ! [N2: real] :
      ( ( member_real @ N2 @ ring_1_Ints_real )
     => ( ( cos_real @ ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) @ N2 ) )
        = one_one_real ) ) ).

% cos_integer_2pi
thf(fact_8272_complex__inverse,axiom,
    ! [A: real,B: real] :
      ( ( invers8013647133539491842omplex @ ( complex2 @ A @ B ) )
      = ( complex2 @ ( divide_divide_real @ A @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ B ) @ ( plus_plus_real @ ( power_power_real @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_inverse
thf(fact_8273_sinh__field__def,axiom,
    ( sinh_real
    = ( ^ [Z5: real] : ( divide_divide_real @ ( minus_minus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% sinh_field_def
thf(fact_8274_sinh__field__def,axiom,
    ( sinh_complex
    = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).

% sinh_field_def
thf(fact_8275_cis__Arg__unique,axiom,
    ! [Z: complex,X4: real] :
      ( ( ( sgn_sgn_complex @ Z )
        = ( cis @ X4 ) )
     => ( ( ord_less_real @ ( uminus_uminus_real @ pi ) @ X4 )
       => ( ( ord_less_eq_real @ X4 @ pi )
         => ( ( arg @ Z )
            = X4 ) ) ) ) ).

% cis_Arg_unique
thf(fact_8276_Arg__correct,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( ( sgn_sgn_complex @ Z )
          = ( cis @ ( arg @ Z ) ) )
        & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ ( arg @ Z ) )
        & ( ord_less_eq_real @ ( arg @ Z ) @ pi ) ) ) ).

% Arg_correct
thf(fact_8277_cosh__ln__real,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( cosh_real @ ( ln_ln_real @ X4 ) )
        = ( divide_divide_real @ ( plus_plus_real @ X4 @ ( inverse_inverse_real @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_ln_real
thf(fact_8278_cosh__double,axiom,
    ! [X4: complex] :
      ( ( cosh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
      = ( plus_plus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cosh_double
thf(fact_8279_cosh__double,axiom,
    ! [X4: real] :
      ( ( cosh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
      = ( plus_plus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cosh_double
thf(fact_8280_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_8281_cosh__zero__iff,axiom,
    ! [X4: real] :
      ( ( ( cosh_real @ X4 )
        = zero_zero_real )
      = ( ( power_power_real @ ( exp_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( uminus_uminus_real @ one_one_real ) ) ) ).

% cosh_zero_iff
thf(fact_8282_cosh__zero__iff,axiom,
    ! [X4: complex] :
      ( ( ( cosh_complex @ X4 )
        = zero_zero_complex )
      = ( ( power_power_complex @ ( exp_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = ( uminus1482373934393186551omplex @ one_one_complex ) ) ) ).

% cosh_zero_iff
thf(fact_8283_push__bit__numeral__minus__1,axiom,
    ! [N2: num] :
      ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ N2 ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( uminus1351360451143612070nteger @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_8284_push__bit__numeral__minus__1,axiom,
    ! [N2: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ N2 ) @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ N2 ) ) ) ) ).

% push_bit_numeral_minus_1
thf(fact_8285_push__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se545348938243370406it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% push_bit_nonnegative_int_iff
thf(fact_8286_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_8287_push__bit__eq__0__iff,axiom,
    ! [N2: nat,A: int] :
      ( ( ( bit_se545348938243370406it_int @ N2 @ A )
        = zero_zero_int )
      = ( A = zero_zero_int ) ) ).

% push_bit_eq_0_iff
thf(fact_8288_push__bit__eq__0__iff,axiom,
    ! [N2: nat,A: nat] :
      ( ( ( bit_se547839408752420682it_nat @ N2 @ A )
        = zero_zero_nat )
      = ( A = zero_zero_nat ) ) ).

% push_bit_eq_0_iff
thf(fact_8289_push__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ zero_zero_int )
      = zero_zero_int ) ).

% push_bit_of_0
thf(fact_8290_push__bit__of__0,axiom,
    ! [N2: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% push_bit_of_0
thf(fact_8291_push__bit__push__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ A ) )
      = ( bit_se545348938243370406it_int @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_8292_push__bit__push__bit,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
      = ( bit_se547839408752420682it_nat @ ( plus_plus_nat @ M @ N2 ) @ A ) ) ).

% push_bit_push_bit
thf(fact_8293_push__bit__and,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( bit_se725231765392027082nd_int @ A @ B ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( bit_se545348938243370406it_int @ N2 @ B ) ) ) ).

% push_bit_and
thf(fact_8294_push__bit__and,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( bit_se727722235901077358nd_nat @ A @ B ) )
      = ( bit_se727722235901077358nd_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( bit_se547839408752420682it_nat @ N2 @ B ) ) ) ).

% push_bit_and
thf(fact_8295_push__bit__xor,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( bit_se6526347334894502574or_int @ A @ B ) )
      = ( bit_se6526347334894502574or_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( bit_se545348938243370406it_int @ N2 @ B ) ) ) ).

% push_bit_xor
thf(fact_8296_push__bit__xor,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( bit_se6528837805403552850or_nat @ A @ B ) )
      = ( bit_se6528837805403552850or_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( bit_se547839408752420682it_nat @ N2 @ B ) ) ) ).

% push_bit_xor
thf(fact_8297_concat__bit__of__zero__1,axiom,
    ! [N2: nat,L: int] :
      ( ( bit_concat_bit @ N2 @ zero_zero_int @ L )
      = ( bit_se545348938243370406it_int @ N2 @ L ) ) ).

% concat_bit_of_zero_1
thf(fact_8298_cosh__0,axiom,
    ( ( cosh_complex @ zero_zero_complex )
    = one_one_complex ) ).

% cosh_0
thf(fact_8299_cosh__0,axiom,
    ( ( cosh_real @ zero_zero_real )
    = one_one_real ) ).

% cosh_0
thf(fact_8300_push__bit__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ ( numeral_numeral_int @ K ) )
      = ( bit_se545348938243370406it_int @ N2 @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).

% push_bit_Suc_numeral
thf(fact_8301_push__bit__Suc__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se547839408752420682it_nat @ ( suc @ N2 ) @ ( numeral_numeral_nat @ K ) )
      = ( bit_se547839408752420682it_nat @ N2 @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% push_bit_Suc_numeral
thf(fact_8302_push__bit__Suc__minus__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se7788150548672797655nteger @ ( suc @ N2 ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( bit_se7788150548672797655nteger @ N2 @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_Suc_minus_numeral
thf(fact_8303_push__bit__Suc__minus__numeral,axiom,
    ! [N2: nat,K: num] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_Suc_minus_numeral
thf(fact_8304_push__bit__numeral,axiom,
    ! [L: num,K: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_int @ K ) )
      = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ).

% push_bit_numeral
thf(fact_8305_push__bit__numeral,axiom,
    ! [L: num,K: num] :
      ( ( bit_se547839408752420682it_nat @ ( numeral_numeral_nat @ L ) @ ( numeral_numeral_nat @ K ) )
      = ( bit_se547839408752420682it_nat @ ( pred_numeral @ L ) @ ( numeral_numeral_nat @ ( bit0 @ K ) ) ) ) ).

% push_bit_numeral
thf(fact_8306_push__bit__minus__one__eq__not__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se7788150548672797655nteger @ N2 @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) )
      = ( bit_ri7632146776885996613nteger @ ( bit_se2119862282449309892nteger @ N2 ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_8307_push__bit__minus__one__eq__not__mask,axiom,
    ! [N2: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ N2 ) ) ) ).

% push_bit_minus_one_eq_not_mask
thf(fact_8308_push__bit__Suc,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ ( suc @ N2 ) @ A )
      = ( bit_se545348938243370406it_int @ N2 @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_8309_push__bit__Suc,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ ( suc @ N2 ) @ A )
      = ( bit_se547839408752420682it_nat @ N2 @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% push_bit_Suc
thf(fact_8310_push__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ one_one_int )
      = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_1
thf(fact_8311_push__bit__of__1,axiom,
    ! [N2: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ one_one_nat )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% push_bit_of_1
thf(fact_8312_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_8313_even__push__bit__iff,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ ( bit_se7788150548672797655nteger @ N2 @ A ) )
      = ( ( N2 != zero_zero_nat )
        | ( dvd_dvd_Code_integer @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_push_bit_iff
thf(fact_8314_even__push__bit__iff,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se545348938243370406it_int @ N2 @ A ) )
      = ( ( N2 != zero_zero_nat )
        | ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_push_bit_iff
thf(fact_8315_even__push__bit__iff,axiom,
    ! [N2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
      = ( ( N2 != zero_zero_nat )
        | ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) ) ) ).

% even_push_bit_iff
thf(fact_8316_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] :
      ( ( bit_se7788150548672797655nteger @ ( numeral_numeral_nat @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ K ) ) )
      = ( bit_se7788150548672797655nteger @ ( pred_numeral @ L ) @ ( uminus1351360451143612070nteger @ ( numera6620942414471956472nteger @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_minus_numeral
thf(fact_8317_push__bit__minus__numeral,axiom,
    ! [L: num,K: num] :
      ( ( bit_se545348938243370406it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) )
      = ( bit_se545348938243370406it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) ) ) ).

% push_bit_minus_numeral
thf(fact_8318_push__bit__of__int,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( ring_1_of_int_int @ K ) )
      = ( ring_1_of_int_int @ ( bit_se545348938243370406it_int @ N2 @ K ) ) ) ).

% push_bit_of_int
thf(fact_8319_of__nat__push__bit,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ M @ N2 ) )
      = ( bit_se545348938243370406it_int @ M @ ( semiri1314217659103216013at_int @ N2 ) ) ) ).

% of_nat_push_bit
thf(fact_8320_of__nat__push__bit,axiom,
    ! [M: nat,N2: nat] :
      ( ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ M @ N2 ) )
      = ( bit_se547839408752420682it_nat @ M @ ( semiri1316708129612266289at_nat @ N2 ) ) ) ).

% of_nat_push_bit
thf(fact_8321_push__bit__of__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( semiri1314217659103216013at_int @ M ) )
      = ( semiri1314217659103216013at_int @ ( bit_se547839408752420682it_nat @ N2 @ M ) ) ) ).

% push_bit_of_nat
thf(fact_8322_push__bit__of__nat,axiom,
    ! [N2: nat,M: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( semiri1316708129612266289at_nat @ M ) )
      = ( semiri1316708129612266289at_nat @ ( bit_se547839408752420682it_nat @ N2 @ M ) ) ) ).

% push_bit_of_nat
thf(fact_8323_push__bit__nat__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se545348938243370406it_int @ N2 @ K ) ) ) ).

% push_bit_nat_eq
thf(fact_8324_push__bit__minus,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( bit_se7788150548672797655nteger @ N2 @ ( uminus1351360451143612070nteger @ A ) )
      = ( uminus1351360451143612070nteger @ ( bit_se7788150548672797655nteger @ N2 @ A ) ) ) ).

% push_bit_minus
thf(fact_8325_push__bit__minus,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ A ) )
      = ( uminus_uminus_int @ ( bit_se545348938243370406it_int @ N2 @ A ) ) ) ).

% push_bit_minus
thf(fact_8326_push__bit__add,axiom,
    ! [N2: nat,A: int,B: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( plus_plus_int @ A @ B ) )
      = ( plus_plus_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( bit_se545348938243370406it_int @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_8327_push__bit__add,axiom,
    ! [N2: nat,A: nat,B: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( plus_plus_nat @ A @ B ) )
      = ( plus_plus_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( bit_se547839408752420682it_nat @ N2 @ B ) ) ) ).

% push_bit_add
thf(fact_8328_cosh__real__pos,axiom,
    ! [X4: real] : ( ord_less_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).

% cosh_real_pos
thf(fact_8329_cosh__real__nonpos__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ Y @ X4 ) ) ) ) ).

% cosh_real_nonpos_le_iff
thf(fact_8330_cosh__real__nonneg__le__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_eq_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) )
          = ( ord_less_eq_real @ X4 @ Y ) ) ) ) ).

% cosh_real_nonneg_le_iff
thf(fact_8331_cosh__real__nonneg,axiom,
    ! [X4: real] : ( ord_less_eq_real @ zero_zero_real @ ( cosh_real @ X4 ) ) ).

% cosh_real_nonneg
thf(fact_8332_cosh__real__ge__1,axiom,
    ! [X4: real] : ( ord_less_eq_real @ one_one_real @ ( cosh_real @ X4 ) ) ).

% cosh_real_ge_1
thf(fact_8333_push__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se2923211474154528505it_int @ N2 @ A ) )
      = ( bit_se2923211474154528505it_int @ ( plus_plus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ M @ A ) ) ) ).

% push_bit_take_bit
thf(fact_8334_push__bit__take__bit,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ M @ ( bit_se2925701944663578781it_nat @ N2 @ A ) )
      = ( bit_se2925701944663578781it_nat @ ( plus_plus_nat @ M @ N2 ) @ ( bit_se547839408752420682it_nat @ M @ A ) ) ) ).

% push_bit_take_bit
thf(fact_8335_take__bit__push__bit,axiom,
    ! [M: nat,N2: nat,A: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ A ) )
      = ( bit_se545348938243370406it_int @ N2 @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N2 ) @ A ) ) ) ).

% take_bit_push_bit
thf(fact_8336_take__bit__push__bit,axiom,
    ! [M: nat,N2: nat,A: nat] :
      ( ( bit_se2925701944663578781it_nat @ M @ ( bit_se547839408752420682it_nat @ N2 @ A ) )
      = ( bit_se547839408752420682it_nat @ N2 @ ( bit_se2925701944663578781it_nat @ ( minus_minus_nat @ M @ N2 ) @ A ) ) ) ).

% take_bit_push_bit
thf(fact_8337_sinh__less__cosh__real,axiom,
    ! [X4: real] : ( ord_less_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ).

% sinh_less_cosh_real
thf(fact_8338_sinh__le__cosh__real,axiom,
    ! [X4: real] : ( ord_less_eq_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) ) ).

% sinh_le_cosh_real
thf(fact_8339_flip__bit__nat__def,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [M6: nat,N: nat] : ( bit_se6528837805403552850or_nat @ N @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).

% flip_bit_nat_def
thf(fact_8340_cosh__real__strict__mono,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ Y )
       => ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) ) ) ) ).

% cosh_real_strict_mono
thf(fact_8341_cosh__real__nonneg__less__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ X4 @ Y ) ) ) ) ).

% cosh_real_nonneg_less_iff
thf(fact_8342_cosh__real__nonpos__less__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ord_less_eq_real @ Y @ zero_zero_real )
       => ( ( ord_less_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) )
          = ( ord_less_real @ Y @ X4 ) ) ) ) ).

% cosh_real_nonpos_less_iff
thf(fact_8343_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_8344_bit__push__bit__iff__nat,axiom,
    ! [M: nat,Q3: nat,N2: nat] :
      ( ( bit_se1148574629649215175it_nat @ ( bit_se547839408752420682it_nat @ M @ Q3 ) @ N2 )
      = ( ( ord_less_eq_nat @ M @ N2 )
        & ( bit_se1148574629649215175it_nat @ Q3 @ ( minus_minus_nat @ N2 @ M ) ) ) ) ).

% bit_push_bit_iff_nat
thf(fact_8345_concat__bit__eq,axiom,
    ( bit_concat_bit
    = ( ^ [N: nat,K3: int,L2: int] : ( plus_plus_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L2 ) ) ) ) ).

% concat_bit_eq
thf(fact_8346_arcosh__cosh__real,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( arcosh_real @ ( cosh_real @ X4 ) )
        = X4 ) ) ).

% arcosh_cosh_real
thf(fact_8347_flip__bit__eq__xor,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N: nat,A3: int] : ( bit_se6526347334894502574or_int @ A3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_8348_flip__bit__eq__xor,axiom,
    ( bit_se2161824704523386999it_nat
    = ( ^ [N: nat,A3: nat] : ( bit_se6528837805403552850or_nat @ A3 @ ( bit_se547839408752420682it_nat @ N @ one_one_nat ) ) ) ) ).

% flip_bit_eq_xor
thf(fact_8349_cosh__add,axiom,
    ! [X4: complex,Y: complex] :
      ( ( cosh_complex @ ( plus_plus_complex @ X4 @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( cosh_complex @ X4 ) @ ( cosh_complex @ Y ) ) @ ( times_times_complex @ ( sinh_complex @ X4 ) @ ( sinh_complex @ Y ) ) ) ) ).

% cosh_add
thf(fact_8350_cosh__add,axiom,
    ! [X4: real,Y: real] :
      ( ( cosh_real @ ( plus_plus_real @ X4 @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( cosh_real @ X4 ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( sinh_real @ X4 ) @ ( sinh_real @ Y ) ) ) ) ).

% cosh_add
thf(fact_8351_sinh__add,axiom,
    ! [X4: complex,Y: complex] :
      ( ( sinh_complex @ ( plus_plus_complex @ X4 @ Y ) )
      = ( plus_plus_complex @ ( times_times_complex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ Y ) ) @ ( times_times_complex @ ( cosh_complex @ X4 ) @ ( sinh_complex @ Y ) ) ) ) ).

% sinh_add
thf(fact_8352_sinh__add,axiom,
    ! [X4: real,Y: real] :
      ( ( sinh_real @ ( plus_plus_real @ X4 @ Y ) )
      = ( plus_plus_real @ ( times_times_real @ ( sinh_real @ X4 ) @ ( cosh_real @ Y ) ) @ ( times_times_real @ ( cosh_real @ X4 ) @ ( sinh_real @ Y ) ) ) ) ).

% sinh_add
thf(fact_8353_cosh__plus__sinh,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( cosh_complex @ X4 ) @ ( sinh_complex @ X4 ) )
      = ( exp_complex @ X4 ) ) ).

% cosh_plus_sinh
thf(fact_8354_cosh__plus__sinh,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( cosh_real @ X4 ) @ ( sinh_real @ X4 ) )
      = ( exp_real @ X4 ) ) ).

% cosh_plus_sinh
thf(fact_8355_sinh__plus__cosh,axiom,
    ! [X4: complex] :
      ( ( plus_plus_complex @ ( sinh_complex @ X4 ) @ ( cosh_complex @ X4 ) )
      = ( exp_complex @ X4 ) ) ).

% sinh_plus_cosh
thf(fact_8356_sinh__plus__cosh,axiom,
    ! [X4: real] :
      ( ( plus_plus_real @ ( sinh_real @ X4 ) @ ( cosh_real @ X4 ) )
      = ( exp_real @ X4 ) ) ).

% sinh_plus_cosh
thf(fact_8357_flip__bit__int__def,axiom,
    ( bit_se2159334234014336723it_int
    = ( ^ [N: nat,K3: int] : ( bit_se6526347334894502574or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% flip_bit_int_def
thf(fact_8358_tanh__def,axiom,
    ( tanh_real
    = ( ^ [X: real] : ( divide_divide_real @ ( sinh_real @ X ) @ ( cosh_real @ X ) ) ) ) ).

% tanh_def
thf(fact_8359_tanh__def,axiom,
    ( tanh_complex
    = ( ^ [X: complex] : ( divide1717551699836669952omplex @ ( sinh_complex @ X ) @ ( cosh_complex @ X ) ) ) ) ).

% tanh_def
thf(fact_8360_push__bit__double,axiom,
    ! [N2: nat,A: int] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( times_times_int @ A @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) )
      = ( times_times_int @ ( bit_se545348938243370406it_int @ N2 @ A ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_8361_push__bit__double,axiom,
    ! [N2: nat,A: nat] :
      ( ( bit_se547839408752420682it_nat @ N2 @ ( times_times_nat @ A @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_nat @ ( bit_se547839408752420682it_nat @ N2 @ A ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% push_bit_double
thf(fact_8362_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1146084159140164899it_int
    = ( ^ [A3: int,N: nat] :
          ( ( bit_se725231765392027082nd_int @ A3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) )
         != zero_zero_int ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_8363_bit__iff__and__push__bit__not__eq__0,axiom,
    ( bit_se1148574629649215175it_nat
    = ( ^ [A3: nat,N: nat] :
          ( ( bit_se727722235901077358nd_nat @ A3 @ ( bit_se547839408752420682it_nat @ N @ one_one_nat ) )
         != zero_zero_nat ) ) ) ).

% bit_iff_and_push_bit_not_eq_0
thf(fact_8364_push__bit__mask__eq,axiom,
    ! [M: nat,N2: nat] :
      ( ( bit_se545348938243370406it_int @ M @ ( bit_se2000444600071755411sk_int @ N2 ) )
      = ( bit_se725231765392027082nd_int @ ( bit_se2000444600071755411sk_int @ ( plus_plus_nat @ N2 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( bit_se2000444600071755411sk_int @ M ) ) ) ) ).

% push_bit_mask_eq
thf(fact_8365_unset__bit__eq__and__not,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N: nat,A3: int] : ( bit_se725231765392027082nd_int @ A3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ) ).

% unset_bit_eq_and_not
thf(fact_8366_unset__bit__int__def,axiom,
    ( bit_se4203085406695923979it_int
    = ( ^ [N: nat,K3: int] : ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ) ).

% unset_bit_int_def
thf(fact_8367_push__bit__int__def,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N: nat,K3: int] : ( times_times_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_int_def
thf(fact_8368_push__bit__nat__def,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N: nat,M6: nat] : ( times_times_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_nat_def
thf(fact_8369_push__bit__eq__mult,axiom,
    ( bit_se545348938243370406it_int
    = ( ^ [N: nat,A3: int] : ( times_times_int @ A3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_eq_mult
thf(fact_8370_push__bit__eq__mult,axiom,
    ( bit_se547839408752420682it_nat
    = ( ^ [N: nat,A3: nat] : ( times_times_nat @ A3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% push_bit_eq_mult
thf(fact_8371_exp__dvdE,axiom,
    ! [N2: nat,A: code_integer] :
      ( ( dvd_dvd_Code_integer @ ( power_8256067586552552935nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ N2 ) @ A )
     => ~ ! [B5: code_integer] :
            ( A
           != ( bit_se7788150548672797655nteger @ N2 @ B5 ) ) ) ).

% exp_dvdE
thf(fact_8372_exp__dvdE,axiom,
    ! [N2: nat,A: int] :
      ( ( dvd_dvd_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) @ A )
     => ~ ! [B5: int] :
            ( A
           != ( bit_se545348938243370406it_int @ N2 @ B5 ) ) ) ).

% exp_dvdE
thf(fact_8373_exp__dvdE,axiom,
    ! [N2: nat,A: nat] :
      ( ( dvd_dvd_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ A )
     => ~ ! [B5: nat] :
            ( A
           != ( bit_se547839408752420682it_nat @ N2 @ B5 ) ) ) ).

% exp_dvdE
thf(fact_8374_sinh__double,axiom,
    ! [X4: complex] :
      ( ( sinh_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ X4 ) )
      = ( times_times_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( sinh_complex @ X4 ) ) @ ( cosh_complex @ X4 ) ) ) ).

% sinh_double
thf(fact_8375_sinh__double,axiom,
    ! [X4: real] :
      ( ( sinh_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( sinh_real @ X4 ) ) @ ( cosh_real @ X4 ) ) ) ).

% sinh_double
thf(fact_8376_push__bit__minus__one,axiom,
    ! [N2: nat] :
      ( ( bit_se545348938243370406it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% push_bit_minus_one
thf(fact_8377_tanh__add,axiom,
    ! [X4: real,Y: real] :
      ( ( ( cosh_real @ X4 )
       != zero_zero_real )
     => ( ( ( cosh_real @ Y )
         != zero_zero_real )
       => ( ( tanh_real @ ( plus_plus_real @ X4 @ Y ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y ) ) @ ( plus_plus_real @ one_one_real @ ( times_times_real @ ( tanh_real @ X4 ) @ ( tanh_real @ Y ) ) ) ) ) ) ) ).

% tanh_add
thf(fact_8378_tanh__add,axiom,
    ! [X4: complex,Y: complex] :
      ( ( ( cosh_complex @ X4 )
       != zero_zero_complex )
     => ( ( ( cosh_complex @ Y )
         != zero_zero_complex )
       => ( ( tanh_complex @ ( plus_plus_complex @ X4 @ Y ) )
          = ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( tanh_complex @ X4 ) @ ( tanh_complex @ Y ) ) @ ( plus_plus_complex @ one_one_complex @ ( times_times_complex @ ( tanh_complex @ X4 ) @ ( tanh_complex @ Y ) ) ) ) ) ) ) ).

% tanh_add
thf(fact_8379_cosh__field__def,axiom,
    ( cosh_real
    = ( ^ [Z5: real] : ( divide_divide_real @ ( plus_plus_real @ ( exp_real @ Z5 ) @ ( exp_real @ ( uminus_uminus_real @ Z5 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% cosh_field_def
thf(fact_8380_cosh__field__def,axiom,
    ( cosh_complex
    = ( ^ [Z5: complex] : ( divide1717551699836669952omplex @ ( plus_plus_complex @ ( exp_complex @ Z5 ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ Z5 ) ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) ) ).

% cosh_field_def
thf(fact_8381_cosh__square__eq,axiom,
    ! [X4: real] :
      ( ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).

% cosh_square_eq
thf(fact_8382_cosh__square__eq,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_complex @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).

% cosh_square_eq
thf(fact_8383_sinh__square__eq,axiom,
    ! [X4: complex] :
      ( ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_complex ) ) ).

% sinh_square_eq
thf(fact_8384_sinh__square__eq,axiom,
    ! [X4: real] :
      ( ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ).

% sinh_square_eq
thf(fact_8385_hyperbolic__pythagoras,axiom,
    ! [X4: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ ( cosh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_complex @ ( sinh_complex @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_complex ) ).

% hyperbolic_pythagoras
thf(fact_8386_hyperbolic__pythagoras,axiom,
    ! [X4: real] :
      ( ( minus_minus_real @ ( power_power_real @ ( cosh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( sinh_real @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = one_one_real ) ).

% hyperbolic_pythagoras
thf(fact_8387_bit__horner__sum__bit__iff,axiom,
    ! [Bs: list_o,N2: nat] :
      ( ( bit_se9216721137139052372nteger @ ( groups3417619833198082522nteger @ zero_n356916108424825756nteger @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) @ Bs ) @ N2 )
      = ( ( ord_less_nat @ N2 @ ( size_size_list_o @ Bs ) )
        & ( nth_o @ Bs @ N2 ) ) ) ).

% bit_horner_sum_bit_iff
thf(fact_8388_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_8389_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_8390_Cauchy__iff2,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X3: nat > real] :
        ! [J3: nat] :
        ? [M8: nat] :
        ! [M6: nat] :
          ( ( ord_less_eq_nat @ M8 @ M6 )
         => ! [N: nat] :
              ( ( ord_less_eq_nat @ M8 @ N )
             => ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ ( X3 @ M6 ) @ ( X3 @ N ) ) ) @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ J3 ) ) ) ) ) ) ) ) ).

% Cauchy_iff2
thf(fact_8391_VEBT__internal_Omembermima_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ! [Uu2: $o,Uv2: $o] :
            ( X4
           != ( vEBT_Leaf @ Uu2 @ Uv2 ) )
       => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( ( Xa = Mi3 )
                  | ( Xa = Ma3 ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
                 => ( ( Xa = Mi3 )
                    | ( Xa = Ma3 )
                    | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(3)
thf(fact_8392_VEBT__internal_Omembermima_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => Y )
       => ( ( ? [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
           => Y )
         => ( ! [Mi3: nat,Ma3: nat] :
                ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
               => ( Y
                  = ( ~ ( ( Xa = Mi3 )
                        | ( Xa = Ma3 ) ) ) ) )
           => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                  ( ? [Vc: vEBT_VEBT] :
                      ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
                 => ( Y
                    = ( ~ ( ( Xa = Mi3 )
                          | ( Xa = Ma3 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) )
             => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                    ( ? [Vd: vEBT_VEBT] :
                        ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                   => ( Y
                      = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(1)
thf(fact_8393_csqrt_Osimps_I1_J,axiom,
    ! [Z: complex] :
      ( ( re @ ( csqrt @ Z ) )
      = ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% csqrt.simps(1)
thf(fact_8394_complex__Re__numeral,axiom,
    ! [V: num] :
      ( ( re @ ( numera6690914467698888265omplex @ V ) )
      = ( numeral_numeral_real @ V ) ) ).

% complex_Re_numeral
thf(fact_8395_Re__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( re @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( re @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Re_divide_numeral
thf(fact_8396_lambda__zero,axiom,
    ( ( ^ [H2: rat] : zero_zero_rat )
    = ( times_times_rat @ zero_zero_rat ) ) ).

% lambda_zero
thf(fact_8397_lambda__zero,axiom,
    ( ( ^ [H2: complex] : zero_zero_complex )
    = ( times_times_complex @ zero_zero_complex ) ) ).

% lambda_zero
thf(fact_8398_lambda__zero,axiom,
    ( ( ^ [H2: real] : zero_zero_real )
    = ( times_times_real @ zero_zero_real ) ) ).

% lambda_zero
thf(fact_8399_lambda__zero,axiom,
    ( ( ^ [H2: nat] : zero_zero_nat )
    = ( times_times_nat @ zero_zero_nat ) ) ).

% lambda_zero
thf(fact_8400_lambda__zero,axiom,
    ( ( ^ [H2: int] : zero_zero_int )
    = ( times_times_int @ zero_zero_int ) ) ).

% lambda_zero
thf(fact_8401_less__set__def,axiom,
    ( ord_less_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_real_o
          @ ^ [X: real] : ( member_real @ X @ A6 )
          @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_8402_less__set__def,axiom,
    ( ord_less_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_8403_less__set__def,axiom,
    ( ord_less_set_complex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( ord_less_complex_o
          @ ^ [X: complex] : ( member_complex @ X @ A6 )
          @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_8404_less__set__def,axiom,
    ( ord_less_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_int_o
          @ ^ [X: int] : ( member_int @ X @ A6 )
          @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_8405_less__set__def,axiom,
    ( ord_le7866589430770878221at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( ord_le549003669493604880_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A6 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ).

% less_set_def
thf(fact_8406_less__eq__set__def,axiom,
    ( ord_less_eq_set_real
    = ( ^ [A6: set_real,B6: set_real] :
          ( ord_less_eq_real_o
          @ ^ [X: real] : ( member_real @ X @ A6 )
          @ ^ [X: real] : ( member_real @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_8407_less__eq__set__def,axiom,
    ( ord_less_eq_set_nat
    = ( ^ [A6: set_nat,B6: set_nat] :
          ( ord_less_eq_nat_o
          @ ^ [X: nat] : ( member_nat @ X @ A6 )
          @ ^ [X: nat] : ( member_nat @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_8408_less__eq__set__def,axiom,
    ( ord_le211207098394363844omplex
    = ( ^ [A6: set_complex,B6: set_complex] :
          ( ord_le4573692005234683329plex_o
          @ ^ [X: complex] : ( member_complex @ X @ A6 )
          @ ^ [X: complex] : ( member_complex @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_8409_less__eq__set__def,axiom,
    ( ord_le3146513528884898305at_nat
    = ( ^ [A6: set_Pr1261947904930325089at_nat,B6: set_Pr1261947904930325089at_nat] :
          ( ord_le704812498762024988_nat_o
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ A6 )
          @ ^ [X: product_prod_nat_nat] : ( member8440522571783428010at_nat @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_8410_less__eq__set__def,axiom,
    ( ord_less_eq_set_int
    = ( ^ [A6: set_int,B6: set_int] :
          ( ord_less_eq_int_o
          @ ^ [X: int] : ( member_int @ X @ A6 )
          @ ^ [X: int] : ( member_int @ X @ B6 ) ) ) ) ).

% less_eq_set_def
thf(fact_8411_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_8412_Collect__subset,axiom,
    ! [A2: set_complex,P: complex > $o] :
      ( ord_le211207098394363844omplex
      @ ( collect_complex
        @ ^ [X: complex] :
            ( ( member_complex @ X @ A2 )
            & ( P @ X ) ) )
      @ A2 ) ).

% Collect_subset
thf(fact_8413_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_8414_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_8415_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_8416_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_8417_subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_le211207098394363844omplex
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B ) ) )
      = ( dvd_dvd_complex @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_8418_subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
      = ( dvd_dvd_real @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_8419_subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
      = ( dvd_dvd_nat @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_8420_subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le7084787975880047091nteger
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B ) ) )
      = ( dvd_dvd_Code_integer @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_8421_subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
      = ( dvd_dvd_int @ A @ B ) ) ).

% subset_divisors_dvd
thf(fact_8422_strict__subset__divisors__dvd,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_set_complex
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ A ) )
        @ ( collect_complex
          @ ^ [C2: complex] : ( dvd_dvd_complex @ C2 @ B ) ) )
      = ( ( dvd_dvd_complex @ A @ B )
        & ~ ( dvd_dvd_complex @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8423_strict__subset__divisors__dvd,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_set_real
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ A ) )
        @ ( collect_real
          @ ^ [C2: real] : ( dvd_dvd_real @ C2 @ B ) ) )
      = ( ( dvd_dvd_real @ A @ B )
        & ~ ( dvd_dvd_real @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8424_strict__subset__divisors__dvd,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_set_nat
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ A ) )
        @ ( collect_nat
          @ ^ [C2: nat] : ( dvd_dvd_nat @ C2 @ B ) ) )
      = ( ( dvd_dvd_nat @ A @ B )
        & ~ ( dvd_dvd_nat @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8425_strict__subset__divisors__dvd,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_set_int
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ A ) )
        @ ( collect_int
          @ ^ [C2: int] : ( dvd_dvd_int @ C2 @ B ) ) )
      = ( ( dvd_dvd_int @ A @ B )
        & ~ ( dvd_dvd_int @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8426_strict__subset__divisors__dvd,axiom,
    ! [A: code_integer,B: code_integer] :
      ( ( ord_le1307284697595431911nteger
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ A ) )
        @ ( collect_Code_integer
          @ ^ [C2: code_integer] : ( dvd_dvd_Code_integer @ C2 @ B ) ) )
      = ( ( dvd_dvd_Code_integer @ A @ B )
        & ~ ( dvd_dvd_Code_integer @ B @ A ) ) ) ).

% strict_subset_divisors_dvd
thf(fact_8427_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_8428_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_8429_numeral__code_I2_J,axiom,
    ! [N2: num] :
      ( ( numera6690914467698888265omplex @ ( bit0 @ N2 ) )
      = ( plus_plus_complex @ ( numera6690914467698888265omplex @ N2 ) @ ( numera6690914467698888265omplex @ N2 ) ) ) ).

% numeral_code(2)
thf(fact_8430_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_8431_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_8432_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_8433_lambda__one,axiom,
    ( ( ^ [X: rat] : X )
    = ( times_times_rat @ one_one_rat ) ) ).

% lambda_one
thf(fact_8434_lambda__one,axiom,
    ( ( ^ [X: complex] : X )
    = ( times_times_complex @ one_one_complex ) ) ).

% lambda_one
thf(fact_8435_lambda__one,axiom,
    ( ( ^ [X: real] : X )
    = ( times_times_real @ one_one_real ) ) ).

% lambda_one
thf(fact_8436_lambda__one,axiom,
    ( ( ^ [X: nat] : X )
    = ( times_times_nat @ one_one_nat ) ) ).

% lambda_one
thf(fact_8437_lambda__one,axiom,
    ( ( ^ [X: int] : X )
    = ( times_times_int @ one_one_int ) ) ).

% lambda_one
thf(fact_8438_nat__less__as__int,axiom,
    ( ord_less_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_less_as_int
thf(fact_8439_nat__leq__as__int,axiom,
    ( ord_less_eq_nat
    = ( ^ [A3: nat,B2: nat] : ( ord_less_eq_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ).

% nat_leq_as_int
thf(fact_8440_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_8441_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_8442_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_8443_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_8444_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_8445_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_8446_power__numeral__even,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_complex @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_8447_power__numeral__even,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_real @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_8448_power__numeral__even,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_nat @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_8449_power__numeral__even,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit0 @ W ) ) )
      = ( times_times_int @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_even
thf(fact_8450_power__numeral__odd,axiom,
    ! [Z: complex,W: num] :
      ( ( power_power_complex @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_complex @ ( times_times_complex @ Z @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_complex @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_8451_power__numeral__odd,axiom,
    ! [Z: real,W: num] :
      ( ( power_power_real @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_real @ ( times_times_real @ Z @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_real @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_8452_power__numeral__odd,axiom,
    ! [Z: nat,W: num] :
      ( ( power_power_nat @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_nat @ ( times_times_nat @ Z @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_nat @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_8453_power__numeral__odd,axiom,
    ! [Z: int,W: num] :
      ( ( power_power_int @ Z @ ( numeral_numeral_nat @ ( bit1 @ W ) ) )
      = ( times_times_int @ ( times_times_int @ Z @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) @ ( power_power_int @ Z @ ( numeral_numeral_nat @ W ) ) ) ) ).

% power_numeral_odd
thf(fact_8454_nat__plus__as__int,axiom,
    ( plus_plus_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_plus_as_int
thf(fact_8455_nat__div__as__int,axiom,
    ( divide_divide_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( divide_divide_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_div_as_int
thf(fact_8456_nat__mod__as__int,axiom,
    ( modulo_modulo_nat
    = ( ^ [A3: nat,B2: nat] : ( nat2 @ ( modulo_modulo_int @ ( semiri1314217659103216013at_int @ A3 ) @ ( semiri1314217659103216013at_int @ B2 ) ) ) ) ) ).

% nat_mod_as_int
thf(fact_8457_complex__Re__le__cmod,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( re @ X4 ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% complex_Re_le_cmod
thf(fact_8458_one__complex_Osimps_I1_J,axiom,
    ( ( re @ one_one_complex )
    = one_one_real ) ).

% one_complex.simps(1)
thf(fact_8459_set__conv__nth,axiom,
    ( set_complex2
    = ( ^ [Xs3: list_complex] :
          ( collect_complex
          @ ^ [Uu3: complex] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_complex @ Xs3 @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_s3451745648224563538omplex @ Xs3 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8460_set__conv__nth,axiom,
    ( set_real2
    = ( ^ [Xs3: list_real] :
          ( collect_real
          @ ^ [Uu3: real] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_real @ Xs3 @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_size_list_real @ Xs3 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8461_set__conv__nth,axiom,
    ( set_list_nat2
    = ( ^ [Xs3: list_list_nat] :
          ( collect_list_nat
          @ ^ [Uu3: list_nat] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_list_nat @ Xs3 @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_s3023201423986296836st_nat @ Xs3 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8462_set__conv__nth,axiom,
    ( set_VEBT_VEBT2
    = ( ^ [Xs3: list_VEBT_VEBT] :
          ( collect_VEBT_VEBT
          @ ^ [Uu3: vEBT_VEBT] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_VEBT_VEBT @ Xs3 @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_s6755466524823107622T_VEBT @ Xs3 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8463_set__conv__nth,axiom,
    ( set_o2
    = ( ^ [Xs3: list_o] :
          ( collect_o
          @ ^ [Uu3: $o] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_o @ Xs3 @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_size_list_o @ Xs3 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8464_set__conv__nth,axiom,
    ( set_nat2
    = ( ^ [Xs3: list_nat] :
          ( collect_nat
          @ ^ [Uu3: nat] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_nat @ Xs3 @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_size_list_nat @ Xs3 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8465_set__conv__nth,axiom,
    ( set_int2
    = ( ^ [Xs3: list_int] :
          ( collect_int
          @ ^ [Uu3: int] :
            ? [I3: nat] :
              ( ( Uu3
                = ( nth_int @ Xs3 @ I3 ) )
              & ( ord_less_nat @ I3 @ ( size_size_list_int @ Xs3 ) ) ) ) ) ) ).

% set_conv_nth
thf(fact_8466_diff__nat__eq__if,axiom,
    ! [Z6: int,Z: int] :
      ( ( ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( nat2 @ Z ) ) )
      & ( ~ ( ord_less_int @ Z6 @ zero_zero_int )
       => ( ( minus_minus_nat @ ( nat2 @ Z ) @ ( nat2 @ Z6 ) )
          = ( if_nat @ ( ord_less_int @ ( minus_minus_int @ Z @ Z6 ) @ zero_zero_int ) @ zero_zero_nat @ ( nat2 @ ( minus_minus_int @ Z @ Z6 ) ) ) ) ) ) ).

% diff_nat_eq_if
thf(fact_8467_abs__Re__le__cmod,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% abs_Re_le_cmod
thf(fact_8468_Re__csqrt,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) ) ).

% Re_csqrt
thf(fact_8469_set__decode__def,axiom,
    ( nat_set_decode
    = ( ^ [X: nat] :
          ( collect_nat
          @ ^ [N: nat] :
              ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( divide_divide_nat @ X @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ) ) ).

% set_decode_def
thf(fact_8470_signed__take__bit__code,axiom,
    ( bit_ri6519982836138164636nteger
    = ( ^ [N: nat,A3: code_integer] : ( if_Code_integer @ ( bit_se9216721137139052372nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A3 ) @ N ) @ ( plus_p5714425477246183910nteger @ ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A3 ) @ ( bit_se7788150548672797655nteger @ ( suc @ N ) @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) ) ) @ ( bit_se1745604003318907178nteger @ ( suc @ N ) @ A3 ) ) ) ) ).

% signed_take_bit_code
thf(fact_8471_signed__take__bit__code,axiom,
    ( bit_ri631733984087533419it_int
    = ( ^ [N: nat,A3: int] : ( if_int @ ( bit_se1146084159140164899it_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A3 ) @ N ) @ ( plus_plus_int @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A3 ) @ ( bit_se545348938243370406it_int @ ( suc @ N ) @ ( uminus_uminus_int @ one_one_int ) ) ) @ ( bit_se2923211474154528505it_int @ ( suc @ N ) @ A3 ) ) ) ) ).

% signed_take_bit_code
thf(fact_8472_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_8473_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_8474_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_8475_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_8476_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_8477_cmod__plus__Re__le__0__iff,axiom,
    ! [Z: complex] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ zero_zero_real )
      = ( ( re @ Z )
        = ( uminus_uminus_real @ ( real_V1022390504157884413omplex @ Z ) ) ) ) ).

% cmod_plus_Re_le_0_iff
thf(fact_8478_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_8479_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_8480_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_8481_VEBT__internal_Onaive__member_Osimps_I3_J,axiom,
    ! [Uy: option4927543243414619207at_nat,V: nat,TreeList2: list_VEBT_VEBT,S: vEBT_VEBT,X4: nat] :
      ( ( vEBT_V5719532721284313246member @ ( vEBT_Node @ Uy @ ( suc @ V ) @ TreeList2 @ S ) @ X4 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.naive_member.simps(3)
thf(fact_8482_CauchyD,axiom,
    ! [X8: nat > complex,E2: real] :
      ( ( topolo6517432010174082258omplex @ X8 )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ? [M9: nat] :
          ! [M2: nat] :
            ( ( ord_less_eq_nat @ M9 @ M2 )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ M9 @ N6 )
               => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M2 ) @ ( X8 @ N6 ) ) ) @ E2 ) ) ) ) ) ).

% CauchyD
thf(fact_8483_CauchyD,axiom,
    ! [X8: nat > real,E2: real] :
      ( ( topolo4055970368930404560y_real @ X8 )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ? [M9: nat] :
          ! [M2: nat] :
            ( ( ord_less_eq_nat @ M9 @ M2 )
           => ! [N6: nat] :
                ( ( ord_less_eq_nat @ M9 @ N6 )
               => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M2 ) @ ( X8 @ N6 ) ) ) @ E2 ) ) ) ) ) ).

% CauchyD
thf(fact_8484_CauchyI,axiom,
    ! [X8: nat > complex] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ? [M10: nat] :
            ! [M5: nat] :
              ( ( ord_less_eq_nat @ M10 @ M5 )
             => ! [N3: nat] :
                  ( ( ord_less_eq_nat @ M10 @ N3 )
                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) )
     => ( topolo6517432010174082258omplex @ X8 ) ) ).

% CauchyI
thf(fact_8485_CauchyI,axiom,
    ! [X8: nat > real] :
      ( ! [E: real] :
          ( ( ord_less_real @ zero_zero_real @ E )
         => ? [M10: nat] :
            ! [M5: nat] :
              ( ( ord_less_eq_nat @ M10 @ M5 )
             => ! [N3: nat] :
                  ( ( ord_less_eq_nat @ M10 @ N3 )
                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) ) @ E ) ) ) )
     => ( topolo4055970368930404560y_real @ X8 ) ) ).

% CauchyI
thf(fact_8486_Cauchy__iff,axiom,
    ( topolo6517432010174082258omplex
    = ( ^ [X3: nat > complex] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [M8: nat] :
            ! [M6: nat] :
              ( ( ord_less_eq_nat @ M8 @ M6 )
             => ! [N: nat] :
                  ( ( ord_less_eq_nat @ M8 @ N )
                 => ( ord_less_real @ ( real_V1022390504157884413omplex @ ( minus_minus_complex @ ( X3 @ M6 ) @ ( X3 @ N ) ) ) @ E3 ) ) ) ) ) ) ).

% Cauchy_iff
thf(fact_8487_Cauchy__iff,axiom,
    ( topolo4055970368930404560y_real
    = ( ^ [X3: nat > real] :
        ! [E3: real] :
          ( ( ord_less_real @ zero_zero_real @ E3 )
         => ? [M8: nat] :
            ! [M6: nat] :
              ( ( ord_less_eq_nat @ M8 @ M6 )
             => ! [N: nat] :
                  ( ( ord_less_eq_nat @ M8 @ N )
                 => ( ord_less_real @ ( real_V7735802525324610683m_real @ ( minus_minus_real @ ( X3 @ M6 ) @ ( X3 @ N ) ) ) @ E3 ) ) ) ) ) ) ).

% Cauchy_iff
thf(fact_8488_VEBT__internal_Omembermima_Osimps_I5_J,axiom,
    ! [V: nat,TreeList2: list_VEBT_VEBT,Vd2: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V ) @ TreeList2 @ Vd2 ) @ X4 )
      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ).

% VEBT_internal.membermima.simps(5)
thf(fact_8489_VEBT__internal_Omembermima_Osimps_I4_J,axiom,
    ! [Mi: nat,Ma: nat,V: nat,TreeList2: list_VEBT_VEBT,Vc2: vEBT_VEBT,X4: nat] :
      ( ( vEBT_VEBT_membermima @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi @ Ma ) ) @ ( suc @ V ) @ TreeList2 @ Vc2 ) @ X4 )
      = ( ( X4 = Mi )
        | ( X4 = Ma )
        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) )
           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList2 @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
          & ( ord_less_nat @ ( vEBT_VEBT_high @ X4 @ ( divide_divide_nat @ ( suc @ V ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList2 ) ) ) ) ) ).

% VEBT_internal.membermima.simps(4)
thf(fact_8490_VEBT__internal_Onaive__member_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
        = Y )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( Y
              = ( ~ ( ( ( Xa = zero_zero_nat )
                     => A5 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B5 )
                        & ( Xa = one_one_nat ) ) ) ) ) ) )
       => ( ( ? [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
           => Y )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( Y
                  = ( ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(1)
thf(fact_8491_VEBT__internal_Onaive__member_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ~ ( ( ( Xa = zero_zero_nat )
                 => A5 )
                & ( ( Xa != zero_zero_nat )
                 => ( ( ( Xa = one_one_nat )
                     => B5 )
                    & ( Xa = one_one_nat ) ) ) ) )
       => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [S3: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
             => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(2)
thf(fact_8492_VEBT__internal_Onaive__member_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ! [A5: $o,B5: $o] :
            ( ( X4
              = ( vEBT_Leaf @ A5 @ B5 ) )
           => ( ( ( Xa = zero_zero_nat )
               => A5 )
              & ( ( Xa != zero_zero_nat )
               => ( ( ( Xa = one_one_nat )
                   => B5 )
                  & ( Xa = one_one_nat ) ) ) ) )
       => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
              ( X4
             != ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [S3: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                   => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                  & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.elims(3)
thf(fact_8493_VEBT__internal_Omembermima_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ! [Mi3: nat,Ma3: nat] :
            ( ? [Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                ( X4
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
           => ~ ( ( Xa = Mi3 )
                | ( Xa = Ma3 ) ) )
       => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT] :
              ( ? [Vc: vEBT_VEBT] :
                  ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
             => ~ ( ( Xa = Mi3 )
                  | ( Xa = Ma3 )
                  | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) )
         => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT] :
                ( ? [Vd: vEBT_VEBT] :
                    ( X4
                    = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
               => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                     => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                    & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.elims(2)
thf(fact_8494_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_8495_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_8496_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_8497_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_8498_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_8499_monoseq__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ 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 @ X4 @ ( plus_plus_nat @ ( times_times_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ).

% monoseq_arctan_series
thf(fact_8500_csqrt_Ocode,axiom,
    ( csqrt
    = ( ^ [Z5: complex] :
          ( complex2 @ ( sqrt @ ( divide_divide_real @ ( plus_plus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          @ ( times_times_real
            @ ( if_real
              @ ( ( im @ Z5 )
                = zero_zero_real )
              @ one_one_real
              @ ( sgn_sgn_real @ ( im @ Z5 ) ) )
            @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z5 ) @ ( re @ Z5 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% csqrt.code
thf(fact_8501_ln__series,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
       => ( ( ln_ln_real @ X4 )
          = ( suminf_real
            @ ^ [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 @ X4 @ one_one_real ) @ ( suc @ N ) ) ) ) ) ) ) ).

% ln_series
thf(fact_8502_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_8503_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_8504_Re__power__real,axiom,
    ! [X4: complex,N2: nat] :
      ( ( ( im @ X4 )
        = zero_zero_real )
     => ( ( re @ ( power_power_complex @ X4 @ N2 ) )
        = ( power_power_real @ ( re @ X4 ) @ N2 ) ) ) ).

% Re_power_real
thf(fact_8505_Im__divide__numeral,axiom,
    ! [Z: complex,W: num] :
      ( ( im @ ( divide1717551699836669952omplex @ Z @ ( numera6690914467698888265omplex @ W ) ) )
      = ( divide_divide_real @ ( im @ Z ) @ ( numeral_numeral_real @ W ) ) ) ).

% Im_divide_numeral
thf(fact_8506_csqrt__of__real__nonneg,axiom,
    ! [X4: complex] :
      ( ( ( im @ X4 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( re @ X4 ) )
       => ( ( csqrt @ X4 )
          = ( real_V4546457046886955230omplex @ ( sqrt @ ( re @ X4 ) ) ) ) ) ) ).

% csqrt_of_real_nonneg
thf(fact_8507_csqrt__minus,axiom,
    ! [X4: complex] :
      ( ( ( ord_less_real @ ( im @ X4 ) @ zero_zero_real )
        | ( ( ( im @ X4 )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( re @ X4 ) ) ) )
     => ( ( csqrt @ ( uminus1482373934393186551omplex @ X4 ) )
        = ( times_times_complex @ imaginary_unit @ ( csqrt @ X4 ) ) ) ) ).

% csqrt_minus
thf(fact_8508_csqrt__of__real__nonpos,axiom,
    ! [X4: complex] :
      ( ( ( im @ X4 )
        = zero_zero_real )
     => ( ( ord_less_eq_real @ ( re @ X4 ) @ zero_zero_real )
       => ( ( csqrt @ X4 )
          = ( times_times_complex @ imaginary_unit @ ( real_V4546457046886955230omplex @ ( sqrt @ ( abs_abs_real @ ( re @ X4 ) ) ) ) ) ) ) ) ).

% csqrt_of_real_nonpos
thf(fact_8509_imaginary__unit_Osimps_I2_J,axiom,
    ( ( im @ imaginary_unit )
    = one_one_real ) ).

% imaginary_unit.simps(2)
thf(fact_8510_abs__Im__le__cmod,axiom,
    ! [X4: complex] : ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X4 ) ) @ ( real_V1022390504157884413omplex @ X4 ) ) ).

% abs_Im_le_cmod
thf(fact_8511_cmod__Im__le__iff,axiom,
    ! [X4: complex,Y: complex] :
      ( ( ( re @ X4 )
        = ( re @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( im @ X4 ) ) @ ( abs_abs_real @ ( im @ Y ) ) ) ) ) ).

% cmod_Im_le_iff
thf(fact_8512_cmod__Re__le__iff,axiom,
    ! [X4: complex,Y: complex] :
      ( ( ( im @ X4 )
        = ( im @ Y ) )
     => ( ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ X4 ) @ ( real_V1022390504157884413omplex @ Y ) )
        = ( ord_less_eq_real @ ( abs_abs_real @ ( re @ X4 ) ) @ ( abs_abs_real @ ( re @ Y ) ) ) ) ) ).

% cmod_Re_le_iff
thf(fact_8513_csqrt__principal,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( csqrt @ Z ) ) )
      | ( ( ( re @ ( csqrt @ Z ) )
          = zero_zero_real )
        & ( ord_less_eq_real @ zero_zero_real @ ( im @ ( csqrt @ Z ) ) ) ) ) ).

% csqrt_principal
thf(fact_8514_cmod__le,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ Z ) @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) ) ).

% cmod_le
thf(fact_8515_monoseq__realpow,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( topolo6980174941875973593q_real @ ( power_power_real @ X4 ) ) ) ) ).

% monoseq_realpow
thf(fact_8516_cmod__power2,axiom,
    ! [Z: complex] :
      ( ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
      = ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% cmod_power2
thf(fact_8517_Im__power2,axiom,
    ! [X4: complex] :
      ( ( im @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ X4 ) ) @ ( im @ X4 ) ) ) ).

% Im_power2
thf(fact_8518_Re__power2,axiom,
    ! [X4: complex] :
      ( ( re @ ( power_power_complex @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( minus_minus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% Re_power2
thf(fact_8519_complex__eq__0,axiom,
    ! [Z: complex] :
      ( ( Z = zero_zero_complex )
      = ( ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = zero_zero_real ) ) ).

% complex_eq_0
thf(fact_8520_norm__complex__def,axiom,
    ( real_V1022390504157884413omplex
    = ( ^ [Z5: complex] : ( sqrt @ ( plus_plus_real @ ( power_power_real @ ( re @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% norm_complex_def
thf(fact_8521_inverse__complex_Osimps_I1_J,axiom,
    ! [X4: complex] :
      ( ( re @ ( invers8013647133539491842omplex @ X4 ) )
      = ( divide_divide_real @ ( re @ X4 ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(1)
thf(fact_8522_complex__neq__0,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
      = ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_neq_0
thf(fact_8523_Re__divide,axiom,
    ! [X4: complex,Y: complex] :
      ( ( re @ ( divide1717551699836669952omplex @ X4 @ Y ) )
      = ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X4 ) @ ( re @ Y ) ) @ ( times_times_real @ ( im @ X4 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_divide
thf(fact_8524_csqrt__unique,axiom,
    ! [W: complex,Z: complex] :
      ( ( ( power_power_complex @ W @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        = Z )
     => ( ( ( ord_less_real @ zero_zero_real @ ( re @ W ) )
          | ( ( ( re @ W )
              = zero_zero_real )
            & ( ord_less_eq_real @ zero_zero_real @ ( im @ W ) ) ) )
       => ( ( csqrt @ Z )
          = W ) ) ) ).

% csqrt_unique
thf(fact_8525_csqrt__square,axiom,
    ! [B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ B ) )
        | ( ( ( re @ B )
            = zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ ( im @ B ) ) ) )
     => ( ( csqrt @ ( power_power_complex @ B @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = B ) ) ).

% csqrt_square
thf(fact_8526_inverse__complex_Osimps_I2_J,axiom,
    ! [X4: complex] :
      ( ( im @ ( invers8013647133539491842omplex @ X4 ) )
      = ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X4 ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X4 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% inverse_complex.simps(2)
thf(fact_8527_Im__divide,axiom,
    ! [X4: complex,Y: complex] :
      ( ( im @ ( divide1717551699836669952omplex @ X4 @ Y ) )
      = ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X4 ) @ ( re @ Y ) ) @ ( times_times_real @ ( re @ X4 ) @ ( im @ Y ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_divide
thf(fact_8528_complex__abs__le__norm,axiom,
    ! [Z: complex] : ( ord_less_eq_real @ ( plus_plus_real @ ( abs_abs_real @ ( re @ Z ) ) @ ( abs_abs_real @ ( im @ Z ) ) ) @ ( times_times_real @ ( sqrt @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( real_V1022390504157884413omplex @ Z ) ) ) ).

% complex_abs_le_norm
thf(fact_8529_complex__unit__circle,axiom,
    ! [Z: complex] :
      ( ( Z != zero_zero_complex )
     => ( ( plus_plus_real @ ( power_power_real @ ( divide_divide_real @ ( re @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( divide_divide_real @ ( im @ Z ) @ ( real_V1022390504157884413omplex @ Z ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
        = one_one_real ) ) ).

% complex_unit_circle
thf(fact_8530_inverse__complex_Ocode,axiom,
    ( invers8013647133539491842omplex
    = ( ^ [X: complex] : ( complex2 @ ( divide_divide_real @ ( re @ X ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( uminus_uminus_real @ ( im @ X ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ X ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% inverse_complex.code
thf(fact_8531_Complex__divide,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [X: complex,Y5: complex] : ( complex2 @ ( divide_divide_real @ ( plus_plus_real @ ( times_times_real @ ( re @ X ) @ ( re @ Y5 ) ) @ ( times_times_real @ ( im @ X ) @ ( im @ Y5 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( divide_divide_real @ ( minus_minus_real @ ( times_times_real @ ( im @ X ) @ ( re @ Y5 ) ) @ ( times_times_real @ ( re @ X ) @ ( im @ Y5 ) ) ) @ ( plus_plus_real @ ( power_power_real @ ( re @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Y5 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% Complex_divide
thf(fact_8532_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_8533_csqrt_Osimps_I2_J,axiom,
    ! [Z: complex] :
      ( ( im @ ( csqrt @ Z ) )
      = ( times_times_real
        @ ( if_real
          @ ( ( im @ Z )
            = zero_zero_real )
          @ one_one_real
          @ ( sgn_sgn_real @ ( im @ Z ) ) )
        @ ( sqrt @ ( divide_divide_real @ ( minus_minus_real @ ( real_V1022390504157884413omplex @ Z ) @ ( re @ Z ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% csqrt.simps(2)
thf(fact_8534_arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( ( arctan @ X4 )
        = ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ) ).

% arctan_series
thf(fact_8535_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_8536_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_8537_summable__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( summable_real
        @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X4 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) ) ) ).

% summable_arctan_series
thf(fact_8538_vebt__buildup_Oelims,axiom,
    ! [X4: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X4 )
        = Y )
     => ( ( ( X4 = zero_zero_nat )
         => ( Y
           != ( vEBT_Leaf @ $false @ $false ) ) )
       => ( ( ( X4
              = ( suc @ zero_zero_nat ) )
           => ( Y
             != ( vEBT_Leaf @ $false @ $false ) ) )
         => ~ ! [Va2: nat] :
                ( ( X4
                  = ( suc @ ( suc @ Va2 ) ) )
               => ~ ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                     => ( Y
                        = ( 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 ) ) )
                     => ( Y
                        = ( 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_8539_Im__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( ( member_complex @ R3 @ real_V2521375963428798218omplex )
     => ( ( im @ ( divide1717551699836669952omplex @ R3 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( uminus_uminus_real @ ( re @ R3 ) ) @ ( im @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Im_Reals_divide
thf(fact_8540_intind,axiom,
    ! [I2: nat,N2: nat,P: nat > $o,X4: nat] :
      ( ( ord_less_nat @ I2 @ N2 )
     => ( ( P @ X4 )
       => ( P @ ( nth_nat @ ( replicate_nat @ N2 @ X4 ) @ I2 ) ) ) ) ).

% intind
thf(fact_8541_intind,axiom,
    ! [I2: nat,N2: nat,P: int > $o,X4: int] :
      ( ( ord_less_nat @ I2 @ N2 )
     => ( ( P @ X4 )
       => ( P @ ( nth_int @ ( replicate_int @ N2 @ X4 ) @ I2 ) ) ) ) ).

% intind
thf(fact_8542_intind,axiom,
    ! [I2: nat,N2: nat,P: vEBT_VEBT > $o,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I2 @ N2 )
     => ( ( P @ X4 )
       => ( P @ ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X4 ) @ I2 ) ) ) ) ).

% intind
thf(fact_8543_replicate__eq__replicate,axiom,
    ! [M: nat,X4: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( ( replicate_VEBT_VEBT @ M @ X4 )
        = ( replicate_VEBT_VEBT @ N2 @ Y ) )
      = ( ( M = N2 )
        & ( ( M != zero_zero_nat )
         => ( X4 = Y ) ) ) ) ).

% replicate_eq_replicate
thf(fact_8544_length__replicate,axiom,
    ! [N2: nat,X4: vEBT_VEBT] :
      ( ( size_s6755466524823107622T_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X4 ) )
      = N2 ) ).

% length_replicate
thf(fact_8545_length__replicate,axiom,
    ! [N2: nat,X4: $o] :
      ( ( size_size_list_o @ ( replicate_o @ N2 @ X4 ) )
      = N2 ) ).

% length_replicate
thf(fact_8546_length__replicate,axiom,
    ! [N2: nat,X4: nat] :
      ( ( size_size_list_nat @ ( replicate_nat @ N2 @ X4 ) )
      = N2 ) ).

% length_replicate
thf(fact_8547_length__replicate,axiom,
    ! [N2: nat,X4: int] :
      ( ( size_size_list_int @ ( replicate_int @ N2 @ X4 ) )
      = N2 ) ).

% length_replicate
thf(fact_8548_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_8549_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_8550_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_8551_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_8552_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_8553_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_8554_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_8555_in__set__replicate,axiom,
    ! [X4: real,N2: nat,Y: real] :
      ( ( member_real @ X4 @ ( set_real2 @ ( replicate_real @ N2 @ Y ) ) )
      = ( ( X4 = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8556_in__set__replicate,axiom,
    ! [X4: complex,N2: nat,Y: complex] :
      ( ( member_complex @ X4 @ ( set_complex2 @ ( replicate_complex @ N2 @ Y ) ) )
      = ( ( X4 = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8557_in__set__replicate,axiom,
    ! [X4: product_prod_nat_nat,N2: nat,Y: product_prod_nat_nat] :
      ( ( member8440522571783428010at_nat @ X4 @ ( set_Pr5648618587558075414at_nat @ ( replic4235873036481779905at_nat @ N2 @ Y ) ) )
      = ( ( X4 = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8558_in__set__replicate,axiom,
    ! [X4: int,N2: nat,Y: int] :
      ( ( member_int @ X4 @ ( set_int2 @ ( replicate_int @ N2 @ Y ) ) )
      = ( ( X4 = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8559_in__set__replicate,axiom,
    ! [X4: nat,N2: nat,Y: nat] :
      ( ( member_nat @ X4 @ ( set_nat2 @ ( replicate_nat @ N2 @ Y ) ) )
      = ( ( X4 = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8560_in__set__replicate,axiom,
    ! [X4: vEBT_VEBT,N2: nat,Y: vEBT_VEBT] :
      ( ( member_VEBT_VEBT @ X4 @ ( set_VEBT_VEBT2 @ ( replicate_VEBT_VEBT @ N2 @ Y ) ) )
      = ( ( X4 = Y )
        & ( N2 != zero_zero_nat ) ) ) ).

% in_set_replicate
thf(fact_8561_nth__replicate,axiom,
    ! [I2: nat,N2: nat,X4: nat] :
      ( ( ord_less_nat @ I2 @ N2 )
     => ( ( nth_nat @ ( replicate_nat @ N2 @ X4 ) @ I2 )
        = X4 ) ) ).

% nth_replicate
thf(fact_8562_nth__replicate,axiom,
    ! [I2: nat,N2: nat,X4: int] :
      ( ( ord_less_nat @ I2 @ N2 )
     => ( ( nth_int @ ( replicate_int @ N2 @ X4 ) @ I2 )
        = X4 ) ) ).

% nth_replicate
thf(fact_8563_nth__replicate,axiom,
    ! [I2: nat,N2: nat,X4: vEBT_VEBT] :
      ( ( ord_less_nat @ I2 @ N2 )
     => ( ( nth_VEBT_VEBT @ ( replicate_VEBT_VEBT @ N2 @ X4 ) @ I2 )
        = X4 ) ) ).

% nth_replicate
thf(fact_8564_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_8565_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_8566_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_8567_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_8568_summable__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test
thf(fact_8569_summable__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
     => ( ( summable_real @ G )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test
thf(fact_8570_summable__comparison__test_H,axiom,
    ! [G: nat > real,N4: nat,F: nat > real] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N4 @ N3 )
           => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_real @ F ) ) ) ).

% summable_comparison_test'
thf(fact_8571_summable__comparison__test_H,axiom,
    ! [G: nat > real,N4: nat,F: nat > complex] :
      ( ( summable_real @ G )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N4 @ N3 )
           => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( G @ N3 ) ) )
       => ( summable_complex @ F ) ) ) ).

% summable_comparison_test'
thf(fact_8572_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_8573_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_8574_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_8575_Reals__power,axiom,
    ! [A: real,N2: nat] :
      ( ( member_real @ A @ real_V470468836141973256s_real )
     => ( member_real @ ( power_power_real @ A @ N2 ) @ real_V470468836141973256s_real ) ) ).

% Reals_power
thf(fact_8576_Reals__power,axiom,
    ! [A: complex,N2: nat] :
      ( ( member_complex @ A @ real_V2521375963428798218omplex )
     => ( member_complex @ ( power_power_complex @ A @ N2 ) @ real_V2521375963428798218omplex ) ) ).

% Reals_power
thf(fact_8577_Reals__divide,axiom,
    ! [A: real,B: real] :
      ( ( member_real @ A @ real_V470468836141973256s_real )
     => ( ( member_real @ B @ real_V470468836141973256s_real )
       => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).

% Reals_divide
thf(fact_8578_Reals__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( member_complex @ A @ real_V2521375963428798218omplex )
     => ( ( member_complex @ B @ real_V2521375963428798218omplex )
       => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).

% Reals_divide
thf(fact_8579_Reals__add,axiom,
    ! [A: real,B: real] :
      ( ( member_real @ A @ real_V470468836141973256s_real )
     => ( ( member_real @ B @ real_V470468836141973256s_real )
       => ( member_real @ ( plus_plus_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ).

% Reals_add
thf(fact_8580_Reals__add,axiom,
    ! [A: complex,B: complex] :
      ( ( member_complex @ A @ real_V2521375963428798218omplex )
     => ( ( member_complex @ B @ real_V2521375963428798218omplex )
       => ( member_complex @ ( plus_plus_complex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ).

% Reals_add
thf(fact_8581_Reals__1,axiom,
    member_real @ one_one_real @ real_V470468836141973256s_real ).

% Reals_1
thf(fact_8582_Reals__1,axiom,
    member_complex @ one_one_complex @ real_V2521375963428798218omplex ).

% Reals_1
thf(fact_8583_Reals__numeral,axiom,
    ! [W: num] : ( member_complex @ ( numera6690914467698888265omplex @ W ) @ real_V2521375963428798218omplex ) ).

% Reals_numeral
thf(fact_8584_Reals__numeral,axiom,
    ! [W: num] : ( member_real @ ( numeral_numeral_real @ W ) @ real_V470468836141973256s_real ) ).

% Reals_numeral
thf(fact_8585_summable__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( summable_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) ) ) ).

% summable_divide
thf(fact_8586_summable__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( summable_complex
        @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) ) ) ).

% summable_divide
thf(fact_8587_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_8588_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_8589_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_8590_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_8591_summable__Suc__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( F @ ( suc @ N ) ) )
      = ( summable_real @ F ) ) ).

% summable_Suc_iff
thf(fact_8592_summable__zero__power,axiom,
    summable_real @ ( power_power_real @ zero_zero_real ) ).

% summable_zero_power
thf(fact_8593_summable__zero__power,axiom,
    summable_int @ ( power_power_int @ zero_zero_int ) ).

% summable_zero_power
thf(fact_8594_summable__zero__power,axiom,
    summable_complex @ ( power_power_complex @ zero_zero_complex ) ).

% summable_zero_power
thf(fact_8595_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_8596_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_8597_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_8598_suminf__divide,axiom,
    ! [F: nat > real,C: real] :
      ( ( summable_real @ F )
     => ( ( suminf_real
          @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C ) )
        = ( divide_divide_real @ ( suminf_real @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_8599_suminf__divide,axiom,
    ! [F: nat > complex,C: complex] :
      ( ( summable_complex @ F )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C ) )
        = ( divide1717551699836669952omplex @ ( suminf_complex @ F ) @ C ) ) ) ).

% suminf_divide
thf(fact_8600_series__comparison__complex,axiom,
    ! [G: nat > complex,N4: nat,F: nat > real] :
      ( ( summable_complex @ G )
     => ( ! [N3: nat] : ( member_complex @ ( G @ N3 ) @ real_V2521375963428798218omplex )
       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( G @ N3 ) ) )
         => ( ! [N3: nat] :
                ( ( ord_less_eq_nat @ N4 @ N3 )
               => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ N3 ) ) @ ( real_V1022390504157884413omplex @ ( G @ N3 ) ) ) )
           => ( summable_real @ F ) ) ) ) ) ).

% series_comparison_complex
thf(fact_8601_series__comparison__complex,axiom,
    ! [G: nat > complex,N4: nat,F: nat > complex] :
      ( ( summable_complex @ G )
     => ( ! [N3: nat] : ( member_complex @ ( G @ N3 ) @ real_V2521375963428798218omplex )
       => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( re @ ( G @ N3 ) ) )
         => ( ! [N3: nat] :
                ( ( ord_less_eq_nat @ N4 @ N3 )
               => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ N3 ) ) @ ( real_V1022390504157884413omplex @ ( G @ N3 ) ) ) )
           => ( summable_complex @ F ) ) ) ) ) ).

% series_comparison_complex
thf(fact_8602_powser__insidea,axiom,
    ! [F: nat > real,X4: real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X4 @ N ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X4 ) )
       => ( summable_real
          @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ) ).

% powser_insidea
thf(fact_8603_powser__insidea,axiom,
    ! [F: nat > complex,X4: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X4 @ N ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X4 ) )
       => ( summable_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ) ).

% powser_insidea
thf(fact_8604_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_8605_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_8606_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_8607_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_8608_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_8609_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_8610_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_8611_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_8612_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_8613_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_8614_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_8615_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_8616_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_8617_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_8618_summable__powser__split__head,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_8619_summable__powser__split__head,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% summable_powser_split_head
thf(fact_8620_powser__split__head_I3_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_8621_powser__split__head_I3_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% powser_split_head(3)
thf(fact_8622_replicate__eqI,axiom,
    ! [Xs: list_real,N2: nat,X4: real] :
      ( ( ( size_size_list_real @ Xs )
        = N2 )
     => ( ! [Y3: real] :
            ( ( member_real @ Y3 @ ( set_real2 @ Xs ) )
           => ( Y3 = X4 ) )
       => ( Xs
          = ( replicate_real @ N2 @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_8623_replicate__eqI,axiom,
    ! [Xs: list_complex,N2: nat,X4: complex] :
      ( ( ( size_s3451745648224563538omplex @ Xs )
        = N2 )
     => ( ! [Y3: complex] :
            ( ( member_complex @ Y3 @ ( set_complex2 @ Xs ) )
           => ( Y3 = X4 ) )
       => ( Xs
          = ( replicate_complex @ N2 @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_8624_replicate__eqI,axiom,
    ! [Xs: list_P6011104703257516679at_nat,N2: nat,X4: product_prod_nat_nat] :
      ( ( ( size_s5460976970255530739at_nat @ Xs )
        = N2 )
     => ( ! [Y3: product_prod_nat_nat] :
            ( ( member8440522571783428010at_nat @ Y3 @ ( set_Pr5648618587558075414at_nat @ Xs ) )
           => ( Y3 = X4 ) )
       => ( Xs
          = ( replic4235873036481779905at_nat @ N2 @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_8625_replicate__eqI,axiom,
    ! [Xs: list_VEBT_VEBT,N2: nat,X4: vEBT_VEBT] :
      ( ( ( size_s6755466524823107622T_VEBT @ Xs )
        = N2 )
     => ( ! [Y3: vEBT_VEBT] :
            ( ( member_VEBT_VEBT @ Y3 @ ( set_VEBT_VEBT2 @ Xs ) )
           => ( Y3 = X4 ) )
       => ( Xs
          = ( replicate_VEBT_VEBT @ N2 @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_8626_replicate__eqI,axiom,
    ! [Xs: list_o,N2: nat,X4: $o] :
      ( ( ( size_size_list_o @ Xs )
        = N2 )
     => ( ! [Y3: $o] :
            ( ( member_o @ Y3 @ ( set_o2 @ Xs ) )
           => ( Y3 = X4 ) )
       => ( Xs
          = ( replicate_o @ N2 @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_8627_replicate__eqI,axiom,
    ! [Xs: list_nat,N2: nat,X4: nat] :
      ( ( ( size_size_list_nat @ Xs )
        = N2 )
     => ( ! [Y3: nat] :
            ( ( member_nat @ Y3 @ ( set_nat2 @ Xs ) )
           => ( Y3 = X4 ) )
       => ( Xs
          = ( replicate_nat @ N2 @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_8628_replicate__eqI,axiom,
    ! [Xs: list_int,N2: nat,X4: int] :
      ( ( ( size_size_list_int @ Xs )
        = N2 )
     => ( ! [Y3: int] :
            ( ( member_int @ Y3 @ ( set_int2 @ Xs ) )
           => ( Y3 = X4 ) )
       => ( Xs
          = ( replicate_int @ N2 @ X4 ) ) ) ) ).

% replicate_eqI
thf(fact_8629_replicate__length__same,axiom,
    ! [Xs: list_VEBT_VEBT,X4: vEBT_VEBT] :
      ( ! [X5: vEBT_VEBT] :
          ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ Xs ) )
         => ( X5 = X4 ) )
     => ( ( replicate_VEBT_VEBT @ ( size_s6755466524823107622T_VEBT @ Xs ) @ X4 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_8630_replicate__length__same,axiom,
    ! [Xs: list_o,X4: $o] :
      ( ! [X5: $o] :
          ( ( member_o @ X5 @ ( set_o2 @ Xs ) )
         => ( X5 = X4 ) )
     => ( ( replicate_o @ ( size_size_list_o @ Xs ) @ X4 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_8631_replicate__length__same,axiom,
    ! [Xs: list_nat,X4: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ ( set_nat2 @ Xs ) )
         => ( X5 = X4 ) )
     => ( ( replicate_nat @ ( size_size_list_nat @ Xs ) @ X4 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_8632_replicate__length__same,axiom,
    ! [Xs: list_int,X4: int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ ( set_int2 @ Xs ) )
         => ( X5 = X4 ) )
     => ( ( replicate_int @ ( size_size_list_int @ Xs ) @ X4 )
        = Xs ) ) ).

% replicate_length_same
thf(fact_8633_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > complex,M: nat,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_complex @ Z @ N ) ) )
      = ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_8634_summable__powser__ignore__initial__segment,axiom,
    ! [F: nat > real,M: nat,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ ( plus_plus_nat @ N @ M ) ) @ ( power_power_real @ Z @ N ) ) )
      = ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ).

% summable_powser_ignore_initial_segment
thf(fact_8635_summable__norm__comparison__test,axiom,
    ! [F: nat > complex,G: nat > real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ 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_8636_summable__rabs__comparison__test,axiom,
    ! [F: nat > real,G: nat > real] :
      ( ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ 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_8637_nonzero__Reals__divide,axiom,
    ! [A: real,B: real] :
      ( ( member_real @ A @ real_V470468836141973256s_real )
     => ( ( member_real @ B @ real_V470468836141973256s_real )
       => ( ( B != zero_zero_real )
         => ( member_real @ ( divide_divide_real @ A @ B ) @ real_V470468836141973256s_real ) ) ) ) ).

% nonzero_Reals_divide
thf(fact_8638_nonzero__Reals__divide,axiom,
    ! [A: complex,B: complex] :
      ( ( member_complex @ A @ real_V2521375963428798218omplex )
     => ( ( member_complex @ B @ real_V2521375963428798218omplex )
       => ( ( B != zero_zero_complex )
         => ( member_complex @ ( divide1717551699836669952omplex @ A @ B ) @ real_V2521375963428798218omplex ) ) ) ) ).

% nonzero_Reals_divide
thf(fact_8639_summable__rabs,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) )
     => ( ord_less_eq_real @ ( abs_abs_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( abs_abs_real @ ( F @ N ) ) ) ) ) ).

% summable_rabs
thf(fact_8640_suminf__pos2,axiom,
    ! [F: nat > real,I2: nat] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
         => ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_8641_suminf__pos2,axiom,
    ! [F: nat > nat,I2: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
         => ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_8642_suminf__pos2,axiom,
    ! [F: nat > int,I2: nat] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
         => ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) ) ) ) ) ).

% suminf_pos2
thf(fact_8643_suminf__pos__iff,axiom,
    ! [F: nat > real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ N3 ) )
       => ( ( ord_less_real @ zero_zero_real @ ( suminf_real @ F ) )
          = ( ? [I3: nat] : ( ord_less_real @ zero_zero_real @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_8644_suminf__pos__iff,axiom,
    ! [F: nat > nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( F @ N3 ) )
       => ( ( ord_less_nat @ zero_zero_nat @ ( suminf_nat @ F ) )
          = ( ? [I3: nat] : ( ord_less_nat @ zero_zero_nat @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_8645_suminf__pos__iff,axiom,
    ! [F: nat > int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( F @ N3 ) )
       => ( ( ord_less_int @ zero_zero_int @ ( suminf_int @ F ) )
          = ( ? [I3: nat] : ( ord_less_int @ zero_zero_int @ ( F @ I3 ) ) ) ) ) ) ).

% suminf_pos_iff
thf(fact_8646_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_8647_summable__geometric,axiom,
    ! [C: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ C ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ C ) ) ) ).

% summable_geometric
thf(fact_8648_complete__algebra__summable__geometric,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ one_one_real )
     => ( summable_real @ ( power_power_real @ X4 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_8649_complete__algebra__summable__geometric,axiom,
    ! [X4: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ one_one_real )
     => ( summable_complex @ ( power_power_complex @ X4 ) ) ) ).

% complete_algebra_summable_geometric
thf(fact_8650_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_8651_summable__norm,axiom,
    ! [F: nat > real] :
      ( ( summable_real
        @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( suminf_real @ F ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( F @ N ) ) ) ) ) ).

% summable_norm
thf(fact_8652_summable__norm,axiom,
    ! [F: nat > complex] :
      ( ( summable_real
        @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( suminf_complex @ F ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( F @ N ) ) ) ) ) ).

% summable_norm
thf(fact_8653_powser__inside,axiom,
    ! [F: nat > real,X4: real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ X4 @ N ) ) )
     => ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ ( real_V7735802525324610683m_real @ X4 ) )
       => ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) ) ) ) ).

% powser_inside
thf(fact_8654_powser__inside,axiom,
    ! [F: nat > complex,X4: complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ X4 @ N ) ) )
     => ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ ( real_V1022390504157884413omplex @ X4 ) )
       => ( summable_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) ) ) ) ).

% powser_inside
thf(fact_8655_summable__exp,axiom,
    ! [X4: complex] :
      ( summable_complex
      @ ^ [N: nat] : ( times_times_complex @ ( invers8013647133539491842omplex @ ( semiri5044797733671781792omplex @ N ) ) @ ( power_power_complex @ X4 @ N ) ) ) ).

% summable_exp
thf(fact_8656_summable__exp,axiom,
    ! [X4: real] :
      ( summable_real
      @ ^ [N: nat] : ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ).

% summable_exp
thf(fact_8657_powser__split__head_I1_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
        = ( plus_plus_complex @ ( F @ zero_zero_nat )
          @ ( times_times_complex
            @ ( suminf_complex
              @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_8658_powser__split__head_I1_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
        = ( plus_plus_real @ ( F @ zero_zero_nat )
          @ ( times_times_real
            @ ( suminf_real
              @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
            @ Z ) ) ) ) ).

% powser_split_head(1)
thf(fact_8659_powser__split__head_I2_J,axiom,
    ! [F: nat > complex,Z: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
     => ( ( times_times_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) ) )
          @ Z )
        = ( minus_minus_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( times_times_complex @ ( F @ N ) @ ( power_power_complex @ Z @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8660_powser__split__head_I2_J,axiom,
    ! [F: nat > real,Z: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
     => ( ( times_times_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) ) )
          @ Z )
        = ( minus_minus_real
          @ ( suminf_real
            @ ^ [N: nat] : ( times_times_real @ ( F @ N ) @ ( power_power_real @ Z @ N ) ) )
          @ ( F @ zero_zero_nat ) ) ) ) ).

% powser_split_head(2)
thf(fact_8661_suminf__exist__split,axiom,
    ! [R3: real,F: nat > real] :
      ( ( ord_less_real @ zero_zero_real @ R3 )
     => ( ( summable_real @ F )
       => ? [N8: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ N8 @ N6 )
           => ( ord_less_real
              @ ( real_V7735802525324610683m_real
                @ ( suminf_real
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N6 ) ) ) )
              @ R3 ) ) ) ) ).

% suminf_exist_split
thf(fact_8662_suminf__exist__split,axiom,
    ! [R3: real,F: nat > complex] :
      ( ( ord_less_real @ zero_zero_real @ R3 )
     => ( ( summable_complex @ F )
       => ? [N8: nat] :
          ! [N6: nat] :
            ( ( ord_less_eq_nat @ N8 @ N6 )
           => ( ord_less_real
              @ ( real_V1022390504157884413omplex
                @ ( suminf_complex
                  @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N6 ) ) ) )
              @ R3 ) ) ) ) ).

% suminf_exist_split
thf(fact_8663_summable__power__series,axiom,
    ! [F: nat > real,Z: real] :
      ( ! [I4: nat] : ( ord_less_eq_real @ ( F @ I4 ) @ one_one_real )
     => ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ I4 ) )
       => ( ( ord_less_eq_real @ zero_zero_real @ Z )
         => ( ( ord_less_real @ Z @ one_one_real )
           => ( summable_real
              @ ^ [I3: nat] : ( times_times_real @ ( F @ I3 ) @ ( power_power_real @ Z @ I3 ) ) ) ) ) ) ) ).

% summable_power_series
thf(fact_8664_Abel__lemma,axiom,
    ! [R3: real,R0: real,A: nat > complex,M7: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ R3 )
     => ( ( ord_less_real @ R3 @ 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 @ R3 @ N ) ) ) ) ) ) ).

% Abel_lemma
thf(fact_8665_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: nat > real] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N4 @ 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_8666_summable__ratio__test,axiom,
    ! [C: real,N4: nat,F: nat > complex] :
      ( ( ord_less_real @ C @ one_one_real )
     => ( ! [N3: nat] :
            ( ( ord_less_eq_nat @ N4 @ 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_8667_Re__Reals__divide,axiom,
    ! [R3: complex,Z: complex] :
      ( ( member_complex @ R3 @ real_V2521375963428798218omplex )
     => ( ( re @ ( divide1717551699836669952omplex @ R3 @ Z ) )
        = ( divide_divide_real @ ( times_times_real @ ( re @ R3 ) @ ( re @ Z ) ) @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% Re_Reals_divide
thf(fact_8668_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_8669_sin__paired,axiom,
    ! [X4: 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 @ X4 @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
      @ ( sin_real @ X4 ) ) ).

% sin_paired
thf(fact_8670_VEBT__internal_Omembermima_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_membermima @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ~ Y
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ none_P5556105721700978146at_nat @ zero_zero_nat @ Ux2 @ Uy2 ) @ Xa ) ) ) )
           => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( Y
                      = ( ( Xa = Mi3 )
                        | ( Xa = Ma3 ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) ) ) )
             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
                   => ( ( Y
                        = ( ( Xa = Mi3 )
                          | ( Xa = Ma3 )
                          | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) @ Xa ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X4
                        = ( vEBT_Node @ none_P5556105721700978146at_nat @ ( suc @ V2 ) @ TreeList3 @ Vd ) )
                     => ( ( Y
                          = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                             => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                            & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( 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 ) @ Xa ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(1)
thf(fact_8671_VEBT__internal_Omembermima_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) )
         => ( ! [Ux2: list_VEBT_VEBT,Uy2: vEBT_VEBT] :
                ( ( X4
                  = ( 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 ) @ Xa ) ) )
           => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
                   => ( ( Xa = Mi3 )
                      | ( Xa = Ma3 ) ) ) )
             => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                    ( ( X4
                      = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
                   => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) @ Xa ) )
                     => ( ( Xa = Mi3 )
                        | ( Xa = Ma3 )
                        | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
               => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                      ( ( X4
                        = ( 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 ) @ Xa ) )
                       => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                           => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                          & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(3)
thf(fact_8672_geometric__deriv__sums,axiom,
    ! [Z: real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ Z ) @ one_one_real )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( power_power_real @ Z @ N ) )
        @ ( divide_divide_real @ one_one_real @ ( power_power_real @ ( minus_minus_real @ one_one_real @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8673_geometric__deriv__sums,axiom,
    ! [Z: complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ Z ) @ one_one_real )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( power_power_complex @ Z @ N ) )
        @ ( divide1717551699836669952omplex @ one_one_complex @ ( power_power_complex @ ( minus_minus_complex @ one_one_complex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% geometric_deriv_sums
thf(fact_8674_powser__sums__zero__iff,axiom,
    ! [A: nat > complex,X4: complex] :
      ( ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( A @ N ) @ ( power_power_complex @ zero_zero_complex @ N ) )
        @ X4 )
      = ( ( A @ zero_zero_nat )
        = X4 ) ) ).

% powser_sums_zero_iff
thf(fact_8675_powser__sums__zero__iff,axiom,
    ! [A: nat > real,X4: real] :
      ( ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( A @ N ) @ ( power_power_real @ zero_zero_real @ N ) )
        @ X4 )
      = ( ( A @ zero_zero_nat )
        = X4 ) ) ).

% powser_sums_zero_iff
thf(fact_8676_sums__le,axiom,
    ! [F: nat > real,G: nat > real,S: real,T2: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_real @ F @ S )
       => ( ( sums_real @ G @ T2 )
         => ( ord_less_eq_real @ S @ T2 ) ) ) ) ).

% sums_le
thf(fact_8677_sums__le,axiom,
    ! [F: nat > nat,G: nat > nat,S: nat,T2: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_nat @ F @ S )
       => ( ( sums_nat @ G @ T2 )
         => ( ord_less_eq_nat @ S @ T2 ) ) ) ) ).

% sums_le
thf(fact_8678_sums__le,axiom,
    ! [F: nat > int,G: nat > int,S: int,T2: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( F @ N3 ) @ ( G @ N3 ) )
     => ( ( sums_int @ F @ S )
       => ( ( sums_int @ G @ T2 )
         => ( ord_less_eq_int @ S @ T2 ) ) ) ) ).

% sums_le
thf(fact_8679_sums__divide,axiom,
    ! [F: nat > real,A: real,C: real] :
      ( ( sums_real @ F @ A )
     => ( sums_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ C )
        @ ( divide_divide_real @ A @ C ) ) ) ).

% sums_divide
thf(fact_8680_sums__divide,axiom,
    ! [F: nat > complex,A: complex,C: complex] :
      ( ( sums_complex @ F @ A )
     => ( sums_complex
        @ ^ [N: nat] : ( divide1717551699836669952omplex @ ( F @ N ) @ C )
        @ ( divide1717551699836669952omplex @ A @ C ) ) ) ).

% sums_divide
thf(fact_8681_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_8682_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_8683_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_8684_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_8685_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_8686_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_8687_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_8688_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_8689_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_8690_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_8691_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_8692_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > complex,S: complex] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ( F @ I4 )
            = zero_zero_complex ) )
     => ( ( sums_complex
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
          @ S )
        = ( sums_complex @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_8693_sums__zero__iff__shift,axiom,
    ! [N2: nat,F: nat > real,S: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ I4 @ N2 )
         => ( ( F @ I4 )
            = zero_zero_real ) )
     => ( ( sums_real
          @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
          @ S )
        = ( sums_real @ F @ S ) ) ) ).

% sums_zero_iff_shift
thf(fact_8694_powser__sums__if,axiom,
    ! [M: nat,Z: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( times_times_complex @ ( if_complex @ ( N = M ) @ one_one_complex @ zero_zero_complex ) @ ( power_power_complex @ Z @ N ) )
      @ ( power_power_complex @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8695_powser__sums__if,axiom,
    ! [M: nat,Z: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( if_real @ ( N = M ) @ one_one_real @ zero_zero_real ) @ ( power_power_real @ Z @ N ) )
      @ ( power_power_real @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8696_powser__sums__if,axiom,
    ! [M: nat,Z: int] :
      ( sums_int
      @ ^ [N: nat] : ( times_times_int @ ( if_int @ ( N = M ) @ one_one_int @ zero_zero_int ) @ ( power_power_int @ Z @ N ) )
      @ ( power_power_int @ Z @ M ) ) ).

% powser_sums_if
thf(fact_8697_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_8698_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_8699_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_8700_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_8701_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_8702_sums__if_H,axiom,
    ! [G: nat > real,X4: real] :
      ( ( sums_real @ G @ X4 )
     => ( sums_real
        @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
        @ X4 ) ) ).

% sums_if'
thf(fact_8703_sums__if,axiom,
    ! [G: nat > real,X4: real,F: nat > real,Y: real] :
      ( ( sums_real @ G @ X4 )
     => ( ( sums_real @ F @ Y )
       => ( sums_real
          @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( F @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( G @ ( divide_divide_nat @ ( minus_minus_nat @ N @ one_one_nat ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
          @ ( plus_plus_real @ X4 @ Y ) ) ) ) ).

% sums_if
thf(fact_8704_cos__paired,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N: nat] : ( times_times_real @ ( divide_divide_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( semiri2265585572941072030t_real @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) @ ( power_power_real @ X4 @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
      @ ( cos_real @ X4 ) ) ).

% cos_paired
thf(fact_8705_VEBT__internal_Omembermima_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_membermima @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Mi3: nat,Ma3: nat,Va3: list_VEBT_VEBT,Vb2: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ zero_zero_nat @ Va3 @ Vb2 ) @ Xa ) )
               => ~ ( ( Xa = Mi3 )
                    | ( Xa = Ma3 ) ) ) )
         => ( ! [Mi3: nat,Ma3: nat,V2: nat,TreeList3: list_VEBT_VEBT,Vc: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V4351362008482014158ma_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ ( some_P7363390416028606310at_nat @ ( product_Pair_nat_nat @ Mi3 @ Ma3 ) ) @ ( suc @ V2 ) @ TreeList3 @ Vc ) @ Xa ) )
                 => ~ ( ( Xa = Mi3 )
                      | ( Xa = Ma3 )
                      | ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) )
           => ~ ! [V2: nat,TreeList3: list_VEBT_VEBT,Vd: vEBT_VEBT] :
                  ( ( X4
                    = ( 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 ) @ Xa ) )
                   => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_VEBT_membermima @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.membermima.pelims(2)
thf(fact_8706_VEBT__internal_Onaive__member_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
               => ( ( ( Xa = zero_zero_nat )
                   => A5 )
                  & ( ( Xa != zero_zero_nat )
                   => ( ( ( Xa = one_one_nat )
                       => B5 )
                      & ( Xa = one_one_nat ) ) ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
                   => ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(3)
thf(fact_8707_VEBT__internal_Onaive__member_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) )
               => ~ ( ( ( Xa = zero_zero_nat )
                     => A5 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B5 )
                        & ( Xa = one_one_nat ) ) ) ) ) )
         => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) )
                 => ~ ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                       => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                      & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(2)
thf(fact_8708_VEBT__internal_Onaive__member_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_V5719532721284313246member @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [A5: $o,B5: $o] :
              ( ( X4
                = ( vEBT_Leaf @ A5 @ B5 ) )
             => ( ( Y
                  = ( ( ( Xa = zero_zero_nat )
                     => A5 )
                    & ( ( Xa != zero_zero_nat )
                     => ( ( ( Xa = one_one_nat )
                         => B5 )
                        & ( Xa = one_one_nat ) ) ) ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ A5 @ B5 ) @ Xa ) ) ) )
         => ( ! [Uu2: option4927543243414619207at_nat,Uv2: list_VEBT_VEBT,Uw2: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) )
               => ( ~ Y
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uu2 @ zero_zero_nat @ Uv2 @ Uw2 ) @ Xa ) ) ) )
           => ~ ! [Uy2: option4927543243414619207at_nat,V2: nat,TreeList3: list_VEBT_VEBT,S3: vEBT_VEBT] :
                  ( ( X4
                    = ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) )
                 => ( ( Y
                      = ( ( ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) )
                         => ( vEBT_V5719532721284313246member @ ( nth_VEBT_VEBT @ TreeList3 @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( vEBT_VEBT_low @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                        & ( ord_less_nat @ ( vEBT_VEBT_high @ Xa @ ( divide_divide_nat @ ( suc @ V2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( size_s6755466524823107622T_VEBT @ TreeList3 ) ) ) )
                   => ~ ( accp_P2887432264394892906BT_nat @ vEBT_V5765760719290551771er_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Uy2 @ ( suc @ V2 ) @ TreeList3 @ S3 ) @ Xa ) ) ) ) ) ) ) ) ).

% VEBT_internal.naive_member.pelims(1)
thf(fact_8709_diffs__equiv,axiom,
    ! [C: nat > complex,X4: complex] :
      ( ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X4 @ N ) ) )
     => ( sums_complex
        @ ^ [N: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N ) @ ( C @ N ) ) @ ( power_power_complex @ X4 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_complex
          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X4 @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_8710_diffs__equiv,axiom,
    ! [C: nat > real,X4: real] :
      ( ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X4 @ N ) ) )
     => ( sums_real
        @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N ) @ ( C @ N ) ) @ ( power_power_real @ X4 @ ( minus_minus_nat @ N @ ( suc @ zero_zero_nat ) ) ) )
        @ ( suminf_real
          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ).

% diffs_equiv
thf(fact_8711_diffs__def,axiom,
    ( diffs_complex
    = ( ^ [C2: nat > complex,N: nat] : ( times_times_complex @ ( semiri8010041392384452111omplex @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_8712_diffs__def,axiom,
    ( diffs_real
    = ( ^ [C2: nat > real,N: nat] : ( times_times_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_8713_diffs__def,axiom,
    ( diffs_int
    = ( ^ [C2: nat > int,N: nat] : ( times_times_int @ ( semiri1314217659103216013at_int @ ( suc @ N ) ) @ ( C2 @ ( suc @ N ) ) ) ) ) ).

% diffs_def
thf(fact_8714_termdiff__converges__all,axiom,
    ! [C: nat > complex,X4: complex] :
      ( ! [X5: complex] :
          ( summable_complex
          @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X5 @ N ) ) )
     => ( summable_complex
        @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X4 @ N ) ) ) ) ).

% termdiff_converges_all
thf(fact_8715_termdiff__converges__all,axiom,
    ! [C: nat > real,X4: real] :
      ( ! [X5: real] :
          ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X5 @ N ) ) )
     => ( summable_real
        @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X4 @ N ) ) ) ) ).

% termdiff_converges_all
thf(fact_8716_termdiff__converges,axiom,
    ! [X4: real,K5: real,C: nat > real] :
      ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X4 ) @ K5 )
     => ( ! [X5: real] :
            ( ( ord_less_real @ ( real_V7735802525324610683m_real @ X5 ) @ K5 )
           => ( summable_real
              @ ^ [N: nat] : ( times_times_real @ ( C @ N ) @ ( power_power_real @ X5 @ N ) ) ) )
       => ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( diffs_real @ C @ N ) @ ( power_power_real @ X4 @ N ) ) ) ) ) ).

% termdiff_converges
thf(fact_8717_termdiff__converges,axiom,
    ! [X4: complex,K5: real,C: nat > complex] :
      ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X4 ) @ K5 )
     => ( ! [X5: complex] :
            ( ( ord_less_real @ ( real_V1022390504157884413omplex @ X5 ) @ K5 )
           => ( summable_complex
              @ ^ [N: nat] : ( times_times_complex @ ( C @ N ) @ ( power_power_complex @ X5 @ N ) ) ) )
       => ( summable_complex
          @ ^ [N: nat] : ( times_times_complex @ ( diffs_complex @ C @ N ) @ ( power_power_complex @ X4 @ N ) ) ) ) ) ).

% termdiff_converges
thf(fact_8718_exp__first__two__terms,axiom,
    ( exp_real
    = ( ^ [X: real] :
          ( plus_plus_real @ ( plus_plus_real @ one_one_real @ X )
          @ ( suminf_real
            @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_8719_exp__first__two__terms,axiom,
    ( exp_complex
    = ( ^ [X: complex] :
          ( plus_plus_complex @ ( plus_plus_complex @ one_one_complex @ X )
          @ ( suminf_complex
            @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( power_power_complex @ X @ ( plus_plus_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ).

% exp_first_two_terms
thf(fact_8720_monoseq__def,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X3: nat > real] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_real @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_real @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8721_monoseq__def,axiom,
    ( topolo3100542954746470799et_int
    = ( ^ [X3: nat > set_int] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_set_int @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_set_int @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8722_monoseq__def,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X3: nat > rat] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_rat @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_rat @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8723_monoseq__def,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X3: nat > num] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_num @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_num @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8724_monoseq__def,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X3: nat > nat] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_nat @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_nat @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8725_monoseq__def,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X3: nat > int] :
          ( ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_int @ ( X3 @ M6 ) @ ( X3 @ N ) ) )
          | ! [M6: nat,N: nat] :
              ( ( ord_less_eq_nat @ M6 @ N )
             => ( ord_less_eq_int @ ( X3 @ N ) @ ( X3 @ M6 ) ) ) ) ) ) ).

% monoseq_def
thf(fact_8726_monoI2,axiom,
    ! [X8: nat > real] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_real @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% monoI2
thf(fact_8727_monoI2,axiom,
    ! [X8: nat > set_int] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_set_int @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
     => ( topolo3100542954746470799et_int @ X8 ) ) ).

% monoI2
thf(fact_8728_monoI2,axiom,
    ! [X8: nat > rat] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_rat @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% monoI2
thf(fact_8729_monoI2,axiom,
    ! [X8: nat > num] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_num @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% monoI2
thf(fact_8730_monoI2,axiom,
    ! [X8: nat > nat] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_nat @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% monoI2
thf(fact_8731_monoI2,axiom,
    ! [X8: nat > int] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_int @ ( X8 @ N3 ) @ ( X8 @ M5 ) ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% monoI2
thf(fact_8732_monoI1,axiom,
    ! [X8: nat > real] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_real @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% monoI1
thf(fact_8733_monoI1,axiom,
    ! [X8: nat > set_int] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_set_int @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
     => ( topolo3100542954746470799et_int @ X8 ) ) ).

% monoI1
thf(fact_8734_monoI1,axiom,
    ! [X8: nat > rat] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_rat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% monoI1
thf(fact_8735_monoI1,axiom,
    ! [X8: nat > num] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_num @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% monoI1
thf(fact_8736_monoI1,axiom,
    ! [X8: nat > nat] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_nat @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% monoI1
thf(fact_8737_monoI1,axiom,
    ! [X8: nat > int] :
      ( ! [M5: nat,N3: nat] :
          ( ( ord_less_eq_nat @ M5 @ N3 )
         => ( ord_less_eq_int @ ( X8 @ M5 ) @ ( X8 @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% monoI1
thf(fact_8738_scaleR__eq__iff,axiom,
    ! [B: real,U: real,A: real] :
      ( ( ( plus_plus_real @ B @ ( real_V1485227260804924795R_real @ U @ A ) )
        = ( plus_plus_real @ A @ ( real_V1485227260804924795R_real @ U @ B ) ) )
      = ( ( A = B )
        | ( U = one_one_real ) ) ) ).

% scaleR_eq_iff
thf(fact_8739_scaleR__power,axiom,
    ! [X4: real,Y: real,N2: nat] :
      ( ( power_power_real @ ( real_V1485227260804924795R_real @ X4 @ Y ) @ N2 )
      = ( real_V1485227260804924795R_real @ ( power_power_real @ X4 @ N2 ) @ ( power_power_real @ Y @ N2 ) ) ) ).

% scaleR_power
thf(fact_8740_scaleR__power,axiom,
    ! [X4: real,Y: complex,N2: nat] :
      ( ( power_power_complex @ ( real_V2046097035970521341omplex @ X4 @ Y ) @ N2 )
      = ( real_V2046097035970521341omplex @ ( power_power_real @ X4 @ N2 ) @ ( power_power_complex @ Y @ N2 ) ) ) ).

% scaleR_power
thf(fact_8741_scaleR__minus1__left,axiom,
    ! [X4: real] :
      ( ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
      = ( uminus_uminus_real @ X4 ) ) ).

% scaleR_minus1_left
thf(fact_8742_scaleR__minus1__left,axiom,
    ! [X4: complex] :
      ( ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ one_one_real ) @ X4 )
      = ( uminus1482373934393186551omplex @ X4 ) ) ).

% scaleR_minus1_left
thf(fact_8743_scaleR__collapse,axiom,
    ! [U: real,A: real] :
      ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ one_one_real @ U ) @ A ) @ ( real_V1485227260804924795R_real @ U @ A ) )
      = A ) ).

% scaleR_collapse
thf(fact_8744_scaleR__times,axiom,
    ! [U: num,W: num,A: complex] :
      ( ( real_V2046097035970521341omplex @ ( numeral_numeral_real @ U ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( real_V2046097035970521341omplex @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).

% scaleR_times
thf(fact_8745_scaleR__times,axiom,
    ! [U: num,W: num,A: real] :
      ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ U ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( real_V1485227260804924795R_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ A ) ) ).

% scaleR_times
thf(fact_8746_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: complex] :
      ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% inverse_scaleR_times
thf(fact_8747_inverse__scaleR__times,axiom,
    ! [V: num,W: num,A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ W ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% inverse_scaleR_times
thf(fact_8748_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: complex] :
      ( ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_complex @ ( numera6690914467698888265omplex @ W ) @ A ) )
      = ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% fraction_scaleR_times
thf(fact_8749_fraction__scaleR__times,axiom,
    ! [U: num,V: num,W: num,A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ V ) ) @ ( times_times_real @ ( numeral_numeral_real @ W ) @ A ) )
      = ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( times_times_real @ ( numeral_numeral_real @ U ) @ ( numeral_numeral_real @ W ) ) @ ( numeral_numeral_real @ V ) ) @ A ) ) ).

% fraction_scaleR_times
thf(fact_8750_scaleR__half__double,axiom,
    ! [A: real] :
      ( ( real_V1485227260804924795R_real @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ A @ A ) )
      = A ) ).

% scaleR_half_double
thf(fact_8751_scaleR__right__distrib,axiom,
    ! [A: real,X4: real,Y: real] :
      ( ( real_V1485227260804924795R_real @ A @ ( plus_plus_real @ X4 @ Y ) )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X4 ) @ ( real_V1485227260804924795R_real @ A @ Y ) ) ) ).

% scaleR_right_distrib
thf(fact_8752_scaleR__left__distrib,axiom,
    ! [A: real,B: real,X4: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ A @ B ) @ X4 )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ X4 ) @ ( real_V1485227260804924795R_real @ B @ X4 ) ) ) ).

% scaleR_left_distrib
thf(fact_8753_scaleR__left_Oadd,axiom,
    ! [X4: real,Y: real,Xa: real] :
      ( ( real_V1485227260804924795R_real @ ( plus_plus_real @ X4 @ Y ) @ Xa )
      = ( plus_plus_real @ ( real_V1485227260804924795R_real @ X4 @ Xa ) @ ( real_V1485227260804924795R_real @ Y @ Xa ) ) ) ).

% scaleR_left.add
thf(fact_8754_of__real__def,axiom,
    ( real_V1803761363581548252l_real
    = ( ^ [R5: real] : ( real_V1485227260804924795R_real @ R5 @ one_one_real ) ) ) ).

% of_real_def
thf(fact_8755_of__real__def,axiom,
    ( real_V4546457046886955230omplex
    = ( ^ [R5: real] : ( real_V2046097035970521341omplex @ R5 @ one_one_complex ) ) ) ).

% of_real_def
thf(fact_8756_scaleR__right__mono,axiom,
    ! [A: real,B: real,X4: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X4 ) @ ( real_V1485227260804924795R_real @ B @ X4 ) ) ) ) ).

% scaleR_right_mono
thf(fact_8757_scaleR__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 @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ C ) ) ) ) ).

% scaleR_right_mono_neg
thf(fact_8758_Real__Vector__Spaces_Ole__add__iff2,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ C @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ B @ A ) @ E2 ) @ D ) ) ) ).

% Real_Vector_Spaces.le_add_iff2
thf(fact_8759_Real__Vector__Spaces_Ole__add__iff1,axiom,
    ! [A: real,E2: real,C: real,B: real,D: real] :
      ( ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ A @ E2 ) @ C ) @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ B @ E2 ) @ D ) )
      = ( ord_less_eq_real @ ( plus_plus_real @ ( real_V1485227260804924795R_real @ ( minus_minus_real @ A @ B ) @ E2 ) @ C ) @ D ) ) ).

% Real_Vector_Spaces.le_add_iff1
thf(fact_8760_zero__le__scaleR__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ B ) )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( A = zero_zero_real ) ) ) ).

% zero_le_scaleR_iff
thf(fact_8761_scaleR__le__0__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ B ) @ zero_zero_real )
      = ( ( ( ord_less_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ B @ zero_zero_real ) )
        | ( ( ord_less_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ B ) )
        | ( A = zero_zero_real ) ) ) ).

% scaleR_le_0_iff
thf(fact_8762_scaleR__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 @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ) ).

% scaleR_nonpos_nonpos
thf(fact_8763_scaleR__nonpos__nonneg,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_eq_real @ A @ zero_zero_real )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X4 ) @ zero_zero_real ) ) ) ).

% scaleR_nonpos_nonneg
thf(fact_8764_scaleR__nonneg__nonpos,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ X4 @ zero_zero_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X4 ) @ zero_zero_real ) ) ) ).

% scaleR_nonneg_nonpos
thf(fact_8765_scaleR__nonneg__nonneg,axiom,
    ! [A: real,X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ A )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ord_less_eq_real @ zero_zero_real @ ( real_V1485227260804924795R_real @ A @ X4 ) ) ) ) ).

% scaleR_nonneg_nonneg
thf(fact_8766_split__scaleR__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 @ ( real_V1485227260804924795R_real @ A @ B ) ) ) ).

% split_scaleR_pos_le
thf(fact_8767_split__scaleR__neg__le,axiom,
    ! [A: real,X4: real] :
      ( ( ( ( ord_less_eq_real @ zero_zero_real @ A )
          & ( ord_less_eq_real @ X4 @ zero_zero_real ) )
        | ( ( ord_less_eq_real @ A @ zero_zero_real )
          & ( ord_less_eq_real @ zero_zero_real @ X4 ) ) )
     => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X4 ) @ zero_zero_real ) ) ).

% split_scaleR_neg_le
thf(fact_8768_scaleR__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 @ ( real_V1485227260804924795R_real @ A @ C ) @ ( real_V1485227260804924795R_real @ B @ D ) ) ) ) ) ) ).

% scaleR_mono'
thf(fact_8769_scaleR__mono,axiom,
    ! [A: real,B: real,X4: real,Y: real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ( ord_less_eq_real @ zero_zero_real @ B )
         => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
           => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X4 ) @ ( real_V1485227260804924795R_real @ B @ Y ) ) ) ) ) ) ).

% scaleR_mono
thf(fact_8770_scaleR__left__le__one__le,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ A @ one_one_real )
       => ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ A @ X4 ) @ X4 ) ) ) ).

% scaleR_left_le_one_le
thf(fact_8771_scaleR__2,axiom,
    ! [X4: real] :
      ( ( real_V1485227260804924795R_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ X4 )
      = ( plus_plus_real @ X4 @ X4 ) ) ).

% scaleR_2
thf(fact_8772_real__vector__affinity__eq,axiom,
    ! [M: real,X4: real,C: real,Y: real] :
      ( ( M != zero_zero_real )
     => ( ( ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X4 ) @ C )
          = Y )
        = ( X4
          = ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) ) ) ) ) ).

% real_vector_affinity_eq
thf(fact_8773_real__vector__eq__affinity,axiom,
    ! [M: real,Y: real,X4: real,C: real] :
      ( ( M != zero_zero_real )
     => ( ( Y
          = ( plus_plus_real @ ( real_V1485227260804924795R_real @ M @ X4 ) @ C ) )
        = ( ( minus_minus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ Y ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ M ) @ C ) )
          = X4 ) ) ) ).

% real_vector_eq_affinity
thf(fact_8774_neg__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
        = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% neg_less_divideR_eq
thf(fact_8775_neg__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
        = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).

% neg_divideR_less_eq
thf(fact_8776_pos__less__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) )
        = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ B ) ) ) ).

% pos_less_divideR_eq
thf(fact_8777_pos__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) @ A )
        = ( ord_less_real @ B @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% pos_divideR_less_eq
thf(fact_8778_summable__exp__generic,axiom,
    ! [X4: real] :
      ( summable_real
      @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ).

% summable_exp_generic
thf(fact_8779_summable__exp__generic,axiom,
    ! [X4: complex] :
      ( summable_complex
      @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X4 @ N ) ) ) ).

% summable_exp_generic
thf(fact_8780_sin__converges,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N ) @ ( power_power_real @ X4 @ N ) )
      @ ( sin_real @ X4 ) ) ).

% sin_converges
thf(fact_8781_sin__converges,axiom,
    ! [X4: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N ) @ ( power_power_complex @ X4 @ N ) )
      @ ( sin_complex @ X4 ) ) ).

% sin_converges
thf(fact_8782_sin__def,axiom,
    ( sin_real
    = ( ^ [X: real] :
          ( suminf_real
          @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( sin_coeff @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% sin_def
thf(fact_8783_sin__def,axiom,
    ( sin_complex
    = ( ^ [X: complex] :
          ( suminf_complex
          @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( sin_coeff @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).

% sin_def
thf(fact_8784_cos__converges,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N ) @ ( power_power_real @ X4 @ N ) )
      @ ( cos_real @ X4 ) ) ).

% cos_converges
thf(fact_8785_cos__converges,axiom,
    ! [X4: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N ) @ ( power_power_complex @ X4 @ N ) )
      @ ( cos_complex @ X4 ) ) ).

% cos_converges
thf(fact_8786_cos__def,axiom,
    ( cos_real
    = ( ^ [X: real] :
          ( suminf_real
          @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% cos_def
thf(fact_8787_cos__def,axiom,
    ( cos_complex
    = ( ^ [X: complex] :
          ( suminf_complex
          @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).

% cos_def
thf(fact_8788_summable__norm__sin,axiom,
    ! [X4: real] :
      ( summable_real
      @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N ) @ ( power_power_real @ X4 @ N ) ) ) ) ).

% summable_norm_sin
thf(fact_8789_summable__norm__sin,axiom,
    ! [X4: complex] :
      ( summable_real
      @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N ) @ ( power_power_complex @ X4 @ N ) ) ) ) ).

% summable_norm_sin
thf(fact_8790_summable__norm__cos,axiom,
    ! [X4: real] :
      ( summable_real
      @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( cos_coeff @ N ) @ ( power_power_real @ X4 @ N ) ) ) ) ).

% summable_norm_cos
thf(fact_8791_summable__norm__cos,axiom,
    ! [X4: complex] :
      ( summable_real
      @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( cos_coeff @ N ) @ ( power_power_complex @ X4 @ N ) ) ) ) ).

% summable_norm_cos
thf(fact_8792_pos__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
        = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_le_minus_divideR_eq
thf(fact_8793_pos__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% pos_minus_divideR_le_eq
thf(fact_8794_neg__le__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
        = ( ord_less_eq_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% neg_le_minus_divideR_eq
thf(fact_8795_neg__minus__divideR__le__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
        = ( ord_less_eq_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divideR_le_eq
thf(fact_8796_neg__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
        = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% neg_minus_divideR_less_eq
thf(fact_8797_neg__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ C @ zero_zero_real )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% neg_less_minus_divideR_eq
thf(fact_8798_pos__minus__divideR__less__eq,axiom,
    ! [C: real,B: real,A: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) @ A )
        = ( ord_less_real @ ( uminus_uminus_real @ B ) @ ( real_V1485227260804924795R_real @ C @ A ) ) ) ) ).

% pos_minus_divideR_less_eq
thf(fact_8799_pos__less__minus__divideR__eq,axiom,
    ! [C: real,A: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ C )
     => ( ( ord_less_real @ A @ ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ C ) @ B ) ) )
        = ( ord_less_real @ ( real_V1485227260804924795R_real @ C @ A ) @ ( uminus_uminus_real @ B ) ) ) ) ).

% pos_less_minus_divideR_eq
thf(fact_8800_exp__converges,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) )
      @ ( exp_real @ X4 ) ) ).

% exp_converges
thf(fact_8801_exp__converges,axiom,
    ! [X4: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X4 @ N ) )
      @ ( exp_complex @ X4 ) ) ).

% exp_converges
thf(fact_8802_exp__def,axiom,
    ( exp_real
    = ( ^ [X: real] :
          ( suminf_real
          @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) ) ) ) ) ).

% exp_def
thf(fact_8803_exp__def,axiom,
    ( exp_complex
    = ( ^ [X: complex] :
          ( suminf_complex
          @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X @ N ) ) ) ) ) ).

% exp_def
thf(fact_8804_summable__norm__exp,axiom,
    ! [X4: real] :
      ( summable_real
      @ ^ [N: nat] : ( real_V7735802525324610683m_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) ) ) ).

% summable_norm_exp
thf(fact_8805_summable__norm__exp,axiom,
    ! [X4: complex] :
      ( summable_real
      @ ^ [N: nat] : ( real_V1022390504157884413omplex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X4 @ N ) ) ) ) ).

% summable_norm_exp
thf(fact_8806_sin__minus__converges,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N: nat] : ( uminus_uminus_real @ ( real_V1485227260804924795R_real @ ( sin_coeff @ N ) @ ( power_power_real @ ( uminus_uminus_real @ X4 ) @ N ) ) )
      @ ( sin_real @ X4 ) ) ).

% sin_minus_converges
thf(fact_8807_sin__minus__converges,axiom,
    ! [X4: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( uminus1482373934393186551omplex @ ( real_V2046097035970521341omplex @ ( sin_coeff @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ N ) ) )
      @ ( sin_complex @ X4 ) ) ).

% sin_minus_converges
thf(fact_8808_cos__minus__converges,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( cos_coeff @ N ) @ ( power_power_real @ ( uminus_uminus_real @ X4 ) @ N ) )
      @ ( cos_real @ X4 ) ) ).

% cos_minus_converges
thf(fact_8809_cos__minus__converges,axiom,
    ! [X4: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( cos_coeff @ N ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ N ) )
      @ ( cos_complex @ X4 ) ) ).

% cos_minus_converges
thf(fact_8810_cosh__def,axiom,
    ( cosh_real
    = ( ^ [X: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).

% cosh_def
thf(fact_8811_cosh__def,axiom,
    ( cosh_complex
    = ( ^ [X: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( plus_plus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).

% cosh_def
thf(fact_8812_sinh__def,axiom,
    ( sinh_real
    = ( ^ [X: real] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_real @ ( exp_real @ X ) @ ( exp_real @ ( uminus_uminus_real @ X ) ) ) ) ) ) ).

% sinh_def
thf(fact_8813_sinh__def,axiom,
    ( sinh_complex
    = ( ^ [X: complex] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( minus_minus_complex @ ( exp_complex @ X ) @ ( exp_complex @ ( uminus1482373934393186551omplex @ X ) ) ) ) ) ) ).

% sinh_def
thf(fact_8814_exp__first__term,axiom,
    ( exp_real
    = ( ^ [X: real] :
          ( plus_plus_real @ one_one_real
          @ ( suminf_real
            @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X @ ( suc @ N ) ) ) ) ) ) ) ).

% exp_first_term
thf(fact_8815_exp__first__term,axiom,
    ( exp_complex
    = ( ^ [X: complex] :
          ( plus_plus_complex @ one_one_complex
          @ ( suminf_complex
            @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( suc @ N ) ) ) @ ( power_power_complex @ X @ ( suc @ N ) ) ) ) ) ) ) ).

% exp_first_term
thf(fact_8816_cosh__converges,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) @ zero_zero_real )
      @ ( cosh_real @ X4 ) ) ).

% cosh_converges
thf(fact_8817_cosh__converges,axiom,
    ! [X4: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X4 @ N ) ) @ zero_zero_complex )
      @ ( cosh_complex @ X4 ) ) ).

% cosh_converges
thf(fact_8818_sinh__converges,axiom,
    ! [X4: real] :
      ( sums_real
      @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X4 @ N ) ) )
      @ ( sinh_real @ X4 ) ) ).

% sinh_converges
thf(fact_8819_sinh__converges,axiom,
    ! [X4: complex] :
      ( sums_complex
      @ ^ [N: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ zero_zero_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X4 @ N ) ) )
      @ ( sinh_complex @ X4 ) ) ).

% sinh_converges
thf(fact_8820_mono__SucI1,axiom,
    ! [X8: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% mono_SucI1
thf(fact_8821_mono__SucI1,axiom,
    ! [X8: nat > set_int] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
     => ( topolo3100542954746470799et_int @ X8 ) ) ).

% mono_SucI1
thf(fact_8822_mono__SucI1,axiom,
    ! [X8: nat > rat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% mono_SucI1
thf(fact_8823_mono__SucI1,axiom,
    ! [X8: nat > num] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% mono_SucI1
thf(fact_8824_mono__SucI1,axiom,
    ! [X8: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% mono_SucI1
thf(fact_8825_mono__SucI1,axiom,
    ! [X8: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X8 @ N3 ) @ ( X8 @ ( suc @ N3 ) ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% mono_SucI1
thf(fact_8826_mono__SucI2,axiom,
    ! [X8: nat > real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
     => ( topolo6980174941875973593q_real @ X8 ) ) ).

% mono_SucI2
thf(fact_8827_mono__SucI2,axiom,
    ! [X8: nat > set_int] :
      ( ! [N3: nat] : ( ord_less_eq_set_int @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
     => ( topolo3100542954746470799et_int @ X8 ) ) ).

% mono_SucI2
thf(fact_8828_mono__SucI2,axiom,
    ! [X8: nat > rat] :
      ( ! [N3: nat] : ( ord_less_eq_rat @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
     => ( topolo4267028734544971653eq_rat @ X8 ) ) ).

% mono_SucI2
thf(fact_8829_mono__SucI2,axiom,
    ! [X8: nat > num] :
      ( ! [N3: nat] : ( ord_less_eq_num @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
     => ( topolo1459490580787246023eq_num @ X8 ) ) ).

% mono_SucI2
thf(fact_8830_mono__SucI2,axiom,
    ! [X8: nat > nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
     => ( topolo4902158794631467389eq_nat @ X8 ) ) ).

% mono_SucI2
thf(fact_8831_mono__SucI2,axiom,
    ! [X8: nat > int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ ( X8 @ ( suc @ N3 ) ) @ ( X8 @ N3 ) )
     => ( topolo4899668324122417113eq_int @ X8 ) ) ).

% mono_SucI2
thf(fact_8832_monoseq__Suc,axiom,
    ( topolo6980174941875973593q_real
    = ( ^ [X3: nat > real] :
          ( ! [N: nat] : ( ord_less_eq_real @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_real @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8833_monoseq__Suc,axiom,
    ( topolo3100542954746470799et_int
    = ( ^ [X3: nat > set_int] :
          ( ! [N: nat] : ( ord_less_eq_set_int @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_set_int @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8834_monoseq__Suc,axiom,
    ( topolo4267028734544971653eq_rat
    = ( ^ [X3: nat > rat] :
          ( ! [N: nat] : ( ord_less_eq_rat @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_rat @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8835_monoseq__Suc,axiom,
    ( topolo1459490580787246023eq_num
    = ( ^ [X3: nat > num] :
          ( ! [N: nat] : ( ord_less_eq_num @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_num @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8836_monoseq__Suc,axiom,
    ( topolo4902158794631467389eq_nat
    = ( ^ [X3: nat > nat] :
          ( ! [N: nat] : ( ord_less_eq_nat @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_nat @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8837_monoseq__Suc,axiom,
    ( topolo4899668324122417113eq_int
    = ( ^ [X3: nat > int] :
          ( ! [N: nat] : ( ord_less_eq_int @ ( X3 @ N ) @ ( X3 @ ( suc @ N ) ) )
          | ! [N: nat] : ( ord_less_eq_int @ ( X3 @ ( suc @ N ) ) @ ( X3 @ N ) ) ) ) ) ).

% monoseq_Suc
thf(fact_8838_of__nat__code,axiom,
    ( semiri8010041392384452111omplex
    = ( ^ [N: nat] :
          ( semiri2816024913162550771omplex
          @ ^ [I3: complex] : ( plus_plus_complex @ I3 @ one_one_complex )
          @ N
          @ zero_zero_complex ) ) ) ).

% of_nat_code
thf(fact_8839_of__nat__code,axiom,
    ( semiri681578069525770553at_rat
    = ( ^ [N: nat] :
          ( semiri7787848453975740701ux_rat
          @ ^ [I3: rat] : ( plus_plus_rat @ I3 @ one_one_rat )
          @ N
          @ zero_zero_rat ) ) ) ).

% of_nat_code
thf(fact_8840_of__nat__code,axiom,
    ( semiri5074537144036343181t_real
    = ( ^ [N: nat] :
          ( semiri7260567687927622513x_real
          @ ^ [I3: real] : ( plus_plus_real @ I3 @ one_one_real )
          @ N
          @ zero_zero_real ) ) ) ).

% of_nat_code
thf(fact_8841_of__nat__code,axiom,
    ( semiri1314217659103216013at_int
    = ( ^ [N: nat] :
          ( semiri8420488043553186161ux_int
          @ ^ [I3: int] : ( plus_plus_int @ I3 @ one_one_int )
          @ N
          @ zero_zero_int ) ) ) ).

% of_nat_code
thf(fact_8842_of__nat__code,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] :
          ( semiri8422978514062236437ux_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ one_one_nat )
          @ N
          @ zero_zero_nat ) ) ) ).

% of_nat_code
thf(fact_8843_Arg__def,axiom,
    ( arg
    = ( ^ [Z5: complex] :
          ( if_real @ ( Z5 = zero_zero_complex ) @ zero_zero_real
          @ ( fChoice_real
            @ ^ [A3: real] :
                ( ( ( sgn_sgn_complex @ Z5 )
                  = ( cis @ A3 ) )
                & ( ord_less_real @ ( uminus_uminus_real @ pi ) @ A3 )
                & ( ord_less_eq_real @ A3 @ pi ) ) ) ) ) ) ).

% Arg_def
thf(fact_8844_set__vebt__def,axiom,
    ( vEBT_set_vebt
    = ( ^ [T: vEBT_VEBT] : ( collect_nat @ ( vEBT_V8194947554948674370ptions @ T ) ) ) ) ).

% set_vebt_def
thf(fact_8845_sin__x__sin__y,axiom,
    ! [X4: complex,Y: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_complex @ X4 @ N ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P5 @ N ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_complex @ ( sin_complex @ X4 ) @ ( sin_complex @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_8846_sin__x__sin__y,axiom,
    ! [X4: real,Y: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( uminus_uminus_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) ) @ ( power_power_real @ X4 @ N ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P5 @ N ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_real @ ( sin_real @ X4 ) @ ( sin_real @ Y ) ) ) ).

% sin_x_sin_y
thf(fact_8847_atMost__eq__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( set_ord_atMost_nat @ X4 )
        = ( set_ord_atMost_nat @ Y ) )
      = ( X4 = Y ) ) ).

% atMost_eq_iff
thf(fact_8848_atMost__eq__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ( set_ord_atMost_int @ X4 )
        = ( set_ord_atMost_int @ Y ) )
      = ( X4 = Y ) ) ).

% atMost_eq_iff
thf(fact_8849_atMost__iff,axiom,
    ! [I2: real,K: real] :
      ( ( member_real @ I2 @ ( set_ord_atMost_real @ K ) )
      = ( ord_less_eq_real @ I2 @ K ) ) ).

% atMost_iff
thf(fact_8850_atMost__iff,axiom,
    ! [I2: set_int,K: set_int] :
      ( ( member_set_int @ I2 @ ( set_or58775011639299419et_int @ K ) )
      = ( ord_less_eq_set_int @ I2 @ K ) ) ).

% atMost_iff
thf(fact_8851_atMost__iff,axiom,
    ! [I2: rat,K: rat] :
      ( ( member_rat @ I2 @ ( set_ord_atMost_rat @ K ) )
      = ( ord_less_eq_rat @ I2 @ K ) ) ).

% atMost_iff
thf(fact_8852_atMost__iff,axiom,
    ! [I2: num,K: num] :
      ( ( member_num @ I2 @ ( set_ord_atMost_num @ K ) )
      = ( ord_less_eq_num @ I2 @ K ) ) ).

% atMost_iff
thf(fact_8853_atMost__iff,axiom,
    ! [I2: nat,K: nat] :
      ( ( member_nat @ I2 @ ( set_ord_atMost_nat @ K ) )
      = ( ord_less_eq_nat @ I2 @ K ) ) ).

% atMost_iff
thf(fact_8854_atMost__iff,axiom,
    ! [I2: int,K: int] :
      ( ( member_int @ I2 @ ( set_ord_atMost_int @ K ) )
      = ( ord_less_eq_int @ I2 @ K ) ) ).

% atMost_iff
thf(fact_8855_atMost__subset__iff,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or58775011639299419et_int @ X4 ) @ ( set_or58775011639299419et_int @ Y ) )
      = ( ord_less_eq_set_int @ X4 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8856_atMost__subset__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ X4 ) @ ( set_ord_atMost_rat @ Y ) )
      = ( ord_less_eq_rat @ X4 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8857_atMost__subset__iff,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ X4 ) @ ( set_ord_atMost_num @ Y ) )
      = ( ord_less_eq_num @ X4 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8858_atMost__subset__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ X4 ) @ ( set_ord_atMost_nat @ Y ) )
      = ( ord_less_eq_nat @ X4 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8859_atMost__subset__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ X4 ) @ ( set_ord_atMost_int @ Y ) )
      = ( ord_less_eq_int @ X4 @ Y ) ) ).

% atMost_subset_iff
thf(fact_8860_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_8861_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_8862_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_8863_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_8864_norm__sum,axiom,
    ! [F: nat > complex,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_8865_norm__sum,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( real_V7735802525324610683m_real @ ( F @ I3 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_8866_norm__sum,axiom,
    ! [F: complex > complex,A2: set_complex] :
      ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ A2 ) )
      @ ( groups5808333547571424918x_real
        @ ^ [I3: complex] : ( real_V1022390504157884413omplex @ ( F @ I3 ) )
        @ A2 ) ) ).

% norm_sum
thf(fact_8867_sum__norm__le,axiom,
    ! [S2: set_real,F: real > complex,G: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups5754745047067104278omplex @ F @ S2 ) ) @ ( groups8097168146408367636l_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8868_sum__norm__le,axiom,
    ! [S2: set_int,F: int > complex,G: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups3049146728041665814omplex @ F @ S2 ) ) @ ( groups8778361861064173332t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8869_sum__norm__le,axiom,
    ! [S2: set_Pr1261947904930325089at_nat,F: product_prod_nat_nat > complex,G: product_prod_nat_nat > real] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups6381953495645901045omplex @ F @ S2 ) ) @ ( groups4567486121110086003t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8870_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > complex,G: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8871_sum__norm__le,axiom,
    ! [S2: set_nat,F: nat > real,G: nat > real] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ S2 ) ) @ ( groups6591440286371151544t_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8872_sum__norm__le,axiom,
    ! [S2: set_complex,F: complex > complex,G: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ S2 )
         => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( F @ X5 ) ) @ ( G @ X5 ) ) )
     => ( ord_less_eq_real @ ( real_V1022390504157884413omplex @ ( groups7754918857620584856omplex @ F @ S2 ) ) @ ( groups5808333547571424918x_real @ G @ S2 ) ) ) ).

% sum_norm_le
thf(fact_8873_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_8874_mod__sum__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups3542108847815614940at_nat @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_8875_mod__sum__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups4538972089207619220nt_int
          @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups4538972089207619220nt_int @ F @ A2 ) @ A ) ) ).

% mod_sum_eq
thf(fact_8876_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8877_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8878_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8879_sum_OatMost__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% sum.atMost_Suc_shift
thf(fact_8880_sum__telescope,axiom,
    ! [F: nat > rat,I2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_rat @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_8881_sum__telescope,axiom,
    ! [F: nat > int,I2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( minus_minus_int @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_int @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_8882_sum__telescope,axiom,
    ! [F: nat > real,I2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( minus_minus_real @ ( F @ I3 ) @ ( F @ ( suc @ I3 ) ) )
        @ ( set_ord_atMost_nat @ I2 ) )
      = ( minus_minus_real @ ( F @ zero_zero_nat ) @ ( F @ ( suc @ I2 ) ) ) ) ).

% sum_telescope
thf(fact_8883_polyfun__eq__coeffs,axiom,
    ! [C: nat > complex,N2: nat,D: nat > complex] :
      ( ( ! [X: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( D @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_8884_polyfun__eq__coeffs,axiom,
    ! [C: nat > real,N2: nat,D: nat > real] :
      ( ( ! [X: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( D @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = ( D @ I3 ) ) ) ) ) ).

% polyfun_eq_coeffs
thf(fact_8885_bounded__imp__summable,axiom,
    ! [A: nat > int,B3: int] :
      ( ! [N3: nat] : ( ord_less_eq_int @ zero_zero_int @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B3 )
       => ( summable_int @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8886_bounded__imp__summable,axiom,
    ! [A: nat > nat,B3: nat] :
      ( ! [N3: nat] : ( ord_less_eq_nat @ zero_zero_nat @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B3 )
       => ( summable_nat @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8887_bounded__imp__summable,axiom,
    ! [A: nat > real,B3: real] :
      ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ A @ ( set_ord_atMost_nat @ N3 ) ) @ B3 )
       => ( summable_real @ A ) ) ) ).

% bounded_imp_summable
thf(fact_8888_atMost__def,axiom,
    ( set_ord_atMost_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X: real] : ( ord_less_eq_real @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8889_atMost__def,axiom,
    ( set_or58775011639299419et_int
    = ( ^ [U2: set_int] :
          ( collect_set_int
          @ ^ [X: set_int] : ( ord_less_eq_set_int @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8890_atMost__def,axiom,
    ( set_ord_atMost_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X: rat] : ( ord_less_eq_rat @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8891_atMost__def,axiom,
    ( set_ord_atMost_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X: num] : ( ord_less_eq_num @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8892_atMost__def,axiom,
    ( set_ord_atMost_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X: nat] : ( ord_less_eq_nat @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8893_atMost__def,axiom,
    ( set_ord_atMost_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X: int] : ( ord_less_eq_int @ X @ U2 ) ) ) ) ).

% atMost_def
thf(fact_8894_sum__choose__lower,axiom,
    ! [R3: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( binomial @ ( plus_plus_nat @ R3 @ K3 ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( suc @ ( plus_plus_nat @ R3 @ N2 ) ) @ N2 ) ) ).

% sum_choose_lower
thf(fact_8895_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_8896_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_8897_polyfun__eq__0,axiom,
    ! [C: nat > complex,N2: nat] :
      ( ( ! [X: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_complex ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_0
thf(fact_8898_polyfun__eq__0,axiom,
    ! [C: nat > real,N2: nat] :
      ( ( ! [X: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = zero_zero_real ) )
      = ( ! [I3: nat] :
            ( ( ord_less_eq_nat @ I3 @ N2 )
           => ( ( C @ I3 )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_0
thf(fact_8899_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > complex,N2: nat,K: nat] :
      ( ! [W2: complex] :
          ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ W2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_complex ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_8900_zero__polynom__imp__zero__coeffs,axiom,
    ! [C: nat > real,N2: nat,K: nat] :
      ( ! [W2: real] :
          ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ W2 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real )
     => ( ( ord_less_eq_nat @ K @ N2 )
       => ( ( C @ K )
          = zero_zero_real ) ) ) ).

% zero_polynom_imp_zero_coeffs
thf(fact_8901_gbinomial__parallel__sum,axiom,
    ! [A: complex,N2: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( gbinomial_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_complex @ ( plus_plus_complex @ ( plus_plus_complex @ A @ ( semiri8010041392384452111omplex @ N2 ) ) @ one_one_complex ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_8902_gbinomial__parallel__sum,axiom,
    ! [A: rat,N2: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( gbinomial_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_rat @ ( plus_plus_rat @ ( plus_plus_rat @ A @ ( semiri681578069525770553at_rat @ N2 ) ) @ one_one_rat ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_8903_gbinomial__parallel__sum,axiom,
    ! [A: real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( gbinomial_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ K3 ) ) @ K3 )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( gbinomial_real @ ( plus_plus_real @ ( plus_plus_real @ A @ ( semiri5074537144036343181t_real @ N2 ) ) @ one_one_real ) @ N2 ) ) ).

% gbinomial_parallel_sum
thf(fact_8904_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_8905_vandermonde,axiom,
    ! [M: nat,N2: nat,R3: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( times_times_nat @ ( binomial @ M @ K3 ) @ ( binomial @ N2 @ ( minus_minus_nat @ R3 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ R3 ) )
      = ( binomial @ ( plus_plus_nat @ M @ N2 ) @ R3 ) ) ).

% vandermonde
thf(fact_8906_sum__gp__basic,axiom,
    ! [X4: rat,N2: nat] :
      ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X4 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X4 @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8907_sum__gp__basic,axiom,
    ! [X4: complex,N2: nat] :
      ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X4 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X4 @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8908_sum__gp__basic,axiom,
    ! [X4: int,N2: nat] :
      ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X4 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X4 ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_int @ one_one_int @ ( power_power_int @ X4 @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8909_sum__gp__basic,axiom,
    ! [X4: real,N2: nat] :
      ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X4 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_ord_atMost_nat @ N2 ) ) )
      = ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( suc @ N2 ) ) ) ) ).

% sum_gp_basic
thf(fact_8910_choose__row__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat @ ( binomial @ N2 ) @ ( set_ord_atMost_nat @ N2 ) )
      = ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ).

% choose_row_sum
thf(fact_8911_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_8912_sum_Oin__pairs__0,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8913_sum_Oin__pairs__0,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8914_sum_Oin__pairs__0,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8915_sum_Oin__pairs__0,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% sum.in_pairs_0
thf(fact_8916_polynomial__product,axiom,
    ! [M: nat,A: nat > rat,N2: nat,B: nat > rat,X4: rat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_rat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_rat ) )
       => ( ( times_times_rat
            @ ( groups2906978787729119204at_rat
              @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2906978787729119204at_rat
              @ ^ [J3: nat] : ( times_times_rat @ ( B @ J3 ) @ ( power_power_rat @ X4 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups2906978787729119204at_rat
            @ ^ [R5: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [K3: nat] : ( times_times_rat @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_rat @ X4 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8917_polynomial__product,axiom,
    ! [M: nat,A: nat > complex,N2: nat,B: nat > complex,X4: complex] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_complex ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_complex ) )
       => ( ( times_times_complex
            @ ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups2073611262835488442omplex
              @ ^ [J3: nat] : ( times_times_complex @ ( B @ J3 ) @ ( power_power_complex @ X4 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups2073611262835488442omplex
            @ ^ [R5: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [K3: nat] : ( times_times_complex @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_complex @ X4 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8918_polynomial__product,axiom,
    ! [M: nat,A: nat > int,N2: nat,B: nat > int,X4: int] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_int ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_int ) )
       => ( ( times_times_int
            @ ( groups3539618377306564664at_int
              @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3539618377306564664at_int
              @ ^ [J3: nat] : ( times_times_int @ ( B @ J3 ) @ ( power_power_int @ X4 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups3539618377306564664at_int
            @ ^ [R5: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [K3: nat] : ( times_times_int @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_int @ X4 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8919_polynomial__product,axiom,
    ! [M: nat,A: nat > real,N2: nat,B: nat > real,X4: real] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_real ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_real ) )
       => ( ( times_times_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups6591440286371151544t_real
              @ ^ [J3: nat] : ( times_times_real @ ( B @ J3 ) @ ( power_power_real @ X4 @ J3 ) )
              @ ( set_ord_atMost_nat @ N2 ) ) )
          = ( groups6591440286371151544t_real
            @ ^ [R5: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [K3: nat] : ( times_times_real @ ( A @ K3 ) @ ( B @ ( minus_minus_nat @ R5 @ K3 ) ) )
                  @ ( set_ord_atMost_nat @ R5 ) )
                @ ( power_power_real @ X4 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product
thf(fact_8920_gbinomial__sum__lower__neg,axiom,
    ! [A: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ A @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ M ) @ ( gbinomial_complex @ ( minus_minus_complex @ A @ one_one_complex ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8921_gbinomial__sum__lower__neg,axiom,
    ! [A: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ A @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ M ) @ ( gbinomial_rat @ ( minus_minus_rat @ A @ one_one_rat ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8922_gbinomial__sum__lower__neg,axiom,
    ! [A: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ A @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ M ) @ ( gbinomial_real @ ( minus_minus_real @ A @ one_one_real ) @ M ) ) ) ).

% gbinomial_sum_lower_neg
thf(fact_8923_binomial__ring,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( power_power_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K3 ) ) @ ( power_power_rat @ A @ K3 ) ) @ ( power_power_rat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8924_binomial__ring,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( power_power_complex @ ( plus_plus_complex @ A @ B ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K3 ) ) @ ( power_power_complex @ A @ K3 ) ) @ ( power_power_complex @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8925_binomial__ring,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( power_power_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K3 ) ) @ ( power_power_int @ A @ K3 ) ) @ ( power_power_int @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8926_binomial__ring,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_ring
thf(fact_8927_binomial__ring,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( power_power_real @ ( plus_plus_real @ A @ B ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K3 ) ) @ ( power_power_real @ A @ K3 ) ) @ ( power_power_real @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% binomial_ring
thf(fact_8928_polynomial__product__nat,axiom,
    ! [M: nat,A: nat > nat,N2: nat,B: nat > nat,X4: nat] :
      ( ! [I4: nat] :
          ( ( ord_less_nat @ M @ I4 )
         => ( ( A @ I4 )
            = zero_zero_nat ) )
     => ( ! [J2: nat] :
            ( ( ord_less_nat @ N2 @ J2 )
           => ( ( B @ J2 )
              = zero_zero_nat ) )
       => ( ( times_times_nat
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( power_power_nat @ X4 @ I3 ) )
              @ ( set_ord_atMost_nat @ M ) )
            @ ( groups3542108847815614940at_nat
              @ ^ [J3: nat] : ( times_times_nat @ ( B @ J3 ) @ ( power_power_nat @ X4 @ 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 @ X4 @ R5 ) )
            @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ) ).

% polynomial_product_nat
thf(fact_8929_choose__square__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [K3: nat] : ( power_power_nat @ ( binomial @ N2 @ K3 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( binomial @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ N2 ) ) ).

% choose_square_sum
thf(fact_8930_pochhammer__binomial__sum,axiom,
    ! [A: rat,B: rat,N2: nat] :
      ( ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ A @ B ) @ N2 )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ A @ K3 ) ) @ ( comm_s4028243227959126397er_rat @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8931_pochhammer__binomial__sum,axiom,
    ! [A: complex,B: complex,N2: nat] :
      ( ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ A @ B ) @ N2 )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ K3 ) ) @ ( comm_s2602460028002588243omplex @ A @ K3 ) ) @ ( comm_s2602460028002588243omplex @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8932_pochhammer__binomial__sum,axiom,
    ! [A: int,B: int,N2: nat] :
      ( ( comm_s4660882817536571857er_int @ ( plus_plus_int @ A @ B ) @ N2 )
      = ( groups3539618377306564664at_int
        @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ K3 ) ) @ ( comm_s4660882817536571857er_int @ A @ K3 ) ) @ ( comm_s4660882817536571857er_int @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8933_pochhammer__binomial__sum,axiom,
    ! [A: real,B: real,N2: nat] :
      ( ( comm_s7457072308508201937r_real @ ( plus_plus_real @ A @ B ) @ N2 )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ K3 ) ) @ ( comm_s7457072308508201937r_real @ A @ K3 ) ) @ ( comm_s7457072308508201937r_real @ B @ ( minus_minus_nat @ N2 @ K3 ) ) )
        @ ( set_ord_atMost_nat @ N2 ) ) ) ).

% pochhammer_binomial_sum
thf(fact_8934_sum__power__add,axiom,
    ! [X4: complex,M: nat,I5: set_nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( power_power_complex @ X4 @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8935_sum__power__add,axiom,
    ! [X4: int,M: nat,I5: set_nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( power_power_int @ X4 @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_int @ ( power_power_int @ X4 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X4 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8936_sum__power__add,axiom,
    ! [X4: real,M: nat,I5: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( power_power_real @ X4 @ ( plus_plus_nat @ M @ I3 ) )
        @ I5 )
      = ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ I5 ) ) ) ).

% sum_power_add
thf(fact_8937_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: nat > complex,H: nat > complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K @ P2 )
       => ( ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_complex @ ( J3 = K ) @ zero_zero_complex @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups2073611262835488442omplex
            @ ^ [J3: nat] : ( if_complex @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8938_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: nat > rat,H: nat > rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K @ P2 )
       => ( ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_rat @ ( J3 = K ) @ zero_zero_rat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups2906978787729119204at_rat
            @ ^ [J3: nat] : ( if_rat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8939_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: nat > int,H: nat > int] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K @ P2 )
       => ( ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_int @ ( J3 = K ) @ zero_zero_int @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups3539618377306564664at_int
            @ ^ [J3: nat] : ( if_int @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8940_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: nat > nat,H: nat > nat] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K @ P2 )
       => ( ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_nat @ ( J3 = K ) @ zero_zero_nat @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups3542108847815614940at_nat
            @ ^ [J3: nat] : ( if_nat @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8941_sum_Ozero__middle,axiom,
    ! [P2: nat,K: nat,G: nat > real,H: nat > real] :
      ( ( ord_less_eq_nat @ one_one_nat @ P2 )
     => ( ( ord_less_eq_nat @ K @ P2 )
       => ( ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( if_real @ ( J3 = K ) @ zero_zero_real @ ( H @ ( minus_minus_nat @ J3 @ ( suc @ zero_zero_nat ) ) ) ) )
            @ ( set_ord_atMost_nat @ P2 ) )
          = ( groups6591440286371151544t_real
            @ ^ [J3: nat] : ( if_real @ ( ord_less_nat @ J3 @ K ) @ ( G @ J3 ) @ ( H @ J3 ) )
            @ ( set_ord_atMost_nat @ ( minus_minus_nat @ P2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% sum.zero_middle
thf(fact_8942_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: complex,X4: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K3 ) @ ( power_power_complex @ X4 @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( uminus1482373934393186551omplex @ A ) @ K3 ) @ ( power_power_complex @ ( uminus1482373934393186551omplex @ X4 ) @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X4 @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8943_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: rat,X4: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X4 @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( uminus_uminus_rat @ A ) @ K3 ) @ ( power_power_rat @ ( uminus_uminus_rat @ X4 ) @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X4 @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8944_gbinomial__partial__sum__poly,axiom,
    ! [M: nat,A: real,X4: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K3 ) @ ( power_power_real @ X4 @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( uminus_uminus_real @ A ) @ K3 ) @ ( power_power_real @ ( uminus_uminus_real @ X4 ) @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X4 @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly
thf(fact_8945_exp__series__add__commuting,axiom,
    ! [X4: complex,Y: complex,N2: nat] :
      ( ( ( times_times_complex @ X4 @ Y )
        = ( times_times_complex @ Y @ X4 ) )
     => ( ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X4 @ Y ) @ N2 ) )
        = ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I3 ) ) @ ( power_power_complex @ X4 @ I3 ) ) @ ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ I3 ) ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ I3 ) ) ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% exp_series_add_commuting
thf(fact_8946_exp__series__add__commuting,axiom,
    ! [X4: real,Y: real,N2: nat] :
      ( ( ( times_times_real @ X4 @ Y )
        = ( times_times_real @ Y @ X4 ) )
     => ( ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( plus_plus_real @ X4 @ Y ) @ N2 ) )
        = ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ I3 ) ) @ ( power_power_real @ X4 @ I3 ) ) @ ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ I3 ) ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ I3 ) ) ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% exp_series_add_commuting
thf(fact_8947_root__polyfun,axiom,
    ! [N2: nat,Z: int,A: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_int @ Z @ N2 )
          = A )
        = ( ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( if_int @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_int @ A ) @ ( if_int @ ( I3 = N2 ) @ one_one_int @ zero_zero_int ) ) @ ( power_power_int @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_int ) ) ) ).

% root_polyfun
thf(fact_8948_root__polyfun,axiom,
    ! [N2: nat,Z: complex,A: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_complex @ Z @ N2 )
          = A )
        = ( ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( if_complex @ ( I3 = zero_zero_nat ) @ ( uminus1482373934393186551omplex @ A ) @ ( if_complex @ ( I3 = N2 ) @ one_one_complex @ zero_zero_complex ) ) @ ( power_power_complex @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_complex ) ) ) ).

% root_polyfun
thf(fact_8949_root__polyfun,axiom,
    ! [N2: nat,Z: code_integer,A: code_integer] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_8256067586552552935nteger @ Z @ N2 )
          = A )
        = ( ( groups7501900531339628137nteger
            @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( if_Code_integer @ ( I3 = zero_zero_nat ) @ ( uminus1351360451143612070nteger @ A ) @ ( if_Code_integer @ ( I3 = N2 ) @ one_one_Code_integer @ zero_z3403309356797280102nteger ) ) @ ( power_8256067586552552935nteger @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_z3403309356797280102nteger ) ) ) ).

% root_polyfun
thf(fact_8950_root__polyfun,axiom,
    ! [N2: nat,Z: rat,A: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_rat @ Z @ N2 )
          = A )
        = ( ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( if_rat @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_rat @ A ) @ ( if_rat @ ( I3 = N2 ) @ one_one_rat @ zero_zero_rat ) ) @ ( power_power_rat @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_rat ) ) ) ).

% root_polyfun
thf(fact_8951_root__polyfun,axiom,
    ! [N2: nat,Z: real,A: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( ( power_power_real @ Z @ N2 )
          = A )
        = ( ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( if_real @ ( I3 = zero_zero_nat ) @ ( uminus_uminus_real @ A ) @ ( if_real @ ( I3 = N2 ) @ one_one_real @ zero_zero_real ) ) @ ( power_power_real @ Z @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          = zero_zero_real ) ) ) ).

% root_polyfun
thf(fact_8952_sum__gp0,axiom,
    ! [X4: rat,N2: nat] :
      ( ( ( X4 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X4 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X4 @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X4 ) ) ) ) ) ).

% sum_gp0
thf(fact_8953_sum__gp0,axiom,
    ! [X4: complex,N2: nat] :
      ( ( ( X4 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X4 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X4 @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X4 ) ) ) ) ) ).

% sum_gp0
thf(fact_8954_sum__gp0,axiom,
    ! [X4: real,N2: nat] :
      ( ( ( X4 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ ( plus_plus_nat @ N2 @ one_one_nat ) ) ) )
      & ( ( X4 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_ord_atMost_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) ) ) ).

% sum_gp0
thf(fact_8955_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ I3 ) ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_linear_sum
thf(fact_8956_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ I3 ) ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_linear_sum
thf(fact_8957_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ I3 ) ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_linear_sum
thf(fact_8958_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ I3 ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_linear_sum
thf(fact_8959_choose__alternating__linear__sum,axiom,
    ! [N2: nat] :
      ( ( N2 != one_one_nat )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ I3 ) ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_linear_sum
thf(fact_8960_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( divide1717551699836669952omplex @ ( gbinomial_complex @ ( semiri8010041392384452111omplex @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8961_gbinomial__sum__nat__pow2,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( divide_divide_real @ ( gbinomial_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ M @ K3 ) ) @ K3 ) @ ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ K3 ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ M ) ) ).

% gbinomial_sum_nat_pow2
thf(fact_8962_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: rat,X4: rat,Y: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ M ) @ A ) @ K3 ) @ ( power_power_rat @ X4 @ K3 ) ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( gbinomial_rat @ ( minus_minus_rat @ ( plus_plus_rat @ ( semiri681578069525770553at_rat @ K3 ) @ A ) @ one_one_rat ) @ K3 ) @ ( power_power_rat @ X4 @ K3 ) ) @ ( power_power_rat @ ( plus_plus_rat @ X4 @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8963_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: complex,X4: complex,Y: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ M ) @ A ) @ K3 ) @ ( power_power_complex @ X4 @ K3 ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( gbinomial_complex @ ( minus_minus_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ K3 ) @ A ) @ one_one_complex ) @ K3 ) @ ( power_power_complex @ X4 @ K3 ) ) @ ( power_power_complex @ ( plus_plus_complex @ X4 @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8964_gbinomial__partial__sum__poly__xpos,axiom,
    ! [M: nat,A: real,X4: real,Y: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ M ) @ A ) @ K3 ) @ ( power_power_real @ X4 @ K3 ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( gbinomial_real @ ( minus_minus_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ K3 ) @ A ) @ one_one_real ) @ K3 ) @ ( power_power_real @ X4 @ K3 ) ) @ ( power_power_real @ ( plus_plus_real @ X4 @ Y ) @ ( minus_minus_nat @ M @ K3 ) ) )
        @ ( set_ord_atMost_nat @ M ) ) ) ).

% gbinomial_partial_sum_poly_xpos
thf(fact_8965_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_8966_choose__linear__sum,axiom,
    ! [N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( times_times_nat @ I3 @ ( binomial @ N2 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( times_times_nat @ N2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) ) ).

% choose_linear_sum
thf(fact_8967_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ ( uminus1482373934393186551omplex @ one_one_complex ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex ) ) ).

% choose_alternating_sum
thf(fact_8968_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups7501900531339628137nteger
          @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( power_8256067586552552935nteger @ ( uminus1351360451143612070nteger @ one_one_Code_integer ) @ I3 ) @ ( semiri4939895301339042750nteger @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_z3403309356797280102nteger ) ) ).

% choose_alternating_sum
thf(fact_8969_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ ( uminus_uminus_rat @ one_one_rat ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat ) ) ).

% choose_alternating_sum
thf(fact_8970_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int ) ) ).

% choose_alternating_sum
thf(fact_8971_choose__alternating__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real ) ) ).

% choose_alternating_sum
thf(fact_8972_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: nat > complex,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ? [M9: real] :
        ! [Z3: complex] :
          ( ( ord_less_eq_real @ M9 @ ( real_V1022390504157884413omplex @ Z3 ) )
         => ( ord_less_eq_real
            @ ( real_V1022390504157884413omplex
              @ ( groups2073611262835488442omplex
                @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z3 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_8973_polyfun__extremal__lemma,axiom,
    ! [E2: real,C: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ E2 )
     => ? [M9: real] :
        ! [Z3: real] :
          ( ( ord_less_eq_real @ M9 @ ( real_V7735802525324610683m_real @ Z3 ) )
         => ( ord_less_eq_real
            @ ( real_V7735802525324610683m_real
              @ ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
                @ ( set_ord_atMost_nat @ N2 ) ) )
            @ ( times_times_real @ E2 @ ( power_power_real @ ( real_V7735802525324610683m_real @ Z3 ) @ ( suc @ N2 ) ) ) ) ) ) ).

% polyfun_extremal_lemma
thf(fact_8974_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2906978787729119204at_rat @ ( gbinomial_rat @ ( plus_plus_rat @ ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( semiri681578069525770553at_rat @ M ) ) @ one_one_rat ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_8975_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups2073611262835488442omplex @ ( gbinomial_complex @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( semiri8010041392384452111omplex @ M ) ) @ one_one_complex ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_8976_gbinomial__r__part__sum,axiom,
    ! [M: nat] :
      ( ( groups6591440286371151544t_real @ ( gbinomial_real @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( semiri5074537144036343181t_real @ M ) ) @ one_one_real ) ) @ ( set_ord_atMost_nat @ M ) )
      = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M ) ) ) ).

% gbinomial_r_part_sum
thf(fact_8977_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] :
                ( if_rat
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8978_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] :
                ( if_complex
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8979_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] :
                ( if_int
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8980_choose__odd__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] :
                ( if_real
                @ ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 )
                @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) )
                @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_odd_sum
thf(fact_8981_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( if_rat @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri681578069525770553at_rat @ ( binomial @ N2 @ I3 ) ) @ zero_zero_rat )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8982_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri8010041392384452111omplex @ ( binomial @ N2 @ I3 ) ) @ zero_zero_complex )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8983_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( if_int @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri1314217659103216013at_int @ ( binomial @ N2 @ I3 ) ) @ zero_zero_int )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8984_choose__even__sum,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( semiri5074537144036343181t_real @ ( binomial @ N2 @ I3 ) ) @ zero_zero_real )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( power_power_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ N2 ) ) ) ).

% choose_even_sum
thf(fact_8985_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_8986_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_8987_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_8988_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_8989_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_8990_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_8991_cos__x__cos__y,axiom,
    ! [X4: complex,Y: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] :
              ( if_complex
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X4 @ N ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P5 @ N ) ) )
              @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_complex @ ( cos_complex @ X4 ) @ ( cos_complex @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_8992_cos__x__cos__y,axiom,
    ! [X4: real,Y: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] :
              ( if_real
              @ ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 )
                & ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) )
              @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X4 @ N ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P5 @ N ) ) )
              @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( times_times_real @ ( cos_real @ X4 ) @ ( cos_real @ Y ) ) ) ).

% cos_x_cos_y
thf(fact_8993_sums__cos__x__plus__y,axiom,
    ! [X4: complex,Y: complex] :
      ( sums_complex
      @ ^ [P5: nat] :
          ( groups2073611262835488442omplex
          @ ^ [N: nat] : ( if_complex @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_complex @ ( real_V2046097035970521341omplex @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_complex @ X4 @ N ) ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ P5 @ N ) ) ) @ zero_zero_complex )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( cos_complex @ ( plus_plus_complex @ X4 @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8994_sums__cos__x__plus__y,axiom,
    ! [X4: real,Y: real] :
      ( sums_real
      @ ^ [P5: nat] :
          ( groups6591440286371151544t_real
          @ ^ [N: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ P5 ) @ ( times_times_real @ ( real_V1485227260804924795R_real @ ( divide_divide_real @ ( ring_1_of_int_real @ ( times_times_int @ ( power_power_int @ ( uminus_uminus_int @ one_one_int ) @ ( divide_divide_nat @ P5 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) @ ( semiri1314217659103216013at_int @ ( binomial @ P5 @ N ) ) ) ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ X4 @ N ) ) @ ( power_power_real @ Y @ ( minus_minus_nat @ P5 @ N ) ) ) @ zero_zero_real )
          @ ( set_ord_atMost_nat @ P5 ) )
      @ ( cos_real @ ( plus_plus_real @ X4 @ Y ) ) ) ).

% sums_cos_x_plus_y
thf(fact_8995_sum__abs__ge__zero,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ zero_zero_real
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_8996_sum__abs__ge__zero,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ zero_zero_int
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs_ge_zero
thf(fact_8997_sum__abs,axiom,
    ! [F: nat > real,A2: set_nat] :
      ( ord_less_eq_real @ ( abs_abs_real @ ( groups6591440286371151544t_real @ F @ A2 ) )
      @ ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( abs_abs_real @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_8998_sum__abs,axiom,
    ! [F: int > int,A2: set_int] :
      ( ord_less_eq_int @ ( abs_abs_int @ ( groups4538972089207619220nt_int @ F @ A2 ) )
      @ ( groups4538972089207619220nt_int
        @ ^ [I3: int] : ( abs_abs_int @ ( F @ I3 ) )
        @ A2 ) ) ).

% sum_abs
thf(fact_8999_convex__sum__bound__le,axiom,
    ! [I5: set_real,X4: real > code_integer,A: real > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X4 @ I4 ) ) )
     => ( ( ( groups7713935264441627589nteger @ X4 @ I5 )
          = one_one_Code_integer )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7713935264441627589nteger
                  @ ^ [I3: real] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9000_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X4: nat > code_integer,A: nat > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X4 @ I4 ) ) )
     => ( ( ( groups7501900531339628137nteger @ X4 @ I5 )
          = one_one_Code_integer )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7501900531339628137nteger
                  @ ^ [I3: nat] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9001_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X4: complex > code_integer,A: complex > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X4 @ I4 ) ) )
     => ( ( ( groups6621422865394947399nteger @ X4 @ I5 )
          = one_one_Code_integer )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups6621422865394947399nteger
                  @ ^ [I3: complex] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9002_convex__sum__bound__le,axiom,
    ! [I5: set_int,X4: int > code_integer,A: int > code_integer,B: code_integer,Delta: code_integer] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I5 )
         => ( ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ ( X4 @ I4 ) ) )
     => ( ( ( groups7873554091576472773nteger @ X4 @ I5 )
          = one_one_Code_integer )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I5 )
             => ( ord_le3102999989581377725nteger @ ( abs_abs_Code_integer @ ( minus_8373710615458151222nteger @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_le3102999989581377725nteger
            @ ( abs_abs_Code_integer
              @ ( minus_8373710615458151222nteger
                @ ( groups7873554091576472773nteger
                  @ ^ [I3: int] : ( times_3573771949741848930nteger @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9003_convex__sum__bound__le,axiom,
    ! [I5: set_real,X4: real > real,A: real > real,B: real,Delta: real] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I4 ) ) )
     => ( ( ( groups8097168146408367636l_real @ X4 @ I5 )
          = one_one_real )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8097168146408367636l_real
                  @ ^ [I3: real] : ( times_times_real @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9004_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X4: complex > real,A: complex > real,B: real,Delta: real] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I4 ) ) )
     => ( ( ( groups5808333547571424918x_real @ X4 @ I5 )
          = one_one_real )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups5808333547571424918x_real
                  @ ^ [I3: complex] : ( times_times_real @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9005_convex__sum__bound__le,axiom,
    ! [I5: set_int,X4: int > real,A: int > real,B: real,Delta: real] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ I5 )
         => ( ord_less_eq_real @ zero_zero_real @ ( X4 @ I4 ) ) )
     => ( ( ( groups8778361861064173332t_real @ X4 @ I5 )
          = one_one_real )
       => ( ! [I4: int] :
              ( ( member_int @ I4 @ I5 )
             => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_real
            @ ( abs_abs_real
              @ ( minus_minus_real
                @ ( groups8778361861064173332t_real
                  @ ^ [I3: int] : ( times_times_real @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9006_convex__sum__bound__le,axiom,
    ! [I5: set_real,X4: real > rat,A: real > rat,B: rat,Delta: rat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X4 @ I4 ) ) )
     => ( ( ( groups1300246762558778688al_rat @ X4 @ I5 )
          = one_one_rat )
       => ( ! [I4: real] :
              ( ( member_real @ I4 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups1300246762558778688al_rat
                  @ ^ [I3: real] : ( times_times_rat @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9007_convex__sum__bound__le,axiom,
    ! [I5: set_nat,X4: nat > rat,A: nat > rat,B: rat,Delta: rat] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X4 @ I4 ) ) )
     => ( ( ( groups2906978787729119204at_rat @ X4 @ I5 )
          = one_one_rat )
       => ( ! [I4: nat] :
              ( ( member_nat @ I4 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9008_convex__sum__bound__le,axiom,
    ! [I5: set_complex,X4: complex > rat,A: complex > rat,B: rat,Delta: rat] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ I5 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( X4 @ I4 ) ) )
     => ( ( ( groups5058264527183730370ex_rat @ X4 @ I5 )
          = one_one_rat )
       => ( ! [I4: complex] :
              ( ( member_complex @ I4 @ I5 )
             => ( ord_less_eq_rat @ ( abs_abs_rat @ ( minus_minus_rat @ ( A @ I4 ) @ B ) ) @ Delta ) )
         => ( ord_less_eq_rat
            @ ( abs_abs_rat
              @ ( minus_minus_rat
                @ ( groups5058264527183730370ex_rat
                  @ ^ [I3: complex] : ( times_times_rat @ ( A @ I3 ) @ ( X4 @ I3 ) )
                  @ I5 )
                @ B ) )
            @ Delta ) ) ) ) ).

% convex_sum_bound_le
thf(fact_9009_Maclaurin__minus__cos__expansion,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X4 @ zero_zero_real )
       => ? [T3: real] :
            ( ( ord_less_real @ X4 @ T3 )
            & ( ord_less_real @ T3 @ zero_zero_real )
            & ( ( cos_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( 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 @ X4 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_minus_cos_expansion
thf(fact_9010_lessThan__eq__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ( set_ord_lessThan_nat @ X4 )
        = ( set_ord_lessThan_nat @ Y ) )
      = ( X4 = Y ) ) ).

% lessThan_eq_iff
thf(fact_9011_lessThan__eq__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ( set_ord_lessThan_int @ X4 )
        = ( set_ord_lessThan_int @ Y ) )
      = ( X4 = Y ) ) ).

% lessThan_eq_iff
thf(fact_9012_lessThan__eq__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ( set_or5984915006950818249n_real @ X4 )
        = ( set_or5984915006950818249n_real @ Y ) )
      = ( X4 = Y ) ) ).

% lessThan_eq_iff
thf(fact_9013_of__nat__id,axiom,
    ( semiri1316708129612266289at_nat
    = ( ^ [N: nat] : N ) ) ).

% of_nat_id
thf(fact_9014_lessThan__iff,axiom,
    ! [I2: rat,K: rat] :
      ( ( member_rat @ I2 @ ( set_ord_lessThan_rat @ K ) )
      = ( ord_less_rat @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_9015_lessThan__iff,axiom,
    ! [I2: num,K: num] :
      ( ( member_num @ I2 @ ( set_ord_lessThan_num @ K ) )
      = ( ord_less_num @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_9016_lessThan__iff,axiom,
    ! [I2: nat,K: nat] :
      ( ( member_nat @ I2 @ ( set_ord_lessThan_nat @ K ) )
      = ( ord_less_nat @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_9017_lessThan__iff,axiom,
    ! [I2: int,K: int] :
      ( ( member_int @ I2 @ ( set_ord_lessThan_int @ K ) )
      = ( ord_less_int @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_9018_lessThan__iff,axiom,
    ! [I2: real,K: real] :
      ( ( member_real @ I2 @ ( set_or5984915006950818249n_real @ K ) )
      = ( ord_less_real @ I2 @ K ) ) ).

% lessThan_iff
thf(fact_9019_lessThan__subset__iff,axiom,
    ! [X4: rat,Y: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_lessThan_rat @ X4 ) @ ( set_ord_lessThan_rat @ Y ) )
      = ( ord_less_eq_rat @ X4 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_9020_lessThan__subset__iff,axiom,
    ! [X4: num,Y: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_lessThan_num @ X4 ) @ ( set_ord_lessThan_num @ Y ) )
      = ( ord_less_eq_num @ X4 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_9021_lessThan__subset__iff,axiom,
    ! [X4: nat,Y: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_lessThan_nat @ X4 ) @ ( set_ord_lessThan_nat @ Y ) )
      = ( ord_less_eq_nat @ X4 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_9022_lessThan__subset__iff,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_lessThan_int @ X4 ) @ ( set_ord_lessThan_int @ Y ) )
      = ( ord_less_eq_int @ X4 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_9023_lessThan__subset__iff,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_eq_set_real @ ( set_or5984915006950818249n_real @ X4 ) @ ( set_or5984915006950818249n_real @ Y ) )
      = ( ord_less_eq_real @ X4 @ Y ) ) ).

% lessThan_subset_iff
thf(fact_9024_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_9025_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_9026_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_9027_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_9028_sumr__cos__zero__one,axiom,
    ! [N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ zero_zero_real @ M6 ) )
        @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = one_one_real ) ).

% sumr_cos_zero_one
thf(fact_9029_sum__diff__distrib,axiom,
    ! [Q: int > nat,P: int > nat,N2: int] :
      ( ! [X5: int] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups4541462559716669496nt_nat @ P @ ( set_ord_lessThan_int @ N2 ) ) @ ( groups4541462559716669496nt_nat @ Q @ ( set_ord_lessThan_int @ N2 ) ) )
        = ( groups4541462559716669496nt_nat
          @ ^ [X: int] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_int @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_9030_sum__diff__distrib,axiom,
    ! [Q: real > nat,P: real > nat,N2: real] :
      ( ! [X5: real] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( 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_9031_sum__diff__distrib,axiom,
    ! [Q: nat > nat,P: nat > nat,N2: nat] :
      ( ! [X5: nat] : ( ord_less_eq_nat @ ( Q @ X5 ) @ ( P @ X5 ) )
     => ( ( minus_minus_nat @ ( groups3542108847815614940at_nat @ P @ ( set_ord_lessThan_nat @ N2 ) ) @ ( groups3542108847815614940at_nat @ Q @ ( set_ord_lessThan_nat @ N2 ) ) )
        = ( groups3542108847815614940at_nat
          @ ^ [X: nat] : ( minus_minus_nat @ ( P @ X ) @ ( Q @ X ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_diff_distrib
thf(fact_9032_lessThan__def,axiom,
    ( set_ord_lessThan_rat
    = ( ^ [U2: rat] :
          ( collect_rat
          @ ^ [X: rat] : ( ord_less_rat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_9033_lessThan__def,axiom,
    ( set_ord_lessThan_num
    = ( ^ [U2: num] :
          ( collect_num
          @ ^ [X: num] : ( ord_less_num @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_9034_lessThan__def,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [U2: nat] :
          ( collect_nat
          @ ^ [X: nat] : ( ord_less_nat @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_9035_lessThan__def,axiom,
    ( set_ord_lessThan_int
    = ( ^ [U2: int] :
          ( collect_int
          @ ^ [X: int] : ( ord_less_int @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_9036_lessThan__def,axiom,
    ( set_or5984915006950818249n_real
    = ( ^ [U2: real] :
          ( collect_real
          @ ^ [X: real] : ( ord_less_real @ X @ U2 ) ) ) ) ).

% lessThan_def
thf(fact_9037_sum__subtractf__nat,axiom,
    ! [A2: set_real,G: real > nat,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups1935376822645274424al_nat
          @ ^ [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_9038_sum__subtractf__nat,axiom,
    ! [A2: set_complex,G: complex > nat,F: complex > nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups5693394587270226106ex_nat
          @ ^ [X: complex] : ( minus_minus_nat @ ( F @ X ) @ ( G @ X ) )
          @ A2 )
        = ( minus_minus_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ ( groups5693394587270226106ex_nat @ G @ A2 ) ) ) ) ).

% sum_subtractf_nat
thf(fact_9039_sum__subtractf__nat,axiom,
    ! [A2: set_int,G: int > nat,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups4541462559716669496nt_nat
          @ ^ [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_9040_sum__subtractf__nat,axiom,
    ! [A2: set_Pr1261947904930325089at_nat,G: product_prod_nat_nat > nat,F: product_prod_nat_nat > nat] :
      ( ! [X5: product_prod_nat_nat] :
          ( ( member8440522571783428010at_nat @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( 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_9041_sum__subtractf__nat,axiom,
    ! [A2: set_nat,G: nat > nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( G @ X5 ) @ ( F @ X5 ) ) )
     => ( ( groups3542108847815614940at_nat
          @ ^ [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_9042_sum__SucD,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( ( groups3542108847815614940at_nat @ F @ A2 )
        = ( suc @ N2 ) )
     => ? [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
          & ( ord_less_nat @ zero_zero_nat @ ( F @ X5 ) ) ) ) ).

% sum_SucD
thf(fact_9043_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_9044_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_9045_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_9046_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_9047_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_9048_lessThan__Suc__atMost,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( set_ord_atMost_nat @ K ) ) ).

% lessThan_Suc_atMost
thf(fact_9049_Iic__subset__Iio__iff,axiom,
    ! [A: rat,B: rat] :
      ( ( ord_less_eq_set_rat @ ( set_ord_atMost_rat @ A ) @ ( set_ord_lessThan_rat @ B ) )
      = ( ord_less_rat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9050_Iic__subset__Iio__iff,axiom,
    ! [A: num,B: num] :
      ( ( ord_less_eq_set_num @ ( set_ord_atMost_num @ A ) @ ( set_ord_lessThan_num @ B ) )
      = ( ord_less_num @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9051_Iic__subset__Iio__iff,axiom,
    ! [A: nat,B: nat] :
      ( ( ord_less_eq_set_nat @ ( set_ord_atMost_nat @ A ) @ ( set_ord_lessThan_nat @ B ) )
      = ( ord_less_nat @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9052_Iic__subset__Iio__iff,axiom,
    ! [A: int,B: int] :
      ( ( ord_less_eq_set_int @ ( set_ord_atMost_int @ A ) @ ( set_ord_lessThan_int @ B ) )
      = ( ord_less_int @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9053_Iic__subset__Iio__iff,axiom,
    ! [A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( set_ord_atMost_real @ A ) @ ( set_or5984915006950818249n_real @ B ) )
      = ( ord_less_real @ A @ B ) ) ).

% Iic_subset_Iio_iff
thf(fact_9054_sum_Onat__diff__reindex,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_9055_sum_Onat__diff__reindex,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
        @ ( set_ord_lessThan_nat @ N2 ) )
      = ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nat_diff_reindex
thf(fact_9056_suminf__le__const,axiom,
    ! [F: nat > int,X4: int] :
      ( ( summable_int @ F )
     => ( ! [N3: nat] : ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X4 )
       => ( ord_less_eq_int @ ( suminf_int @ F ) @ X4 ) ) ) ).

% suminf_le_const
thf(fact_9057_suminf__le__const,axiom,
    ! [F: nat > nat,X4: nat] :
      ( ( summable_nat @ F )
     => ( ! [N3: nat] : ( ord_less_eq_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X4 )
       => ( ord_less_eq_nat @ ( suminf_nat @ F ) @ X4 ) ) ) ).

% suminf_le_const
thf(fact_9058_suminf__le__const,axiom,
    ! [F: nat > real,X4: real] :
      ( ( summable_real @ F )
     => ( ! [N3: nat] : ( ord_less_eq_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N3 ) ) @ X4 )
       => ( ord_less_eq_real @ ( suminf_real @ F ) @ X4 ) ) ) ).

% suminf_le_const
thf(fact_9059_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
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_9060_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_9061_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_9062_sum_OlessThan__Suc__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.lessThan_Suc_shift
thf(fact_9063_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_9064_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_9065_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_9066_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_9067_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_9068_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_9069_summableI__nonneg__bounded,axiom,
    ! [F: nat > int,X4: 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 ) ) @ X4 )
       => ( summable_int @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_9070_summableI__nonneg__bounded,axiom,
    ! [F: nat > nat,X4: 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 ) ) @ X4 )
       => ( summable_nat @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_9071_summableI__nonneg__bounded,axiom,
    ! [F: nat > real,X4: 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 ) ) @ X4 )
       => ( summable_real @ F ) ) ) ).

% summableI_nonneg_bounded
thf(fact_9072_sums__iff__shift,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ S )
      = ( sums_real @ F @ ( plus_plus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_iff_shift
thf(fact_9073_sums__iff__shift_H,axiom,
    ! [F: nat > real,N2: nat,S: real] :
      ( ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      = ( sums_real @ F @ S ) ) ).

% sums_iff_shift'
thf(fact_9074_sums__split__initial__segment,axiom,
    ! [F: nat > real,S: real,N2: nat] :
      ( ( sums_real @ F @ S )
     => ( sums_real
        @ ^ [I3: nat] : ( F @ ( plus_plus_nat @ I3 @ N2 ) )
        @ ( minus_minus_real @ S @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% sums_split_initial_segment
thf(fact_9075_sum__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = C ) ) )
        = zero_zero_complex ) ) ).

% sum_nth_roots
thf(fact_9076_power__diff__1__eq,axiom,
    ! [X4: rat,N2: nat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X4 @ N2 ) @ one_one_rat )
      = ( times_times_rat @ ( minus_minus_rat @ X4 @ one_one_rat ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_9077_power__diff__1__eq,axiom,
    ! [X4: complex,N2: nat] :
      ( ( minus_minus_complex @ ( power_power_complex @ X4 @ N2 ) @ one_one_complex )
      = ( times_times_complex @ ( minus_minus_complex @ X4 @ one_one_complex ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_9078_power__diff__1__eq,axiom,
    ! [X4: int,N2: nat] :
      ( ( minus_minus_int @ ( power_power_int @ X4 @ N2 ) @ one_one_int )
      = ( times_times_int @ ( minus_minus_int @ X4 @ one_one_int ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_9079_power__diff__1__eq,axiom,
    ! [X4: real,N2: nat] :
      ( ( minus_minus_real @ ( power_power_real @ X4 @ N2 ) @ one_one_real )
      = ( times_times_real @ ( minus_minus_real @ X4 @ one_one_real ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_1_eq
thf(fact_9080_one__diff__power__eq,axiom,
    ! [X4: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X4 @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X4 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_9081_one__diff__power__eq,axiom,
    ! [X4: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X4 @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X4 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_9082_one__diff__power__eq,axiom,
    ! [X4: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X4 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X4 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_9083_one__diff__power__eq,axiom,
    ! [X4: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X4 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq
thf(fact_9084_geometric__sum,axiom,
    ! [X4: rat,N2: nat] :
      ( ( X4 != one_one_rat )
     => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X4 @ N2 ) @ one_one_rat ) @ ( minus_minus_rat @ X4 @ one_one_rat ) ) ) ) ).

% geometric_sum
thf(fact_9085_geometric__sum,axiom,
    ! [X4: complex,N2: nat] :
      ( ( X4 != one_one_complex )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X4 @ N2 ) @ one_one_complex ) @ ( minus_minus_complex @ X4 @ one_one_complex ) ) ) ) ).

% geometric_sum
thf(fact_9086_geometric__sum,axiom,
    ! [X4: real,N2: nat] :
      ( ( X4 != one_one_real )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) )
        = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X4 @ N2 ) @ one_one_real ) @ ( minus_minus_real @ X4 @ one_one_real ) ) ) ) ).

% geometric_sum
thf(fact_9087_sum_OatMost__shift,axiom,
    ! [G: nat > rat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_rat @ ( G @ zero_zero_nat )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_9088_sum_OatMost__shift,axiom,
    ! [G: nat > int,N2: nat] :
      ( ( groups3539618377306564664at_int @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int @ ( G @ zero_zero_nat )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_9089_sum_OatMost__shift,axiom,
    ! [G: nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_nat @ ( G @ zero_zero_nat )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_9090_sum_OatMost__shift,axiom,
    ! [G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real @ ( G @ zero_zero_nat )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum.atMost_shift
thf(fact_9091_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_9092_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_9093_sum__roots__unity,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ one_one_nat @ N2 )
     => ( ( groups7754918857620584856omplex
          @ ^ [X: complex] : X
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = one_one_complex ) ) )
        = zero_zero_complex ) ) ).

% sum_roots_unity
thf(fact_9094_sum__less__suminf,axiom,
    ! [F: nat > int,N2: nat] :
      ( ( summable_int @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_int @ zero_zero_int @ ( F @ M5 ) ) )
       => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_9095_sum__less__suminf,axiom,
    ! [F: nat > nat,N2: nat] :
      ( ( summable_nat @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_nat @ zero_zero_nat @ ( F @ M5 ) ) )
       => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_9096_sum__less__suminf,axiom,
    ! [F: nat > real,N2: nat] :
      ( ( summable_real @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_real @ zero_zero_real @ ( F @ M5 ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_less_suminf
thf(fact_9097_sum__gp__strict,axiom,
    ! [X4: rat,N2: nat] :
      ( ( ( X4 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri681578069525770553at_rat @ N2 ) ) )
      & ( ( X4 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_rat @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X4 @ N2 ) ) @ ( minus_minus_rat @ one_one_rat @ X4 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_9098_sum__gp__strict,axiom,
    ! [X4: complex,N2: nat] :
      ( ( ( X4 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri8010041392384452111omplex @ N2 ) ) )
      & ( ( X4 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide1717551699836669952omplex @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X4 @ N2 ) ) @ ( minus_minus_complex @ one_one_complex @ X4 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_9099_sum__gp__strict,axiom,
    ! [X4: real,N2: nat] :
      ( ( ( X4 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( semiri5074537144036343181t_real @ N2 ) ) )
      & ( ( X4 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_ord_lessThan_nat @ N2 ) )
          = ( divide_divide_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ N2 ) ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) ) ) ).

% sum_gp_strict
thf(fact_9100_lemma__termdiff1,axiom,
    ! [Z: rat,H: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( minus_minus_rat @ ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ P5 ) ) @ ( power_power_rat @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2906978787729119204at_rat
        @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ Z @ P5 ) @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_9101_lemma__termdiff1,axiom,
    ! [Z: complex,H: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( minus_minus_complex @ ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ P5 ) ) @ ( power_power_complex @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups2073611262835488442omplex
        @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ Z @ P5 ) @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_9102_lemma__termdiff1,axiom,
    ! [Z: int,H: int,M: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( minus_minus_int @ ( times_times_int @ ( power_power_int @ ( plus_plus_int @ Z @ H ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ P5 ) ) @ ( power_power_int @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups3539618377306564664at_int
        @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ Z @ P5 ) @ ( minus_minus_int @ ( power_power_int @ ( plus_plus_int @ Z @ H ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_int @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_9103_lemma__termdiff1,axiom,
    ! [Z: real,H: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( minus_minus_real @ ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ P5 ) ) @ ( power_power_real @ Z @ M ) )
        @ ( set_ord_lessThan_nat @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ Z @ P5 ) @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ ( minus_minus_nat @ M @ P5 ) ) @ ( power_power_real @ Z @ ( minus_minus_nat @ M @ P5 ) ) ) )
        @ ( set_ord_lessThan_nat @ M ) ) ) ).

% lemma_termdiff1
thf(fact_9104_diff__power__eq__sum,axiom,
    ! [X4: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X4 @ ( suc @ N2 ) ) @ ( power_power_rat @ Y @ ( suc @ N2 ) ) )
      = ( times_times_rat @ ( minus_minus_rat @ X4 @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [P5: nat] : ( times_times_rat @ ( power_power_rat @ X4 @ P5 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_9105_diff__power__eq__sum,axiom,
    ! [X4: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X4 @ ( suc @ N2 ) ) @ ( power_power_complex @ Y @ ( suc @ N2 ) ) )
      = ( times_times_complex @ ( minus_minus_complex @ X4 @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [P5: nat] : ( times_times_complex @ ( power_power_complex @ X4 @ P5 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_9106_diff__power__eq__sum,axiom,
    ! [X4: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X4 @ ( suc @ N2 ) ) @ ( power_power_int @ Y @ ( suc @ N2 ) ) )
      = ( times_times_int @ ( minus_minus_int @ X4 @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [P5: nat] : ( times_times_int @ ( power_power_int @ X4 @ P5 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_9107_diff__power__eq__sum,axiom,
    ! [X4: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X4 @ ( suc @ N2 ) ) @ ( power_power_real @ Y @ ( suc @ N2 ) ) )
      = ( times_times_real @ ( minus_minus_real @ X4 @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [P5: nat] : ( times_times_real @ ( power_power_real @ X4 @ P5 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ P5 ) ) )
          @ ( set_ord_lessThan_nat @ ( suc @ N2 ) ) ) ) ) ).

% diff_power_eq_sum
thf(fact_9108_power__diff__sumr2,axiom,
    ! [X4: rat,N2: nat,Y: rat] :
      ( ( minus_minus_rat @ ( power_power_rat @ X4 @ N2 ) @ ( power_power_rat @ Y @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ X4 @ Y )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( power_power_rat @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_rat @ X4 @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_9109_power__diff__sumr2,axiom,
    ! [X4: complex,N2: nat,Y: complex] :
      ( ( minus_minus_complex @ ( power_power_complex @ X4 @ N2 ) @ ( power_power_complex @ Y @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ X4 @ Y )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( power_power_complex @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_complex @ X4 @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_9110_power__diff__sumr2,axiom,
    ! [X4: int,N2: nat,Y: int] :
      ( ( minus_minus_int @ ( power_power_int @ X4 @ N2 ) @ ( power_power_int @ Y @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ X4 @ Y )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( power_power_int @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_int @ X4 @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_9111_power__diff__sumr2,axiom,
    ! [X4: real,N2: nat,Y: real] :
      ( ( minus_minus_real @ ( power_power_real @ X4 @ N2 ) @ ( power_power_real @ Y @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ X4 @ Y )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ Y @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) ) @ ( power_power_real @ X4 @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% power_diff_sumr2
thf(fact_9112_polyfun__linear__factor__root,axiom,
    ! [C: nat > rat,A: rat,N2: nat] :
      ( ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_rat )
     => ~ ! [B5: nat > rat] :
            ~ ! [Z3: rat] :
                ( ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
                  @ ( groups2906978787729119204at_rat
                    @ ^ [I3: nat] : ( times_times_rat @ ( B5 @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_9113_polyfun__linear__factor__root,axiom,
    ! [C: nat > complex,A: complex,N2: nat] :
      ( ( ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_complex )
     => ~ ! [B5: nat > complex] :
            ~ ! [Z3: complex] :
                ( ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
                  @ ( groups2073611262835488442omplex
                    @ ^ [I3: nat] : ( times_times_complex @ ( B5 @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_9114_polyfun__linear__factor__root,axiom,
    ! [C: nat > int,A: int,N2: nat] :
      ( ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_int )
     => ~ ! [B5: nat > int] :
            ~ ! [Z3: int] :
                ( ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_int @ ( minus_minus_int @ Z3 @ A )
                  @ ( groups3539618377306564664at_int
                    @ ^ [I3: nat] : ( times_times_int @ ( B5 @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_9115_polyfun__linear__factor__root,axiom,
    ! [C: nat > real,A: real,N2: nat] :
      ( ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) )
        = zero_zero_real )
     => ~ ! [B5: nat > real] :
            ~ ! [Z3: real] :
                ( ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
                  @ ( set_ord_atMost_nat @ N2 ) )
                = ( times_times_real @ ( minus_minus_real @ Z3 @ A )
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( B5 @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_linear_factor_root
thf(fact_9116_polyfun__linear__factor,axiom,
    ! [C: nat > rat,N2: nat,A: rat] :
    ? [B5: nat > rat] :
    ! [Z3: rat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_rat
        @ ( times_times_rat @ ( minus_minus_rat @ Z3 @ A )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( B5 @ I3 ) @ ( power_power_rat @ Z3 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( times_times_rat @ ( C @ I3 ) @ ( power_power_rat @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_9117_polyfun__linear__factor,axiom,
    ! [C: nat > complex,N2: nat,A: complex] :
    ? [B5: nat > complex] :
    ! [Z3: complex] :
      ( ( groups2073611262835488442omplex
        @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_complex
        @ ( times_times_complex @ ( minus_minus_complex @ Z3 @ A )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( B5 @ I3 ) @ ( power_power_complex @ Z3 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_9118_polyfun__linear__factor,axiom,
    ! [C: nat > int,N2: nat,A: int] :
    ? [B5: nat > int] :
    ! [Z3: int] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_int
        @ ( times_times_int @ ( minus_minus_int @ Z3 @ A )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( B5 @ I3 ) @ ( power_power_int @ Z3 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( times_times_int @ ( C @ I3 ) @ ( power_power_int @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_9119_polyfun__linear__factor,axiom,
    ! [C: nat > real,N2: nat,A: real] :
    ? [B5: nat > real] :
    ! [Z3: real] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( plus_plus_real
        @ ( times_times_real @ ( minus_minus_real @ Z3 @ A )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( B5 @ I3 ) @ ( power_power_real @ Z3 @ I3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ A @ I3 ) )
          @ ( set_ord_atMost_nat @ N2 ) ) ) ) ).

% polyfun_linear_factor
thf(fact_9120_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_9121_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_9122_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_9123_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_9124_sum__less__suminf2,axiom,
    ! [F: nat > int,N2: nat,I2: nat] :
      ( ( summable_int @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_eq_int @ zero_zero_int @ ( F @ M5 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I2 )
         => ( ( ord_less_int @ zero_zero_int @ ( F @ I2 ) )
           => ( ord_less_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_int @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_9125_sum__less__suminf2,axiom,
    ! [F: nat > nat,N2: nat,I2: nat] :
      ( ( summable_nat @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ M5 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I2 )
         => ( ( ord_less_nat @ zero_zero_nat @ ( F @ I2 ) )
           => ( ord_less_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_nat @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_9126_sum__less__suminf2,axiom,
    ! [F: nat > real,N2: nat,I2: nat] :
      ( ( summable_real @ F )
     => ( ! [M5: nat] :
            ( ( ord_less_eq_nat @ N2 @ M5 )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ M5 ) ) )
       => ( ( ord_less_eq_nat @ N2 @ I2 )
         => ( ( ord_less_real @ zero_zero_real @ ( F @ I2 ) )
           => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ N2 ) ) @ ( suminf_real @ F ) ) ) ) ) ) ).

% sum_less_suminf2
thf(fact_9127_one__diff__power__eq_H,axiom,
    ! [X4: rat,N2: nat] :
      ( ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X4 @ N2 ) )
      = ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X4 )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( power_power_rat @ X4 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_9128_one__diff__power__eq_H,axiom,
    ! [X4: complex,N2: nat] :
      ( ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X4 @ N2 ) )
      = ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X4 )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( power_power_complex @ X4 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_9129_one__diff__power__eq_H,axiom,
    ! [X4: int,N2: nat] :
      ( ( minus_minus_int @ one_one_int @ ( power_power_int @ X4 @ N2 ) )
      = ( times_times_int @ ( minus_minus_int @ one_one_int @ X4 )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( power_power_int @ X4 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_9130_one__diff__power__eq_H,axiom,
    ! [X4: real,N2: nat] :
      ( ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ N2 ) )
      = ( times_times_real @ ( minus_minus_real @ one_one_real @ X4 )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( power_power_real @ X4 @ ( minus_minus_nat @ N2 @ ( suc @ I3 ) ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% one_diff_power_eq'
thf(fact_9131_Maclaurin__zero,axiom,
    ! [X4: real,N2: nat,Diff: nat > complex > real] :
      ( ( X4 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_complex ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_complex ) ) ) ) ).

% Maclaurin_zero
thf(fact_9132_Maclaurin__zero,axiom,
    ! [X4: real,N2: nat,Diff: nat > real > real] :
      ( ( X4 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_real ) ) ) ) ).

% Maclaurin_zero
thf(fact_9133_Maclaurin__zero,axiom,
    ! [X4: real,N2: nat,Diff: nat > rat > real] :
      ( ( X4 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_rat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_rat ) ) ) ) ).

% Maclaurin_zero
thf(fact_9134_Maclaurin__zero,axiom,
    ! [X4: real,N2: nat,Diff: nat > nat > real] :
      ( ( X4 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_nat ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_nat ) ) ) ) ).

% Maclaurin_zero
thf(fact_9135_Maclaurin__zero,axiom,
    ! [X4: real,N2: nat,Diff: nat > int > real] :
      ( ( X4 = zero_zero_real )
     => ( ( N2 != zero_zero_nat )
       => ( ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_int ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          = ( Diff @ zero_zero_nat @ zero_zero_int ) ) ) ) ).

% Maclaurin_zero
thf(fact_9136_Maclaurin__lemma,axiom,
    ! [H: real,F: real > real,J: nat > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ? [B7: real] :
          ( ( F @ H )
          = ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( J @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
              @ ( set_ord_lessThan_nat @ N2 ) )
            @ ( times_times_real @ B7 @ ( divide_divide_real @ ( power_power_real @ H @ N2 ) @ ( semiri2265585572941072030t_real @ N2 ) ) ) ) ) ) ).

% Maclaurin_lemma
thf(fact_9137_sum__split__even__odd,axiom,
    ! [F: nat > real,G: nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( if_real @ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ ( F @ I3 ) @ ( G @ I3 ) )
        @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( F @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) @ one_one_nat ) )
          @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ).

% sum_split_even_odd
thf(fact_9138_sum__mono,axiom,
    ! [K5: set_real,F: real > rat,G: real > rat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ K5 ) @ ( groups1300246762558778688al_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9139_sum__mono,axiom,
    ! [K5: set_nat,F: nat > rat,G: nat > rat] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ K5 ) @ ( groups2906978787729119204at_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9140_sum__mono,axiom,
    ! [K5: set_complex,F: complex > rat,G: complex > rat] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ K5 ) @ ( groups5058264527183730370ex_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9141_sum__mono,axiom,
    ! [K5: set_int,F: int > rat,G: int > rat] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ K5 )
         => ( ord_less_eq_rat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ K5 ) @ ( groups3906332499630173760nt_rat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9142_sum__mono,axiom,
    ! [K5: set_real,F: real > nat,G: real > nat] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ K5 ) @ ( groups1935376822645274424al_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9143_sum__mono,axiom,
    ! [K5: set_complex,F: complex > nat,G: complex > nat] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ K5 ) @ ( groups5693394587270226106ex_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9144_sum__mono,axiom,
    ! [K5: set_int,F: int > nat,G: int > nat] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ K5 )
         => ( ord_less_eq_nat @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ K5 ) @ ( groups4541462559716669496nt_nat @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9145_sum__mono,axiom,
    ! [K5: set_real,F: real > int,G: real > int] :
      ( ! [I4: real] :
          ( ( member_real @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups1932886352136224148al_int @ F @ K5 ) @ ( groups1932886352136224148al_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9146_sum__mono,axiom,
    ! [K5: set_nat,F: nat > int,G: nat > int] :
      ( ! [I4: nat] :
          ( ( member_nat @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups3539618377306564664at_int @ F @ K5 ) @ ( groups3539618377306564664at_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9147_sum__mono,axiom,
    ! [K5: set_complex,F: complex > int,G: complex > int] :
      ( ! [I4: complex] :
          ( ( member_complex @ I4 @ K5 )
         => ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) )
     => ( ord_less_eq_int @ ( groups5690904116761175830ex_int @ F @ K5 ) @ ( groups5690904116761175830ex_int @ G @ K5 ) ) ) ).

% sum_mono
thf(fact_9148_sum_Odistrib,axiom,
    ! [G: nat > nat,H: nat > nat,A2: set_nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : ( plus_plus_nat @ ( G @ X ) @ ( H @ X ) )
        @ A2 )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ G @ A2 ) @ ( groups3542108847815614940at_nat @ H @ A2 ) ) ) ).

% sum.distrib
thf(fact_9149_sum_Odistrib,axiom,
    ! [G: nat > real,H: nat > real,A2: set_nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [X: nat] : ( plus_plus_real @ ( G @ X ) @ ( H @ X ) )
        @ A2 )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ G @ A2 ) @ ( groups6591440286371151544t_real @ H @ A2 ) ) ) ).

% sum.distrib
thf(fact_9150_sum_Odistrib,axiom,
    ! [G: complex > complex,H: complex > complex,A2: set_complex] :
      ( ( groups7754918857620584856omplex
        @ ^ [X: complex] : ( plus_plus_complex @ ( G @ X ) @ ( H @ X ) )
        @ A2 )
      = ( plus_plus_complex @ ( groups7754918857620584856omplex @ G @ A2 ) @ ( groups7754918857620584856omplex @ H @ A2 ) ) ) ).

% sum.distrib
thf(fact_9151_sum_Odistrib,axiom,
    ! [G: int > int,H: int > int,A2: set_int] :
      ( ( groups4538972089207619220nt_int
        @ ^ [X: int] : ( plus_plus_int @ ( G @ X ) @ ( H @ X ) )
        @ A2 )
      = ( plus_plus_int @ ( groups4538972089207619220nt_int @ G @ A2 ) @ ( groups4538972089207619220nt_int @ H @ A2 ) ) ) ).

% sum.distrib
thf(fact_9152_sum__divide__distrib,axiom,
    ! [F: nat > real,A2: set_nat,R3: real] :
      ( ( divide_divide_real @ ( groups6591440286371151544t_real @ F @ A2 ) @ R3 )
      = ( groups6591440286371151544t_real
        @ ^ [N: nat] : ( divide_divide_real @ ( F @ N ) @ R3 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_9153_sum__divide__distrib,axiom,
    ! [F: complex > complex,A2: set_complex,R3: complex] :
      ( ( divide1717551699836669952omplex @ ( groups7754918857620584856omplex @ F @ A2 ) @ R3 )
      = ( groups7754918857620584856omplex
        @ ^ [N: complex] : ( divide1717551699836669952omplex @ ( F @ N ) @ R3 )
        @ A2 ) ) ).

% sum_divide_distrib
thf(fact_9154_Maclaurin__exp__le,axiom,
    ! [X4: real,N2: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X4 ) )
      & ( ( exp_real @ X4 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X4 @ N2 ) ) ) ) ) ).

% Maclaurin_exp_le
thf(fact_9155_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > rat,X4: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X4 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X4 @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [K3: nat] : ( times_times_rat @ ( times_times_rat @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_rat @ Y @ K3 ) ) @ ( power_power_rat @ X4 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_9156_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > complex,X4: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X4 @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( groups2073611262835488442omplex
                @ ^ [K3: nat] : ( times_times_complex @ ( times_times_complex @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_complex @ Y @ K3 ) ) @ ( power_power_complex @ X4 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_9157_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > int,X4: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X4 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X4 @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( groups3539618377306564664at_int
                @ ^ [K3: nat] : ( times_times_int @ ( times_times_int @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_int @ Y @ K3 ) ) @ ( power_power_int @ X4 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_9158_polyfun__diff__alt,axiom,
    ! [N2: nat,A: nat > real,X4: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X4 @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( groups6591440286371151544t_real
                @ ^ [K3: nat] : ( times_times_real @ ( times_times_real @ ( A @ ( plus_plus_nat @ ( plus_plus_nat @ J3 @ K3 ) @ one_one_nat ) ) @ ( power_power_real @ Y @ K3 ) ) @ ( power_power_real @ X4 @ J3 ) )
                @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ J3 ) ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff_alt
thf(fact_9159_exp__first__terms,axiom,
    ! [K: nat] :
      ( exp_complex
      = ( ^ [X: complex] :
            ( plus_plus_complex
            @ ( groups2073611262835488442omplex
              @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_complex @ X @ N ) )
              @ ( set_ord_lessThan_nat @ K ) )
            @ ( suminf_complex
              @ ^ [N: nat] : ( real_V2046097035970521341omplex @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N @ K ) ) ) @ ( power_power_complex @ X @ ( plus_plus_nat @ N @ K ) ) ) ) ) ) ) ).

% exp_first_terms
thf(fact_9160_exp__first__terms,axiom,
    ! [K: nat] :
      ( exp_real
      = ( ^ [X: real] :
            ( plus_plus_real
            @ ( groups6591440286371151544t_real
              @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N ) ) @ ( power_power_real @ X @ N ) )
              @ ( set_ord_lessThan_nat @ K ) )
            @ ( suminf_real
              @ ^ [N: nat] : ( real_V1485227260804924795R_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ ( plus_plus_nat @ N @ K ) ) ) @ ( power_power_real @ X @ ( plus_plus_nat @ N @ K ) ) ) ) ) ) ) ).

% exp_first_terms
thf(fact_9161_Maclaurin__sin__bound,axiom,
    ! [X4: real,N2: nat] :
      ( ord_less_eq_real
      @ ( abs_abs_real
        @ ( minus_minus_real @ ( sin_real @ X4 )
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) )
      @ ( times_times_real @ ( inverse_inverse_real @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( abs_abs_real @ X4 ) @ N2 ) ) ) ).

% Maclaurin_sin_bound
thf(fact_9162_sum__pos__lt__pair,axiom,
    ! [F: nat > real,K: nat] :
      ( ( summable_real @ F )
     => ( ! [D3: nat] : ( ord_less_real @ zero_zero_real @ ( plus_plus_real @ ( F @ ( plus_plus_nat @ K @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) ) ) @ ( F @ ( plus_plus_nat @ K @ ( plus_plus_nat @ ( times_times_nat @ ( suc @ ( suc @ zero_zero_nat ) ) @ D3 ) @ one_one_nat ) ) ) ) )
       => ( ord_less_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_lessThan_nat @ K ) ) @ ( suminf_real @ F ) ) ) ) ).

% sum_pos_lt_pair
thf(fact_9163_sum__nonpos,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8097168146408367636l_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_9164_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups5808333547571424918x_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_9165_sum__nonpos,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_real @ ( F @ X5 ) @ zero_zero_real ) )
     => ( ord_less_eq_real @ ( groups8778361861064173332t_real @ F @ A2 ) @ zero_zero_real ) ) ).

% sum_nonpos
thf(fact_9166_sum__nonpos,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_9167_sum__nonpos,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_9168_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_9169_sum__nonpos,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_rat @ ( F @ X5 ) @ zero_zero_rat ) )
     => ( ord_less_eq_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) @ zero_zero_rat ) ) ).

% sum_nonpos
thf(fact_9170_sum__nonpos,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_9171_sum__nonpos,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_9172_sum__nonpos,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_nat @ ( F @ X5 ) @ zero_zero_nat ) )
     => ( ord_less_eq_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) @ zero_zero_nat ) ) ).

% sum_nonpos
thf(fact_9173_sum__nonneg,axiom,
    ! [A2: set_real,F: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8097168146408367636l_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9174_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > real] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups5808333547571424918x_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9175_sum__nonneg,axiom,
    ! [A2: set_int,F: int > real] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
     => ( ord_less_eq_real @ zero_zero_real @ ( groups8778361861064173332t_real @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9176_sum__nonneg,axiom,
    ! [A2: set_real,F: real > rat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups1300246762558778688al_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9177_sum__nonneg,axiom,
    ! [A2: set_nat,F: nat > rat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups2906978787729119204at_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9178_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > rat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups5058264527183730370ex_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9179_sum__nonneg,axiom,
    ! [A2: set_int,F: int > rat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_rat @ zero_zero_rat @ ( F @ X5 ) ) )
     => ( ord_less_eq_rat @ zero_zero_rat @ ( groups3906332499630173760nt_rat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9180_sum__nonneg,axiom,
    ! [A2: set_real,F: real > nat] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups1935376822645274424al_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9181_sum__nonneg,axiom,
    ! [A2: set_complex,F: complex > nat] :
      ( ! [X5: complex] :
          ( ( member_complex @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups5693394587270226106ex_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9182_sum__nonneg,axiom,
    ! [A2: set_int,F: int > nat] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups4541462559716669496nt_nat @ F @ A2 ) ) ) ).

% sum_nonneg
thf(fact_9183_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > nat,G: nat > nat] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A2 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups3542108847815614940at_nat @ F @ A2 )
          = ( groups3542108847815614940at_nat @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_9184_sum__cong__Suc,axiom,
    ! [A2: set_nat,F: nat > real,G: nat > real] :
      ( ~ ( member_nat @ zero_zero_nat @ A2 )
     => ( ! [X5: nat] :
            ( ( member_nat @ ( suc @ X5 ) @ A2 )
           => ( ( F @ ( suc @ X5 ) )
              = ( G @ ( suc @ X5 ) ) ) )
       => ( ( groups6591440286371151544t_real @ F @ A2 )
          = ( groups6591440286371151544t_real @ G @ A2 ) ) ) ) ).

% sum_cong_Suc
thf(fact_9185_Maclaurin__exp__lt,axiom,
    ! [X4: real,N2: nat] :
      ( ( X4 != zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
            & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X4 ) )
            & ( ( exp_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( divide_divide_real @ ( power_power_real @ X4 @ M6 ) @ ( semiri2265585572941072030t_real @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( exp_real @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X4 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_exp_lt
thf(fact_9186_lemma__termdiff2,axiom,
    ! [H: rat,Z: rat,N2: nat] :
      ( ( H != zero_zero_rat )
     => ( ( minus_minus_rat @ ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H ) @ N2 ) @ ( power_power_rat @ Z @ N2 ) ) @ H ) @ ( times_times_rat @ ( semiri681578069525770553at_rat @ N2 ) @ ( power_power_rat @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_rat @ H
          @ ( groups2906978787729119204at_rat
            @ ^ [P5: nat] :
                ( groups2906978787729119204at_rat
                @ ^ [Q5: nat] : ( times_times_rat @ ( power_power_rat @ ( plus_plus_rat @ Z @ H ) @ Q5 ) @ ( power_power_rat @ Z @ ( 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 ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_9187_lemma__termdiff2,axiom,
    ! [H: complex,Z: complex,N2: nat] :
      ( ( H != zero_zero_complex )
     => ( ( minus_minus_complex @ ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ N2 ) @ ( power_power_complex @ Z @ N2 ) ) @ H ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ ( power_power_complex @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_complex @ H
          @ ( groups2073611262835488442omplex
            @ ^ [P5: nat] :
                ( groups2073611262835488442omplex
                @ ^ [Q5: nat] : ( times_times_complex @ ( power_power_complex @ ( plus_plus_complex @ Z @ H ) @ Q5 ) @ ( power_power_complex @ Z @ ( 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 ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_9188_lemma__termdiff2,axiom,
    ! [H: real,Z: real,N2: nat] :
      ( ( H != zero_zero_real )
     => ( ( minus_minus_real @ ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ N2 ) @ ( power_power_real @ Z @ N2 ) ) @ H ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ Z @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) )
        = ( times_times_real @ H
          @ ( groups6591440286371151544t_real
            @ ^ [P5: nat] :
                ( groups6591440286371151544t_real
                @ ^ [Q5: nat] : ( times_times_real @ ( power_power_real @ ( plus_plus_real @ Z @ H ) @ Q5 ) @ ( power_power_real @ Z @ ( 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 ) ) @ P5 ) ) )
            @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ).

% lemma_termdiff2
thf(fact_9189_Maclaurin__sin__expansion,axiom,
    ! [X4: real,N2: nat] :
    ? [T3: real] :
      ( ( sin_real @ X4 )
      = ( plus_plus_real
        @ ( groups6591440286371151544t_real
          @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
          @ ( set_ord_lessThan_nat @ N2 ) )
        @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( 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 @ X4 @ N2 ) ) ) ) ).

% Maclaurin_sin_expansion
thf(fact_9190_Maclaurin__sin__expansion2,axiom,
    ! [X4: real,N2: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X4 ) )
      & ( ( sin_real @ X4 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( 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 @ X4 @ N2 ) ) ) ) ) ).

% Maclaurin_sin_expansion2
thf(fact_9191_Maclaurin__cos__expansion,axiom,
    ! [X4: real,N2: nat] :
    ? [T3: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X4 ) )
      & ( ( cos_real @ X4 )
        = ( plus_plus_real
          @ ( groups6591440286371151544t_real
            @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
            @ ( set_ord_lessThan_nat @ N2 ) )
          @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( 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 @ X4 @ N2 ) ) ) ) ) ).

% Maclaurin_cos_expansion
thf(fact_9192_Maclaurin__sin__expansion4,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ? [T3: real] :
          ( ( ord_less_real @ zero_zero_real @ T3 )
          & ( ord_less_eq_real @ T3 @ X4 )
          & ( ( sin_real @ X4 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( 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 @ X4 @ N2 ) ) ) ) ) ) ).

% Maclaurin_sin_expansion4
thf(fact_9193_Maclaurin__sin__expansion3,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X4 )
            & ( ( sin_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( sin_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( sin_real @ ( plus_plus_real @ T3 @ ( 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 @ X4 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_sin_expansion3
thf(fact_9194_Maclaurin__cos__expansion2,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ? [T3: real] :
            ( ( ord_less_real @ zero_zero_real @ T3 )
            & ( ord_less_real @ T3 @ X4 )
            & ( ( cos_real @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( cos_coeff @ M6 ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( cos_real @ ( plus_plus_real @ T3 @ ( 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 @ X4 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_cos_expansion2
thf(fact_9195_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
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N2 )
              = one_one_complex ) ) ) ) ).

% bij_betw_roots_unity
thf(fact_9196_sum__gp,axiom,
    ! [N2: nat,M: nat,X4: rat] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_rat ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X4 = one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri681578069525770553at_rat @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X4 != one_one_rat )
           => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_rat @ ( minus_minus_rat @ ( power_power_rat @ X4 @ M ) @ ( power_power_rat @ X4 @ ( suc @ N2 ) ) ) @ ( minus_minus_rat @ one_one_rat @ X4 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_9197_sum__gp,axiom,
    ! [N2: nat,M: nat,X4: complex] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_complex ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X4 = one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri8010041392384452111omplex @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X4 != one_one_complex )
           => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide1717551699836669952omplex @ ( minus_minus_complex @ ( power_power_complex @ X4 @ M ) @ ( power_power_complex @ X4 @ ( suc @ N2 ) ) ) @ ( minus_minus_complex @ one_one_complex @ X4 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_9198_sum__gp,axiom,
    ! [N2: nat,M: nat,X4: real] :
      ( ( ( ord_less_nat @ N2 @ M )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
          = zero_zero_real ) )
      & ( ~ ( ord_less_nat @ N2 @ M )
       => ( ( ( X4 = one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( semiri5074537144036343181t_real @ ( minus_minus_nat @ ( plus_plus_nat @ N2 @ one_one_nat ) @ M ) ) ) )
          & ( ( X4 != one_one_real )
           => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
              = ( divide_divide_real @ ( minus_minus_real @ ( power_power_real @ X4 @ M ) @ ( power_power_real @ X4 @ ( suc @ N2 ) ) ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) ) ) ) ) ).

% sum_gp
thf(fact_9199_gchoose__row__sum__weighted,axiom,
    ! [R3: rat,M: nat] :
      ( ( groups2906978787729119204at_rat
        @ ^ [K3: nat] : ( times_times_rat @ ( gbinomial_rat @ R3 @ K3 ) @ ( minus_minus_rat @ ( divide_divide_rat @ R3 @ ( 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 @ R3 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_9200_gchoose__row__sum__weighted,axiom,
    ! [R3: complex,M: nat] :
      ( ( groups2073611262835488442omplex
        @ ^ [K3: nat] : ( times_times_complex @ ( gbinomial_complex @ R3 @ K3 ) @ ( minus_minus_complex @ ( divide1717551699836669952omplex @ R3 @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( semiri8010041392384452111omplex @ K3 ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ M ) )
      = ( times_times_complex @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ ( suc @ M ) ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) @ ( gbinomial_complex @ R3 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_9201_gchoose__row__sum__weighted,axiom,
    ! [R3: real,M: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [K3: nat] : ( times_times_real @ ( gbinomial_real @ R3 @ K3 ) @ ( minus_minus_real @ ( divide_divide_real @ R3 @ ( 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 @ R3 @ ( suc @ M ) ) ) ) ).

% gchoose_row_sum_weighted
thf(fact_9202_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_9203_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_9204_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_9205_atLeastAtMost__iff,axiom,
    ! [I2: set_int,L: set_int,U: set_int] :
      ( ( member_set_int @ I2 @ ( set_or370866239135849197et_int @ L @ U ) )
      = ( ( ord_less_eq_set_int @ L @ I2 )
        & ( ord_less_eq_set_int @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_9206_atLeastAtMost__iff,axiom,
    ! [I2: rat,L: rat,U: rat] :
      ( ( member_rat @ I2 @ ( set_or633870826150836451st_rat @ L @ U ) )
      = ( ( ord_less_eq_rat @ L @ I2 )
        & ( ord_less_eq_rat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_9207_atLeastAtMost__iff,axiom,
    ! [I2: num,L: num,U: num] :
      ( ( member_num @ I2 @ ( set_or7049704709247886629st_num @ L @ U ) )
      = ( ( ord_less_eq_num @ L @ I2 )
        & ( ord_less_eq_num @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_9208_atLeastAtMost__iff,axiom,
    ! [I2: nat,L: nat,U: nat] :
      ( ( member_nat @ I2 @ ( set_or1269000886237332187st_nat @ L @ U ) )
      = ( ( ord_less_eq_nat @ L @ I2 )
        & ( ord_less_eq_nat @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_9209_atLeastAtMost__iff,axiom,
    ! [I2: int,L: int,U: int] :
      ( ( member_int @ I2 @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( ( ord_less_eq_int @ L @ I2 )
        & ( ord_less_eq_int @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_9210_atLeastAtMost__iff,axiom,
    ! [I2: real,L: real,U: real] :
      ( ( member_real @ I2 @ ( set_or1222579329274155063t_real @ L @ U ) )
      = ( ( ord_less_eq_real @ L @ I2 )
        & ( ord_less_eq_real @ I2 @ U ) ) ) ).

% atLeastAtMost_iff
thf(fact_9211_Icc__eq__Icc,axiom,
    ! [L: set_int,H: set_int,L3: set_int,H3: set_int] :
      ( ( ( set_or370866239135849197et_int @ L @ H )
        = ( set_or370866239135849197et_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_set_int @ L @ H )
          & ~ ( ord_less_eq_set_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_9212_Icc__eq__Icc,axiom,
    ! [L: rat,H: rat,L3: rat,H3: rat] :
      ( ( ( set_or633870826150836451st_rat @ L @ H )
        = ( set_or633870826150836451st_rat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_rat @ L @ H )
          & ~ ( ord_less_eq_rat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_9213_Icc__eq__Icc,axiom,
    ! [L: num,H: num,L3: num,H3: num] :
      ( ( ( set_or7049704709247886629st_num @ L @ H )
        = ( set_or7049704709247886629st_num @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_num @ L @ H )
          & ~ ( ord_less_eq_num @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_9214_Icc__eq__Icc,axiom,
    ! [L: nat,H: nat,L3: nat,H3: nat] :
      ( ( ( set_or1269000886237332187st_nat @ L @ H )
        = ( set_or1269000886237332187st_nat @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_nat @ L @ H )
          & ~ ( ord_less_eq_nat @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_9215_Icc__eq__Icc,axiom,
    ! [L: int,H: int,L3: int,H3: int] :
      ( ( ( set_or1266510415728281911st_int @ L @ H )
        = ( set_or1266510415728281911st_int @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_int @ L @ H )
          & ~ ( ord_less_eq_int @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_9216_Icc__eq__Icc,axiom,
    ! [L: real,H: real,L3: real,H3: real] :
      ( ( ( set_or1222579329274155063t_real @ L @ H )
        = ( set_or1222579329274155063t_real @ L3 @ H3 ) )
      = ( ( ( L = L3 )
          & ( H = H3 ) )
        | ( ~ ( ord_less_eq_real @ L @ H )
          & ~ ( ord_less_eq_real @ L3 @ H3 ) ) ) ) ).

% Icc_eq_Icc
thf(fact_9217_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_9218_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_9219_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_9220_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_9221_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_9222_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_9223_Icc__subset__Iic__iff,axiom,
    ! [L: set_int,H: set_int,H3: set_int] :
      ( ( ord_le4403425263959731960et_int @ ( set_or370866239135849197et_int @ L @ H ) @ ( set_or58775011639299419et_int @ H3 ) )
      = ( ~ ( ord_less_eq_set_int @ L @ H )
        | ( ord_less_eq_set_int @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_9224_Icc__subset__Iic__iff,axiom,
    ! [L: rat,H: rat,H3: rat] :
      ( ( ord_less_eq_set_rat @ ( set_or633870826150836451st_rat @ L @ H ) @ ( set_ord_atMost_rat @ H3 ) )
      = ( ~ ( ord_less_eq_rat @ L @ H )
        | ( ord_less_eq_rat @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_9225_Icc__subset__Iic__iff,axiom,
    ! [L: num,H: num,H3: num] :
      ( ( ord_less_eq_set_num @ ( set_or7049704709247886629st_num @ L @ H ) @ ( set_ord_atMost_num @ H3 ) )
      = ( ~ ( ord_less_eq_num @ L @ H )
        | ( ord_less_eq_num @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_9226_Icc__subset__Iic__iff,axiom,
    ! [L: nat,H: nat,H3: nat] :
      ( ( ord_less_eq_set_nat @ ( set_or1269000886237332187st_nat @ L @ H ) @ ( set_ord_atMost_nat @ H3 ) )
      = ( ~ ( ord_less_eq_nat @ L @ H )
        | ( ord_less_eq_nat @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_9227_Icc__subset__Iic__iff,axiom,
    ! [L: int,H: int,H3: int] :
      ( ( ord_less_eq_set_int @ ( set_or1266510415728281911st_int @ L @ H ) @ ( set_ord_atMost_int @ H3 ) )
      = ( ~ ( ord_less_eq_int @ L @ H )
        | ( ord_less_eq_int @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_9228_Icc__subset__Iic__iff,axiom,
    ! [L: real,H: real,H3: real] :
      ( ( ord_less_eq_set_real @ ( set_or1222579329274155063t_real @ L @ H ) @ ( set_ord_atMost_real @ H3 ) )
      = ( ~ ( ord_less_eq_real @ L @ H )
        | ( ord_less_eq_real @ H @ H3 ) ) ) ).

% Icc_subset_Iic_iff
thf(fact_9229_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_9230_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_9231_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_9232_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_9233_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_9234_not__Iic__eq__Icc,axiom,
    ! [H3: int,L: int,H: int] :
      ( ( set_ord_atMost_int @ H3 )
     != ( set_or1266510415728281911st_int @ L @ H ) ) ).

% not_Iic_eq_Icc
thf(fact_9235_not__Iic__eq__Icc,axiom,
    ! [H3: real,L: real,H: real] :
      ( ( set_ord_atMost_real @ H3 )
     != ( set_or1222579329274155063t_real @ L @ H ) ) ).

% not_Iic_eq_Icc
thf(fact_9236_not__Iic__le__Icc,axiom,
    ! [H: int,L3: int,H3: int] :
      ~ ( ord_less_eq_set_int @ ( set_ord_atMost_int @ H ) @ ( set_or1266510415728281911st_int @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_9237_not__Iic__le__Icc,axiom,
    ! [H: real,L3: real,H3: real] :
      ~ ( ord_less_eq_set_real @ ( set_ord_atMost_real @ H ) @ ( set_or1222579329274155063t_real @ L3 @ H3 ) ) ).

% not_Iic_le_Icc
thf(fact_9238_all__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N2 )
           => ( P @ M6 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less
thf(fact_9239_ex__nat__less,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_eq_nat @ M6 @ N2 )
            & ( P @ M6 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less
thf(fact_9240_atMost__atLeast0,axiom,
    ( set_ord_atMost_nat
    = ( set_or1269000886237332187st_nat @ zero_zero_nat ) ) ).

% atMost_atLeast0
thf(fact_9241_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
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_9242_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
        @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_Suc_ivl
thf(fact_9243_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
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_9244_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
        @ ^ [I3: nat] : ( G @ ( plus_plus_nat @ I3 @ K ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.shift_bounds_cl_nat_ivl
thf(fact_9245_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_9246_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_9247_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_9248_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_9249_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_9250_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_9251_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > nat,N2: nat,M: nat] :
      ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_9252_sum_OatLeastAtMost__rev,axiom,
    ! [G: nat > real,N2: nat,M: nat] :
      ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ N2 @ M ) )
      = ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( G @ ( minus_minus_nat @ ( plus_plus_nat @ M @ N2 ) @ I3 ) )
        @ ( set_or1269000886237332187st_nat @ N2 @ M ) ) ) ).

% sum.atLeastAtMost_rev
thf(fact_9253_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_9254_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_9255_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_9256_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_9257_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_9258_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_9259_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_9260_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_9261_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_9262_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_9263_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_9264_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_9265_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_9266_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_9267_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_9268_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_9269_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_9270_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
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_9271_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
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_9272_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
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_9273_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
            @ ^ [I3: nat] : ( G @ ( suc @ I3 ) )
            @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ) ) ).

% sum.Suc_reindex_ivl
thf(fact_9274_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > rat] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( minus_minus_rat @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_rat @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_9275_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > int] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( minus_minus_int @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_int @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_9276_sum__Suc__diff,axiom,
    ! [M: nat,N2: nat,F: nat > real] :
      ( ( ord_less_eq_nat @ M @ ( suc @ N2 ) )
     => ( ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( minus_minus_real @ ( F @ ( suc @ I3 ) ) @ ( F @ I3 ) )
          @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( minus_minus_real @ ( F @ ( suc @ N2 ) ) @ ( F @ M ) ) ) ) ).

% sum_Suc_diff
thf(fact_9277_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_9278_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_9279_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_9280_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_9281_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( groups3542108847815614940at_nat @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups3542108847815614940at_nat
        @ ^ [J3: nat] :
            ( groups3542108847815614940at_nat
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_9282_sum_Onested__swap_H,axiom,
    ! [A: nat > nat > real,N2: nat] :
      ( ( groups6591440286371151544t_real
        @ ^ [I3: nat] : ( groups6591440286371151544t_real @ ( A @ I3 ) @ ( set_ord_lessThan_nat @ I3 ) )
        @ ( set_ord_atMost_nat @ N2 ) )
      = ( groups6591440286371151544t_real
        @ ^ [J3: nat] :
            ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( A @ I3 @ J3 )
            @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
        @ ( set_ord_lessThan_nat @ N2 ) ) ) ).

% sum.nested_swap'
thf(fact_9283_sum__atLeastAtMost__code,axiom,
    ! [F: nat > complex,A: nat,B: nat] :
      ( ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1517530859248394432omplex
        @ ^ [A3: nat] : ( plus_plus_complex @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_zero_complex ) ) ).

% sum_atLeastAtMost_code
thf(fact_9284_sum__atLeastAtMost__code,axiom,
    ! [F: nat > rat,A: nat,B: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo1949268297981939178at_rat
        @ ^ [A3: nat] : ( plus_plus_rat @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_zero_rat ) ) ).

% sum_atLeastAtMost_code
thf(fact_9285_sum__atLeastAtMost__code,axiom,
    ! [F: nat > int,A: nat,B: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2581907887559384638at_int
        @ ^ [A3: nat] : ( plus_plus_int @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_zero_int ) ) ).

% sum_atLeastAtMost_code
thf(fact_9286_sum__atLeastAtMost__code,axiom,
    ! [F: nat > nat,A: nat,B: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo2584398358068434914at_nat
        @ ^ [A3: nat] : ( plus_plus_nat @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_zero_nat ) ) ).

% sum_atLeastAtMost_code
thf(fact_9287_sum__atLeastAtMost__code,axiom,
    ! [F: nat > real,A: nat,B: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_fo3111899725591712190t_real
        @ ^ [A3: nat] : ( plus_plus_real @ ( F @ A3 ) )
        @ A
        @ B
        @ zero_zero_real ) ) ).

% sum_atLeastAtMost_code
thf(fact_9288_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > rat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups2906978787729119204at_rat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( 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 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_9289_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > int,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3539618377306564664at_int @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( 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 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_9290_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > nat,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups3542108847815614940at_nat @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( 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 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_9291_sum_Oub__add__nat,axiom,
    ! [M: nat,N2: nat,G: nat > real,P2: nat] :
      ( ( ord_less_eq_nat @ M @ ( plus_plus_nat @ N2 @ one_one_nat ) )
     => ( ( groups6591440286371151544t_real @ G @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ N2 @ P2 ) ) )
        = ( 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 @ P2 ) ) ) ) ) ) ).

% sum.ub_add_nat
thf(fact_9292_sum__up__index__split,axiom,
    ! [F: nat > rat,M: nat,N2: nat] :
      ( ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_rat @ ( groups2906978787729119204at_rat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups2906978787729119204at_rat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_9293_sum__up__index__split,axiom,
    ! [F: nat > int,M: nat,N2: nat] :
      ( ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_int @ ( groups3539618377306564664at_int @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3539618377306564664at_int @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_9294_sum__up__index__split,axiom,
    ! [F: nat > nat,M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_nat @ ( groups3542108847815614940at_nat @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups3542108847815614940at_nat @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_9295_sum__up__index__split,axiom,
    ! [F: nat > real,M: nat,N2: nat] :
      ( ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ ( plus_plus_nat @ M @ N2 ) ) )
      = ( plus_plus_real @ ( groups6591440286371151544t_real @ F @ ( set_ord_atMost_nat @ M ) ) @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ ( suc @ M ) @ ( plus_plus_nat @ M @ N2 ) ) ) ) ) ).

% sum_up_index_split
thf(fact_9296_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_9297_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_9298_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_9299_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_9300_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_9301_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_9302_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_9303_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X4: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_9304_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X4: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups3539618377306564664at_int @ ( power_power_int @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_int @ ( power_power_int @ X4 @ M ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X4 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_9305_sum__power__shift,axiom,
    ! [M: nat,N2: nat,X4: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) )
        = ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_ord_atMost_nat @ ( minus_minus_nat @ N2 @ M ) ) ) ) ) ) ).

% sum_power_shift
thf(fact_9306_summable__partial__sum__bound,axiom,
    ! [F: nat > complex,E2: real] :
      ( ( summable_complex @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M2 )
                 => ! [N6: nat] : ( ord_less_real @ ( real_V1022390504157884413omplex @ ( groups2073611262835488442omplex @ F @ ( set_or1269000886237332187st_nat @ M2 @ N6 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_9307_summable__partial__sum__bound,axiom,
    ! [F: nat > real,E2: real] :
      ( ( summable_real @ F )
     => ( ( ord_less_real @ zero_zero_real @ E2 )
       => ~ ! [N8: nat] :
              ~ ! [M2: nat] :
                  ( ( ord_less_eq_nat @ N8 @ M2 )
                 => ! [N6: nat] : ( ord_less_real @ ( real_V7735802525324610683m_real @ ( groups6591440286371151544t_real @ F @ ( set_or1269000886237332187st_nat @ M2 @ N6 ) ) ) @ E2 ) ) ) ) ).

% summable_partial_sum_bound
thf(fact_9308_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X4: rat] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_rat @ ( minus_minus_rat @ one_one_rat @ X4 ) @ ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_rat @ ( power_power_rat @ X4 @ M ) @ ( power_power_rat @ X4 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_9309_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X4: complex] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_complex @ ( minus_minus_complex @ one_one_complex @ X4 ) @ ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_complex @ ( power_power_complex @ X4 @ M ) @ ( power_power_complex @ X4 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_9310_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X4: int] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_int @ ( minus_minus_int @ one_one_int @ X4 ) @ ( groups3539618377306564664at_int @ ( power_power_int @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_int @ ( power_power_int @ X4 @ M ) @ ( power_power_int @ X4 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_9311_sum__gp__multiplied,axiom,
    ! [M: nat,N2: nat,X4: real] :
      ( ( ord_less_eq_nat @ M @ N2 )
     => ( ( times_times_real @ ( minus_minus_real @ one_one_real @ X4 ) @ ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) )
        = ( minus_minus_real @ ( power_power_real @ X4 @ M ) @ ( power_power_real @ X4 @ ( suc @ N2 ) ) ) ) ) ).

% sum_gp_multiplied
thf(fact_9312_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
        @ ^ [I3: nat] : ( plus_plus_rat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_9313_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
        @ ^ [I3: nat] : ( plus_plus_int @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_9314_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
        @ ^ [I3: nat] : ( plus_plus_nat @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_9315_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
        @ ^ [I3: nat] : ( plus_plus_real @ ( G @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) @ ( G @ ( suc @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ I3 ) ) ) )
        @ ( set_or1269000886237332187st_nat @ M @ N2 ) ) ) ).

% sum.in_pairs
thf(fact_9316_polyfun__eq__const,axiom,
    ! [C: nat > complex,N2: nat,K: complex] :
      ( ( ! [X: complex] :
            ( ( groups2073611262835488442omplex
              @ ^ [I3: nat] : ( times_times_complex @ ( C @ I3 ) @ ( power_power_complex @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X )
              = zero_zero_complex ) ) ) ) ).

% polyfun_eq_const
thf(fact_9317_polyfun__eq__const,axiom,
    ! [C: nat > real,N2: nat,K: real] :
      ( ( ! [X: real] :
            ( ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( C @ I3 ) @ ( power_power_real @ X @ I3 ) )
              @ ( set_ord_atMost_nat @ N2 ) )
            = K ) )
      = ( ( ( C @ zero_zero_nat )
          = K )
        & ! [X: nat] :
            ( ( member_nat @ X @ ( set_or1269000886237332187st_nat @ one_one_nat @ N2 ) )
           => ( ( C @ X )
              = zero_zero_real ) ) ) ) ).

% polyfun_eq_const
thf(fact_9318_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_9319_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_9320_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_9321_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_9322_double__arith__series,axiom,
    ! [A: rat,D: rat,N2: nat] :
      ( ( times_times_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) )
        @ ( groups2906978787729119204at_rat
          @ ^ [I3: nat] : ( plus_plus_rat @ A @ ( times_times_rat @ ( semiri681578069525770553at_rat @ I3 ) @ 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_9323_double__arith__series,axiom,
    ! [A: extended_enat,D: extended_enat,N2: nat] :
      ( ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) )
        @ ( groups7108830773950497114d_enat
          @ ^ [I3: nat] : ( plus_p3455044024723400733d_enat @ A @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_7803423173614009249d_enat @ ( plus_p3455044024723400733d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ one_on7984719198319812577d_enat ) @ ( plus_p3455044024723400733d_enat @ ( times_7803423173614009249d_enat @ ( numera1916890842035813515d_enat @ ( bit0 @ one ) ) @ A ) @ ( times_7803423173614009249d_enat @ ( semiri4216267220026989637d_enat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_9324_double__arith__series,axiom,
    ! [A: complex,D: complex,N2: nat] :
      ( ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) )
        @ ( groups2073611262835488442omplex
          @ ^ [I3: nat] : ( plus_plus_complex @ A @ ( times_times_complex @ ( semiri8010041392384452111omplex @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_complex @ ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) @ ( plus_plus_complex @ ( times_times_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) @ A ) @ ( times_times_complex @ ( semiri8010041392384452111omplex @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_9325_double__arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) )
        @ ( groups3539618377306564664at_int
          @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_9326_double__arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) )
        @ ( groups3542108847815614940at_nat
          @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_9327_double__arith__series,axiom,
    ! [A: real,D: real,N2: nat] :
      ( ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
        @ ( groups6591440286371151544t_real
          @ ^ [I3: nat] : ( plus_plus_real @ A @ ( times_times_real @ ( semiri5074537144036343181t_real @ I3 ) @ D ) )
          @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) ) )
      = ( times_times_real @ ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) @ ( plus_plus_real @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ A ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ D ) ) ) ) ).

% double_arith_series
thf(fact_9328_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_9329_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_9330_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_9331_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_9332_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_9333_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_9334_arith__series__nat,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ I3 @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( suc @ N2 ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ N2 @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series_nat
thf(fact_9335_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_9336_arith__series,axiom,
    ! [A: code_integer,D: code_integer,N2: nat] :
      ( ( groups7501900531339628137nteger
        @ ^ [I3: nat] : ( plus_p5714425477246183910nteger @ A @ ( times_3573771949741848930nteger @ ( semiri4939895301339042750nteger @ I3 ) @ 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_9337_arith__series,axiom,
    ! [A: int,D: int,N2: nat] :
      ( ( groups3539618377306564664at_int
        @ ^ [I3: nat] : ( plus_plus_int @ A @ ( times_times_int @ ( semiri1314217659103216013at_int @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_int @ ( times_times_int @ ( plus_plus_int @ ( semiri1314217659103216013at_int @ N2 ) @ one_one_int ) @ ( plus_plus_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ A ) @ ( times_times_int @ ( semiri1314217659103216013at_int @ N2 ) @ D ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_9338_arith__series,axiom,
    ! [A: nat,D: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ A @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ I3 ) @ D ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
      = ( divide_divide_nat @ ( times_times_nat @ ( plus_plus_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ one_one_nat ) @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ A ) @ ( times_times_nat @ ( semiri1316708129612266289at_nat @ N2 ) @ D ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% arith_series
thf(fact_9339_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_9340_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_9341_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_9342_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_9343_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_9344_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_9345_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_9346_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_9347_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_9348_sum__gp__offset,axiom,
    ! [X4: rat,M: nat,N2: nat] :
      ( ( ( X4 = one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_rat @ ( semiri681578069525770553at_rat @ N2 ) @ one_one_rat ) ) )
      & ( ( X4 != one_one_rat )
       => ( ( groups2906978787729119204at_rat @ ( power_power_rat @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_rat @ ( times_times_rat @ ( power_power_rat @ X4 @ M ) @ ( minus_minus_rat @ one_one_rat @ ( power_power_rat @ X4 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_rat @ one_one_rat @ X4 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_9349_sum__gp__offset,axiom,
    ! [X4: complex,M: nat,N2: nat] :
      ( ( ( X4 = one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_complex @ ( semiri8010041392384452111omplex @ N2 ) @ one_one_complex ) ) )
      & ( ( X4 != one_one_complex )
       => ( ( groups2073611262835488442omplex @ ( power_power_complex @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide1717551699836669952omplex @ ( times_times_complex @ ( power_power_complex @ X4 @ M ) @ ( minus_minus_complex @ one_one_complex @ ( power_power_complex @ X4 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_complex @ one_one_complex @ X4 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_9350_sum__gp__offset,axiom,
    ! [X4: real,M: nat,N2: nat] :
      ( ( ( X4 = one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( plus_plus_real @ ( semiri5074537144036343181t_real @ N2 ) @ one_one_real ) ) )
      & ( ( X4 != one_one_real )
       => ( ( groups6591440286371151544t_real @ ( power_power_real @ X4 ) @ ( set_or1269000886237332187st_nat @ M @ ( plus_plus_nat @ M @ N2 ) ) )
          = ( divide_divide_real @ ( times_times_real @ ( power_power_real @ X4 @ M ) @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( suc @ N2 ) ) ) ) @ ( minus_minus_real @ one_one_real @ X4 ) ) ) ) ) ).

% sum_gp_offset
thf(fact_9351_polyfun__diff,axiom,
    ! [N2: nat,A: nat > rat,X4: rat,Y: rat] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_rat
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ X4 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2906978787729119204at_rat
            @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_rat @ ( minus_minus_rat @ X4 @ Y )
          @ ( groups2906978787729119204at_rat
            @ ^ [J3: nat] :
                ( times_times_rat
                @ ( groups2906978787729119204at_rat
                  @ ^ [I3: nat] : ( times_times_rat @ ( A @ I3 ) @ ( power_power_rat @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_rat @ X4 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_9352_polyfun__diff,axiom,
    ! [N2: nat,A: nat > complex,X4: complex,Y: complex] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_complex
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ X4 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups2073611262835488442omplex
            @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_complex @ ( minus_minus_complex @ X4 @ Y )
          @ ( groups2073611262835488442omplex
            @ ^ [J3: nat] :
                ( times_times_complex
                @ ( groups2073611262835488442omplex
                  @ ^ [I3: nat] : ( times_times_complex @ ( A @ I3 ) @ ( power_power_complex @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_complex @ X4 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_9353_polyfun__diff,axiom,
    ! [N2: nat,A: nat > int,X4: int,Y: int] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_int
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ X4 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups3539618377306564664at_int
            @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_int @ ( minus_minus_int @ X4 @ Y )
          @ ( groups3539618377306564664at_int
            @ ^ [J3: nat] :
                ( times_times_int
                @ ( groups3539618377306564664at_int
                  @ ^ [I3: nat] : ( times_times_int @ ( A @ I3 ) @ ( power_power_int @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_int @ X4 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_9354_polyfun__diff,axiom,
    ! [N2: nat,A: nat > real,X4: real,Y: real] :
      ( ( ord_less_eq_nat @ one_one_nat @ N2 )
     => ( ( minus_minus_real
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ X4 @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) )
          @ ( groups6591440286371151544t_real
            @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ I3 ) )
            @ ( set_ord_atMost_nat @ N2 ) ) )
        = ( times_times_real @ ( minus_minus_real @ X4 @ Y )
          @ ( groups6591440286371151544t_real
            @ ^ [J3: nat] :
                ( times_times_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( A @ I3 ) @ ( power_power_real @ Y @ ( minus_minus_nat @ ( minus_minus_nat @ I3 @ J3 ) @ one_one_nat ) ) )
                  @ ( set_or1269000886237332187st_nat @ ( suc @ J3 ) @ N2 ) )
                @ ( power_power_real @ X4 @ J3 ) )
            @ ( set_ord_lessThan_nat @ N2 ) ) ) ) ) ).

% polyfun_diff
thf(fact_9355_pochhammer__times__pochhammer__half,axiom,
    ! [Z: rat,N2: nat] :
      ( ( times_times_rat @ ( comm_s4028243227959126397er_rat @ Z @ ( suc @ N2 ) ) @ ( comm_s4028243227959126397er_rat @ ( plus_plus_rat @ Z @ ( divide_divide_rat @ one_one_rat @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups73079841787564623at_rat
        @ ^ [K3: nat] : ( plus_plus_rat @ Z @ ( divide_divide_rat @ ( semiri681578069525770553at_rat @ K3 ) @ ( numeral_numeral_rat @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_9356_pochhammer__times__pochhammer__half,axiom,
    ! [Z: complex,N2: nat] :
      ( ( times_times_complex @ ( comm_s2602460028002588243omplex @ Z @ ( suc @ N2 ) ) @ ( comm_s2602460028002588243omplex @ ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ one_one_complex @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups6464643781859351333omplex
        @ ^ [K3: nat] : ( plus_plus_complex @ Z @ ( divide1717551699836669952omplex @ ( semiri8010041392384452111omplex @ K3 ) @ ( numera6690914467698888265omplex @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_9357_pochhammer__times__pochhammer__half,axiom,
    ! [Z: real,N2: nat] :
      ( ( times_times_real @ ( comm_s7457072308508201937r_real @ Z @ ( suc @ N2 ) ) @ ( comm_s7457072308508201937r_real @ ( plus_plus_real @ Z @ ( divide_divide_real @ one_one_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( suc @ N2 ) ) )
      = ( groups129246275422532515t_real
        @ ^ [K3: nat] : ( plus_plus_real @ Z @ ( divide_divide_real @ ( semiri5074537144036343181t_real @ K3 ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
        @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ).

% pochhammer_times_pochhammer_half
thf(fact_9358_vebt__buildup_Opelims,axiom,
    ! [X4: nat,Y: vEBT_VEBT] :
      ( ( ( vEBT_vebt_buildup @ X4 )
        = Y )
     => ( ( accp_nat @ vEBT_v4011308405150292612up_rel @ X4 )
       => ( ( ( X4 = zero_zero_nat )
           => ( ( Y
                = ( vEBT_Leaf @ $false @ $false ) )
             => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ zero_zero_nat ) ) )
         => ( ( ( X4
                = ( suc @ zero_zero_nat ) )
             => ( ( Y
                  = ( vEBT_Leaf @ $false @ $false ) )
               => ~ ( accp_nat @ vEBT_v4011308405150292612up_rel @ ( suc @ zero_zero_nat ) ) ) )
           => ~ ! [Va2: nat] :
                  ( ( X4
                    = ( suc @ ( suc @ Va2 ) ) )
                 => ( ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( suc @ ( suc @ Va2 ) ) )
                       => ( Y
                          = ( 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 ) ) )
                       => ( Y
                          = ( 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_9359_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_9360_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_9361_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_9362_arctan__def,axiom,
    ( arctan
    = ( ^ [Y5: 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 )
                = Y5 ) ) ) ) ) ).

% arctan_def
thf(fact_9363_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups708209901874060359at_nat
        @ ^ [Uu3: nat] : one_one_nat
        @ A2 )
      = one_one_nat ) ).

% prod.neutral_const
thf(fact_9364_prod_Oneutral__const,axiom,
    ! [A2: set_nat] :
      ( ( groups705719431365010083at_int
        @ ^ [Uu3: nat] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_9365_prod_Oneutral__const,axiom,
    ! [A2: set_int] :
      ( ( groups1705073143266064639nt_int
        @ ^ [Uu3: int] : one_one_int
        @ A2 )
      = one_one_int ) ).

% prod.neutral_const
thf(fact_9366_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_9367_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_9368_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_9369_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_9370_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_9371_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_9372_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_9373_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_9374_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_9375_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_9376_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_9377_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_9378_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_9379_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_9380_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_9381_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_9382_mod__prod__eq,axiom,
    ! [F: nat > nat,A: nat,A2: set_nat] :
      ( ( modulo_modulo_nat
        @ ( groups708209901874060359at_nat
          @ ^ [I3: nat] : ( modulo_modulo_nat @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_9383_mod__prod__eq,axiom,
    ! [F: nat > int,A: int,A2: set_nat] :
      ( ( modulo_modulo_int
        @ ( groups705719431365010083at_int
          @ ^ [I3: nat] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups705719431365010083at_int @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_9384_mod__prod__eq,axiom,
    ! [F: int > int,A: int,A2: set_int] :
      ( ( modulo_modulo_int
        @ ( groups1705073143266064639nt_int
          @ ^ [I3: int] : ( modulo_modulo_int @ ( F @ I3 ) @ A )
          @ A2 )
        @ A )
      = ( modulo_modulo_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ A ) ) ).

% mod_prod_eq
thf(fact_9385_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > complex,A2: set_real] :
      ( ( ( groups713298508707869441omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9386_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > complex,A2: set_nat] :
      ( ( ( groups6464643781859351333omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9387_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > complex,A2: set_complex] :
      ( ( ( groups3708469109370488835omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9388_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > complex,A2: set_int] :
      ( ( ( groups7440179247065528705omplex @ G @ A2 )
       != one_one_complex )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_complex ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9389_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > real,A2: set_real] :
      ( ( ( groups1681761925125756287l_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9390_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > real,A2: set_nat] :
      ( ( ( groups129246275422532515t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9391_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: complex > real,A2: set_complex] :
      ( ( ( groups766887009212190081x_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A5: complex] :
            ( ( member_complex @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9392_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: int > real,A2: set_int] :
      ( ( ( groups2316167850115554303t_real @ G @ A2 )
       != one_one_real )
     => ~ ! [A5: int] :
            ( ( member_int @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_real ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9393_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: real > rat,A2: set_real] :
      ( ( ( groups4061424788464935467al_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A5: real] :
            ( ( member_real @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9394_prod_Onot__neutral__contains__not__neutral,axiom,
    ! [G: nat > rat,A2: set_nat] :
      ( ( ( groups73079841787564623at_rat @ G @ A2 )
       != one_one_rat )
     => ~ ! [A5: nat] :
            ( ( member_nat @ A5 @ A2 )
           => ( ( G @ A5 )
              = one_one_rat ) ) ) ).

% prod.not_neutral_contains_not_neutral
thf(fact_9395_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ( G @ X5 )
            = one_one_nat ) )
     => ( ( groups708209901874060359at_nat @ G @ A2 )
        = one_one_nat ) ) ).

% prod.neutral
thf(fact_9396_prod_Oneutral,axiom,
    ! [A2: set_nat,G: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ( G @ X5 )
            = one_one_int ) )
     => ( ( groups705719431365010083at_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_9397_prod_Oneutral,axiom,
    ! [A2: set_int,G: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ( G @ X5 )
            = one_one_int ) )
     => ( ( groups1705073143266064639nt_int @ G @ A2 )
        = one_one_int ) ) ).

% prod.neutral
thf(fact_9398_prod__power__distrib,axiom,
    ! [F: nat > nat,A2: set_nat,N2: nat] :
      ( ( power_power_nat @ ( groups708209901874060359at_nat @ F @ A2 ) @ N2 )
      = ( groups708209901874060359at_nat
        @ ^ [X: nat] : ( power_power_nat @ ( F @ X ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_9399_prod__power__distrib,axiom,
    ! [F: nat > int,A2: set_nat,N2: nat] :
      ( ( power_power_int @ ( groups705719431365010083at_int @ F @ A2 ) @ N2 )
      = ( groups705719431365010083at_int
        @ ^ [X: nat] : ( power_power_int @ ( F @ X ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_9400_prod__power__distrib,axiom,
    ! [F: int > int,A2: set_int,N2: nat] :
      ( ( power_power_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ N2 )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : ( power_power_int @ ( F @ X ) @ N2 )
        @ A2 ) ) ).

% prod_power_distrib
thf(fact_9401_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_nat @ zero_zero_nat @ ( F @ X5 ) ) )
     => ( ord_less_eq_nat @ zero_zero_nat @ ( groups708209901874060359at_nat @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_9402_prod__nonneg,axiom,
    ! [A2: set_nat,F: nat > int] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups705719431365010083at_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_9403_prod__nonneg,axiom,
    ! [A2: set_int,F: int > int] :
      ( ! [X5: int] :
          ( ( member_int @ X5 @ A2 )
         => ( ord_less_eq_int @ zero_zero_int @ ( F @ X5 ) ) )
     => ( ord_less_eq_int @ zero_zero_int @ ( groups1705073143266064639nt_int @ F @ A2 ) ) ) ).

% prod_nonneg
thf(fact_9404_prod__mono,axiom,
    ! [A2: set_int,F: int > int,G: int > int] :
      ( ! [I4: int] :
          ( ( member_int @ I4 @ A2 )
         => ( ( ord_less_eq_int @ zero_zero_int @ ( F @ I4 ) )
            & ( ord_less_eq_int @ ( F @ I4 ) @ ( G @ I4 ) ) ) )
     => ( ord_less_eq_int @ ( groups1705073143266064639nt_int @ F @ A2 ) @ ( groups1705073143266064639nt_int @ G @ A2 ) ) ) ).

% prod_mono
thf(fact_9405_ln__neg__is__const,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ X4 @ zero_zero_real )
     => ( ( ln_ln_real @ X4 )
        = ( the_real
          @ ^ [X: real] : $false ) ) ) ).

% ln_neg_is_const
thf(fact_9406_periodic__finite__ex,axiom,
    ! [D: int,P: int > $o] :
      ( ( ord_less_int @ zero_zero_int @ D )
     => ( ! [X5: int,K2: int] :
            ( ( P @ X5 )
            = ( P @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D ) ) ) )
       => ( ( ? [X3: int] : ( P @ X3 ) )
          = ( ? [X: int] :
                ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D ) )
                & ( P @ X ) ) ) ) ) ) ).

% periodic_finite_ex
thf(fact_9407_bset_I3_J,axiom,
    ! [D4: int,T2: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T2 @ one_one_int ) @ B3 )
       => ! [X2: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X2 = T2 )
             => ( ( minus_minus_int @ X2 @ D4 )
                = T2 ) ) ) ) ) ).

% bset(3)
thf(fact_9408_bset_I4_J,axiom,
    ! [D4: int,T2: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T2 @ B3 )
       => ! [X2: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X2 != T2 )
             => ( ( minus_minus_int @ X2 @ D4 )
               != T2 ) ) ) ) ) ).

% bset(4)
thf(fact_9409_bset_I5_J,axiom,
    ! [D4: int,B3: set_int,T2: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X2
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_int @ X2 @ T2 )
           => ( ord_less_int @ ( minus_minus_int @ X2 @ D4 ) @ T2 ) ) ) ) ).

% bset(5)
thf(fact_9410_bset_I7_J,axiom,
    ! [D4: int,T2: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T2 @ B3 )
       => ! [X2: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_int @ T2 @ X2 )
             => ( ord_less_int @ T2 @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ).

% bset(7)
thf(fact_9411_aset_I3_J,axiom,
    ! [D4: int,T2: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T2 @ one_one_int ) @ A2 )
       => ! [X2: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X2 = T2 )
             => ( ( plus_plus_int @ X2 @ D4 )
                = T2 ) ) ) ) ) ).

% aset(3)
thf(fact_9412_aset_I4_J,axiom,
    ! [D4: int,T2: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T2 @ A2 )
       => ! [X2: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( X2 != T2 )
             => ( ( plus_plus_int @ X2 @ D4 )
               != T2 ) ) ) ) ) ).

% aset(4)
thf(fact_9413_aset_I5_J,axiom,
    ! [D4: int,T2: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ T2 @ A2 )
       => ! [X2: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_int @ X2 @ T2 )
             => ( ord_less_int @ ( plus_plus_int @ X2 @ D4 ) @ T2 ) ) ) ) ) ).

% aset(5)
thf(fact_9414_aset_I7_J,axiom,
    ! [D4: int,A2: set_int,T2: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X2
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_int @ T2 @ X2 )
           => ( ord_less_int @ T2 @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ).

% aset(7)
thf(fact_9415_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_9416_bset_I6_J,axiom,
    ! [D4: int,B3: set_int,T2: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ B3 )
                 => ( X2
                   != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_eq_int @ X2 @ T2 )
           => ( ord_less_eq_int @ ( minus_minus_int @ X2 @ D4 ) @ T2 ) ) ) ) ).

% bset(6)
thf(fact_9417_bset_I8_J,axiom,
    ! [D4: int,T2: int,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( minus_minus_int @ T2 @ one_one_int ) @ B3 )
       => ! [X2: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ B3 )
                   => ( X2
                     != ( plus_plus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_eq_int @ T2 @ X2 )
             => ( ord_less_eq_int @ T2 @ ( minus_minus_int @ X2 @ D4 ) ) ) ) ) ) ).

% bset(8)
thf(fact_9418_aset_I6_J,axiom,
    ! [D4: int,T2: int,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ( member_int @ ( plus_plus_int @ T2 @ one_one_int ) @ A2 )
       => ! [X2: int] :
            ( ! [Xa2: int] :
                ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
               => ! [Xb2: int] :
                    ( ( member_int @ Xb2 @ A2 )
                   => ( X2
                     != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
           => ( ( ord_less_eq_int @ X2 @ T2 )
             => ( ord_less_eq_int @ ( plus_plus_int @ X2 @ D4 ) @ T2 ) ) ) ) ) ).

% aset(6)
thf(fact_9419_aset_I8_J,axiom,
    ! [D4: int,A2: set_int,T2: int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ! [X2: int] :
          ( ! [Xa2: int] :
              ( ( member_int @ Xa2 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
             => ! [Xb2: int] :
                  ( ( member_int @ Xb2 @ A2 )
                 => ( X2
                   != ( minus_minus_int @ Xb2 @ Xa2 ) ) ) )
         => ( ( ord_less_eq_int @ T2 @ X2 )
           => ( ord_less_eq_int @ T2 @ ( plus_plus_int @ X2 @ D4 ) ) ) ) ) ).

% aset(8)
thf(fact_9420_cppi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,A2: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ Z3 @ X5 )
           => ( ( P @ X5 )
              = ( P6 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa3: int] :
                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ A2 )
                     => ( X5
                       != ( minus_minus_int @ Xb3 @ Xa3 ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( plus_plus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K2: int] :
                ( ( P6 @ X5 )
                = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X3: int] : ( P @ X3 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y5: int] :
                        ( ( member_int @ Y5 @ A2 )
                        & ( P @ ( minus_minus_int @ Y5 @ X ) ) ) ) ) ) ) ) ) ) ).

% cppi
thf(fact_9421_cpmi,axiom,
    ! [D4: int,P: int > $o,P6: int > $o,B3: set_int] :
      ( ( ord_less_int @ zero_zero_int @ D4 )
     => ( ? [Z3: int] :
          ! [X5: int] :
            ( ( ord_less_int @ X5 @ Z3 )
           => ( ( P @ X5 )
              = ( P6 @ X5 ) ) )
       => ( ! [X5: int] :
              ( ! [Xa3: int] :
                  ( ( member_int @ Xa3 @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                 => ! [Xb3: int] :
                      ( ( member_int @ Xb3 @ B3 )
                     => ( X5
                       != ( plus_plus_int @ Xb3 @ Xa3 ) ) ) )
             => ( ( P @ X5 )
               => ( P @ ( minus_minus_int @ X5 @ D4 ) ) ) )
         => ( ! [X5: int,K2: int] :
                ( ( P6 @ X5 )
                = ( P6 @ ( minus_minus_int @ X5 @ ( times_times_int @ K2 @ D4 ) ) ) )
           => ( ( ? [X3: int] : ( P @ X3 ) )
              = ( ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ( P6 @ X ) )
                | ? [X: int] :
                    ( ( member_int @ X @ ( set_or1266510415728281911st_int @ one_one_int @ D4 ) )
                    & ? [Y5: int] :
                        ( ( member_int @ Y5 @ B3 )
                        & ( P @ ( plus_plus_int @ Y5 @ X ) ) ) ) ) ) ) ) ) ) ).

% cpmi
thf(fact_9422_arccos__def,axiom,
    ( arccos
    = ( ^ [Y5: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ zero_zero_real @ X )
              & ( ord_less_eq_real @ X @ pi )
              & ( ( cos_real @ X )
                = Y5 ) ) ) ) ) ).

% arccos_def
thf(fact_9423_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_9424_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_9425_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_9426_pi__half,axiom,
    ( ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
    = ( the_real
      @ ^ [X: real] :
          ( ( ord_less_eq_real @ zero_zero_real @ X )
          & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
          & ( ( cos_real @ X )
            = zero_zero_real ) ) ) ) ).

% pi_half
thf(fact_9427_pi__def,axiom,
    ( pi
    = ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) )
      @ ( the_real
        @ ^ [X: real] :
            ( ( ord_less_eq_real @ zero_zero_real @ X )
            & ( ord_less_eq_real @ X @ ( numeral_numeral_real @ ( bit0 @ one ) ) )
            & ( ( cos_real @ X )
              = zero_zero_real ) ) ) ) ) ).

% pi_def
thf(fact_9428_Sum__Icc__int,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ N2 )
     => ( ( groups4538972089207619220nt_int
          @ ^ [X: int] : X
          @ ( set_or1266510415728281911st_int @ M @ N2 ) )
        = ( divide_divide_int @ ( minus_minus_int @ ( times_times_int @ N2 @ ( plus_plus_int @ N2 @ one_one_int ) ) @ ( times_times_int @ M @ ( minus_minus_int @ M @ one_one_int ) ) ) @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ).

% Sum_Icc_int
thf(fact_9429_arcsin__def,axiom,
    ( arcsin
    = ( ^ [Y5: real] :
          ( the_real
          @ ^ [X: real] :
              ( ( ord_less_eq_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ X )
              & ( ord_less_eq_real @ X @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
              & ( ( sin_real @ X )
                = Y5 ) ) ) ) ) ).

% arcsin_def
thf(fact_9430_divmod__nat__if,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N: nat] :
          ( if_Pro6206227464963214023at_nat
          @ ( ( N = zero_zero_nat )
            | ( ord_less_nat @ M6 @ N ) )
          @ ( product_Pair_nat_nat @ zero_zero_nat @ M6 )
          @ ( produc2626176000494625587at_nat
            @ ^ [Q5: nat] : ( product_Pair_nat_nat @ ( suc @ Q5 ) )
            @ ( divmod_nat @ ( minus_minus_nat @ M6 @ N ) @ N ) ) ) ) ) ).

% divmod_nat_if
thf(fact_9431_complex__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( times_times_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( plus_plus_real @ ( power_power_real @ ( re @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( power_power_real @ ( im @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ).

% complex_mult_cnj
thf(fact_9432_prod__int__eq,axiom,
    ! [I2: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ J ) ) ) ) ).

% prod_int_eq
thf(fact_9433_prod__int__plus__eq,axiom,
    ! [I2: nat,J: nat] :
      ( ( groups705719431365010083at_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ I2 @ ( plus_plus_nat @ I2 @ J ) ) )
      = ( groups1705073143266064639nt_int
        @ ^ [X: int] : X
        @ ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ I2 ) @ ( semiri1314217659103216013at_int @ ( plus_plus_nat @ I2 @ J ) ) ) ) ) ).

% prod_int_plus_eq
thf(fact_9434_Re__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_gt_0
thf(fact_9435_Re__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_lt_0
thf(fact_9436_Re__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Re_complex_div_ge_0
thf(fact_9437_Re__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Re_complex_div_le_0
thf(fact_9438_Im__complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_gt_0
thf(fact_9439_Im__complex__div__lt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_lt_0
thf(fact_9440_Im__complex__div__ge__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
      = ( ord_less_eq_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ).

% Im_complex_div_ge_0
thf(fact_9441_Im__complex__div__le__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ord_less_eq_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) @ zero_zero_real )
      = ( ord_less_eq_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) @ zero_zero_real ) ) ).

% Im_complex_div_le_0
thf(fact_9442_complex__mod__mult__cnj,axiom,
    ! [Z: complex] :
      ( ( real_V1022390504157884413omplex @ ( times_times_complex @ Z @ ( cnj @ Z ) ) )
      = ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% complex_mod_mult_cnj
thf(fact_9443_complex__div__gt__0,axiom,
    ! [A: complex,B: complex] :
      ( ( ( ord_less_real @ zero_zero_real @ ( re @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( re @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) )
      & ( ( ord_less_real @ zero_zero_real @ ( im @ ( divide1717551699836669952omplex @ A @ B ) ) )
        = ( ord_less_real @ zero_zero_real @ ( im @ ( times_times_complex @ A @ ( cnj @ B ) ) ) ) ) ) ).

% complex_div_gt_0
thf(fact_9444_complex__norm__square,axiom,
    ! [Z: complex] :
      ( ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ Z ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) )
      = ( times_times_complex @ Z @ ( cnj @ Z ) ) ) ).

% complex_norm_square
thf(fact_9445_divmod__nat__def,axiom,
    ( divmod_nat
    = ( ^ [M6: nat,N: nat] : ( product_Pair_nat_nat @ ( divide_divide_nat @ M6 @ N ) @ ( modulo_modulo_nat @ M6 @ N ) ) ) ) ).

% divmod_nat_def
thf(fact_9446_complex__add__cnj,axiom,
    ! [Z: complex] :
      ( ( plus_plus_complex @ Z @ ( cnj @ Z ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ Z ) ) ) ) ).

% complex_add_cnj
thf(fact_9447_complex__diff__cnj,axiom,
    ! [Z: complex] :
      ( ( minus_minus_complex @ Z @ ( cnj @ Z ) )
      = ( times_times_complex @ ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( im @ Z ) ) ) @ imaginary_unit ) ) ).

% complex_diff_cnj
thf(fact_9448_complex__div__cnj,axiom,
    ( divide1717551699836669952omplex
    = ( ^ [A3: complex,B2: complex] : ( divide1717551699836669952omplex @ ( times_times_complex @ A3 @ ( cnj @ B2 ) ) @ ( real_V4546457046886955230omplex @ ( power_power_real @ ( real_V1022390504157884413omplex @ B2 ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ).

% complex_div_cnj
thf(fact_9449_cnj__add__mult__eq__Re,axiom,
    ! [Z: complex,W: complex] :
      ( ( plus_plus_complex @ ( times_times_complex @ Z @ ( cnj @ W ) ) @ ( times_times_complex @ ( cnj @ Z ) @ W ) )
      = ( real_V4546457046886955230omplex @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ ( re @ ( times_times_complex @ Z @ ( cnj @ W ) ) ) ) ) ) ).

% cnj_add_mult_eq_Re
thf(fact_9450_set__encode__def,axiom,
    ( nat_set_encode
    = ( groups3542108847815614940at_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ).

% set_encode_def
thf(fact_9451_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_9452_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_9453_Sum__Ico__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ M @ N2 ) )
      = ( divide_divide_nat @ ( minus_minus_nat @ ( times_times_nat @ N2 @ ( minus_minus_nat @ N2 @ one_one_nat ) ) @ ( times_times_nat @ M @ ( minus_minus_nat @ M @ one_one_nat ) ) ) @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% Sum_Ico_nat
thf(fact_9454_ex__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ? [M6: nat] :
            ( ( ord_less_nat @ M6 @ N2 )
            & ( P @ M6 ) ) )
      = ( ? [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
            & ( P @ X ) ) ) ) ).

% ex_nat_less_eq
thf(fact_9455_all__nat__less__eq,axiom,
    ! [N2: nat,P: nat > $o] :
      ( ( ! [M6: nat] :
            ( ( ord_less_nat @ M6 @ N2 )
           => ( P @ M6 ) ) )
      = ( ! [X: nat] :
            ( ( member_nat @ X @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
           => ( P @ X ) ) ) ) ).

% all_nat_less_eq
thf(fact_9456_atLeastLessThanSuc__atLeastAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ L @ ( suc @ U ) )
      = ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% atLeastLessThanSuc_atLeastAtMost
thf(fact_9457_lessThan__atLeast0,axiom,
    ( set_ord_lessThan_nat
    = ( set_or4665077453230672383an_nat @ zero_zero_nat ) ) ).

% lessThan_atLeast0
thf(fact_9458_prod__Suc__fact,axiom,
    ! [N2: nat] :
      ( ( groups708209901874060359at_nat @ suc @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
      = ( semiri1408675320244567234ct_nat @ N2 ) ) ).

% prod_Suc_fact
thf(fact_9459_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_9460_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_9461_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_9462_Chebyshev__sum__upper__nat,axiom,
    ! [N2: nat,A: nat > nat,B: nat > nat] :
      ( ! [I4: nat,J2: nat] :
          ( ( ord_less_eq_nat @ I4 @ J2 )
         => ( ( ord_less_nat @ J2 @ N2 )
           => ( ord_less_eq_nat @ ( A @ I4 ) @ ( A @ J2 ) ) ) )
     => ( ! [I4: nat,J2: nat] :
            ( ( ord_less_eq_nat @ I4 @ J2 )
           => ( ( ord_less_nat @ J2 @ N2 )
             => ( ord_less_eq_nat @ ( B @ J2 ) @ ( B @ I4 ) ) ) )
       => ( ord_less_eq_nat
          @ ( times_times_nat @ N2
            @ ( groups3542108847815614940at_nat
              @ ^ [I3: nat] : ( times_times_nat @ ( A @ I3 ) @ ( B @ I3 ) )
              @ ( 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_9463_atLeastLessThanPlusOne__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ L @ ( plus_plus_int @ U @ one_one_int ) )
      = ( set_or1266510415728281911st_int @ L @ U ) ) ).

% atLeastLessThanPlusOne_atLeastAtMost_int
thf(fact_9464_int__ge__less__than2__def,axiom,
    ( int_ge_less_than2
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z5: int] :
                ( ( ord_less_eq_int @ D5 @ Z5 )
                & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).

% int_ge_less_than2_def
thf(fact_9465_int__ge__less__than__def,axiom,
    ( int_ge_less_than
    = ( ^ [D5: int] :
          ( collec213857154873943460nt_int
          @ ( produc4947309494688390418_int_o
            @ ^ [Z7: int,Z5: int] :
                ( ( ord_less_eq_int @ D5 @ Z7 )
                & ( ord_less_int @ Z7 @ Z5 ) ) ) ) ) ) ).

% int_ge_less_than_def
thf(fact_9466_upto_Opinduct,axiom,
    ! [A0: int,A12: int,P: int > int > $o] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ A0 @ A12 ) )
     => ( ! [I4: int,J2: int] :
            ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I4 @ J2 ) )
           => ( ( ( ord_less_eq_int @ I4 @ J2 )
               => ( P @ ( plus_plus_int @ I4 @ one_one_int ) @ J2 ) )
             => ( P @ I4 @ J2 ) ) )
       => ( P @ A0 @ A12 ) ) ) ).

% upto.pinduct
thf(fact_9467_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_9468_or__int__rec,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( 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_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_int_rec
thf(fact_9469_or__nonnegative__int__iff,axiom,
    ! [K: int,L: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ K @ L ) )
      = ( ( ord_less_eq_int @ zero_zero_int @ K )
        & ( ord_less_eq_int @ zero_zero_int @ L ) ) ) ).

% or_nonnegative_int_iff
thf(fact_9470_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_9471_or__minus__numerals_I6_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).

% or_minus_numerals(6)
thf(fact_9472_or__minus__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) ) ).

% or_minus_numerals(2)
thf(fact_9473_or__minus__minus__numerals,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).

% or_minus_minus_numerals
thf(fact_9474_and__minus__minus__numerals,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se1409905431419307370or_int @ ( minus_minus_int @ ( numeral_numeral_int @ M ) @ one_one_int ) @ ( minus_minus_int @ ( numeral_numeral_int @ N2 ) @ one_one_int ) ) ) ) ).

% and_minus_minus_numerals
thf(fact_9475_divmod__integer_H__def,axiom,
    ( unique3479559517661332726nteger
    = ( ^ [M6: num,N: num] : ( produc1086072967326762835nteger @ ( divide6298287555418463151nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) @ ( modulo364778990260209775nteger @ ( numera6620942414471956472nteger @ M6 ) @ ( numera6620942414471956472nteger @ N ) ) ) ) ) ).

% divmod_integer'_def
thf(fact_9476_bit__or__int__iff,axiom,
    ! [K: int,L: int,N2: nat] :
      ( ( bit_se1146084159140164899it_int @ ( bit_se1409905431419307370or_int @ K @ L ) @ N2 )
      = ( ( bit_se1146084159140164899it_int @ K @ N2 )
        | ( bit_se1146084159140164899it_int @ L @ N2 ) ) ) ).

% bit_or_int_iff
thf(fact_9477_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_9478_less__eq__integer__code_I1_J,axiom,
    ord_le3102999989581377725nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ).

% less_eq_integer_code(1)
thf(fact_9479_OR__lower,axiom,
    ! [X4: int,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_eq_int @ zero_zero_int @ Y )
       => ( ord_less_eq_int @ zero_zero_int @ ( bit_se1409905431419307370or_int @ X4 @ Y ) ) ) ) ).

% OR_lower
thf(fact_9480_or__greater__eq,axiom,
    ! [L: int,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ L )
     => ( ord_less_eq_int @ K @ ( bit_se1409905431419307370or_int @ K @ L ) ) ) ).

% or_greater_eq
thf(fact_9481_plus__and__or,axiom,
    ! [X4: int,Y: int] :
      ( ( plus_plus_int @ ( bit_se725231765392027082nd_int @ X4 @ Y ) @ ( bit_se1409905431419307370or_int @ X4 @ Y ) )
      = ( plus_plus_int @ X4 @ Y ) ) ).

% plus_and_or
thf(fact_9482_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_9483_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_9484_or__int__def,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] : ( bit_ri7919022796975470100ot_int @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ ( bit_ri7919022796975470100ot_int @ L2 ) ) ) ) ) ).

% or_int_def
thf(fact_9485_or__not__numerals_I1_J,axiom,
    ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
    = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(1)
thf(fact_9486_xor__int__def,axiom,
    ( bit_se6526347334894502574or_int
    = ( ^ [K3: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se725231765392027082nd_int @ K3 @ ( bit_ri7919022796975470100ot_int @ L2 ) ) @ ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ K3 ) @ L2 ) ) ) ) ).

% xor_int_def
thf(fact_9487_concat__bit__def,axiom,
    ( bit_concat_bit
    = ( ^ [N: nat,K3: int,L2: int] : ( bit_se1409905431419307370or_int @ ( bit_se2923211474154528505it_int @ N @ K3 ) @ ( bit_se545348938243370406it_int @ N @ L2 ) ) ) ) ).

% concat_bit_def
thf(fact_9488_set__bit__int__def,axiom,
    ( bit_se7879613467334960850it_int
    = ( ^ [N: nat,K3: int] : ( bit_se1409905431419307370or_int @ K3 @ ( bit_se545348938243370406it_int @ N @ one_one_int ) ) ) ) ).

% set_bit_int_def
thf(fact_9489_one__natural_Orsp,axiom,
    one_one_nat = one_one_nat ).

% one_natural.rsp
thf(fact_9490_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_9491_or__not__numerals_I4_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ one_one_int ) ) ).

% or_not_numerals(4)
thf(fact_9492_or__not__numerals_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).

% or_not_numerals(2)
thf(fact_9493_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_9494_or__not__numerals_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) ) ).

% or_not_numerals(3)
thf(fact_9495_or__not__numerals_I7_J,axiom,
    ! [M: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ one_one_int ) )
      = ( bit_ri7919022796975470100ot_int @ zero_zero_int ) ) ).

% or_not_numerals(7)
thf(fact_9496_or__not__numerals_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% or_not_numerals(6)
thf(fact_9497_OR__upper,axiom,
    ! [X4: int,N2: nat,Y: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ X4 )
     => ( ( ord_less_int @ X4 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
       => ( ( ord_less_int @ Y @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) )
         => ( ord_less_int @ ( bit_se1409905431419307370or_int @ X4 @ Y ) @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N2 ) ) ) ) ) ).

% OR_upper
thf(fact_9498_or__not__numerals_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit0 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(5)
thf(fact_9499_or__not__numerals_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(9)
thf(fact_9500_or__not__numerals_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ ( bit1 @ M ) ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( plus_plus_int @ one_one_int @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ) ).

% or_not_numerals(8)
thf(fact_9501_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_9502_or__minus__numerals_I1_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(1)
thf(fact_9503_or__minus__numerals_I5_J,axiom,
    ! [N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ one_one_int )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ one @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(5)
thf(fact_9504_or__nat__numerals_I4_J,axiom,
    ! [X4: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit1 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% or_nat_numerals(4)
thf(fact_9505_or__nat__numerals_I2_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit1 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(2)
thf(fact_9506_or__nat__numerals_I3_J,axiom,
    ! [X4: num] :
      ( ( bit_se1412395901928357646or_nat @ ( numeral_numeral_nat @ ( bit0 @ X4 ) ) @ ( suc @ zero_zero_nat ) )
      = ( numeral_numeral_nat @ ( bit1 @ X4 ) ) ) ).

% or_nat_numerals(3)
thf(fact_9507_or__nat__numerals_I1_J,axiom,
    ! [Y: num] :
      ( ( bit_se1412395901928357646or_nat @ ( suc @ zero_zero_nat ) @ ( numeral_numeral_nat @ ( bit0 @ Y ) ) )
      = ( numeral_numeral_nat @ ( bit1 @ Y ) ) ) ).

% or_nat_numerals(1)
thf(fact_9508_or__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).

% or_minus_numerals(8)
thf(fact_9509_or__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bit0 @ N2 ) ) ) ) ) ).

% or_minus_numerals(4)
thf(fact_9510_or__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(3)
thf(fact_9511_or__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se1409905431419307370or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ ( bitM @ N2 ) ) ) ) ) ).

% or_minus_numerals(7)
thf(fact_9512_modulo__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( modulo364778990260209775nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( code_integer_of_int @ ( modulo_modulo_int @ Xa @ X4 ) ) ) ).

% modulo_integer.abs_eq
thf(fact_9513_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_9514_less__integer__code_I1_J,axiom,
    ~ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ zero_z3403309356797280102nteger ) ).

% less_integer_code(1)
thf(fact_9515_less__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( ord_le6747313008572928689nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( ord_less_int @ Xa @ X4 ) ) ).

% less_integer.abs_eq
thf(fact_9516_or__not__num__neg_Osimps_I1_J,axiom,
    ( ( bit_or_not_num_neg @ one @ one )
    = one ) ).

% or_not_num_neg.simps(1)
thf(fact_9517_less__eq__integer_Oabs__eq,axiom,
    ! [Xa: int,X4: int] :
      ( ( ord_le3102999989581377725nteger @ ( code_integer_of_int @ Xa ) @ ( code_integer_of_int @ X4 ) )
      = ( ord_less_eq_int @ Xa @ X4 ) ) ).

% less_eq_integer.abs_eq
thf(fact_9518_set__bit__nat__def,axiom,
    ( bit_se7882103937844011126it_nat
    = ( ^ [M6: nat,N: nat] : ( bit_se1412395901928357646or_nat @ N @ ( bit_se547839408752420682it_nat @ M6 @ one_one_nat ) ) ) ) ).

% set_bit_nat_def
thf(fact_9519_or__not__num__neg_Osimps_I4_J,axiom,
    ! [N2: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ one )
      = ( bit0 @ one ) ) ).

% or_not_num_neg.simps(4)
thf(fact_9520_or__not__num__neg_Osimps_I6_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit1 @ M ) )
      = ( bit0 @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(6)
thf(fact_9521_or__not__num__neg_Osimps_I3_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit1 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(3)
thf(fact_9522_or__not__num__neg_Osimps_I7_J,axiom,
    ! [N2: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ one )
      = one ) ).

% or_not_num_neg.simps(7)
thf(fact_9523_or__not__num__neg_Osimps_I5_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit0 @ N2 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(5)
thf(fact_9524_or__not__num__neg_Osimps_I9_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit1 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(9)
thf(fact_9525_or__nat__def,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N: nat] : ( nat2 @ ( bit_se1409905431419307370or_int @ ( semiri1314217659103216013at_int @ M6 ) @ ( semiri1314217659103216013at_int @ N ) ) ) ) ) ).

% or_nat_def
thf(fact_9526_or__not__num__neg_Osimps_I2_J,axiom,
    ! [M: num] :
      ( ( bit_or_not_num_neg @ one @ ( bit0 @ M ) )
      = ( bit1 @ M ) ) ).

% or_not_num_neg.simps(2)
thf(fact_9527_or__not__num__neg_Osimps_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_or_not_num_neg @ ( bit1 @ N2 ) @ ( bit0 @ M ) )
      = ( bitM @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ).

% or_not_num_neg.simps(8)
thf(fact_9528_or__not__num__neg_Oelims,axiom,
    ! [X4: num,Xa: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X4 @ Xa )
        = Y )
     => ( ( ( X4 = one )
         => ( ( Xa = one )
           => ( Y != one ) ) )
       => ( ( ( X4 = one )
           => ! [M5: num] :
                ( ( Xa
                  = ( bit0 @ M5 ) )
               => ( Y
                 != ( bit1 @ M5 ) ) ) )
         => ( ( ( X4 = one )
             => ! [M5: num] :
                  ( ( Xa
                    = ( bit1 @ M5 ) )
                 => ( Y
                   != ( bit1 @ M5 ) ) ) )
           => ( ( ? [N3: num] :
                    ( X4
                    = ( bit0 @ N3 ) )
               => ( ( Xa = one )
                 => ( Y
                   != ( bit0 @ one ) ) ) )
             => ( ! [N3: num] :
                    ( ( X4
                      = ( bit0 @ N3 ) )
                   => ! [M5: num] :
                        ( ( Xa
                          = ( bit0 @ M5 ) )
                       => ( Y
                         != ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) )
               => ( ! [N3: num] :
                      ( ( X4
                        = ( bit0 @ N3 ) )
                     => ! [M5: num] :
                          ( ( Xa
                            = ( bit1 @ M5 ) )
                         => ( Y
                           != ( bit0 @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) )
                 => ( ( ? [N3: num] :
                          ( X4
                          = ( bit1 @ N3 ) )
                     => ( ( Xa = one )
                       => ( Y != one ) ) )
                   => ( ! [N3: num] :
                          ( ( X4
                            = ( bit1 @ N3 ) )
                         => ! [M5: num] :
                              ( ( Xa
                                = ( bit0 @ M5 ) )
                             => ( Y
                               != ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) )
                     => ~ ! [N3: num] :
                            ( ( X4
                              = ( bit1 @ N3 ) )
                           => ! [M5: num] :
                                ( ( Xa
                                  = ( bit1 @ M5 ) )
                               => ( Y
                                 != ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.elims
thf(fact_9529_numeral__or__not__num__eq,axiom,
    ! [M: num,N2: num] :
      ( ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) )
      = ( uminus_uminus_int @ ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) ) ) ) ).

% numeral_or_not_num_eq
thf(fact_9530_int__numeral__not__or__num__neg,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ N2 @ M ) ) ) ) ).

% int_numeral_not_or_num_neg
thf(fact_9531_int__numeral__or__not__num__neg,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se1409905431419307370or_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit_or_not_num_neg @ M @ N2 ) ) ) ) ).

% int_numeral_or_not_num_neg
thf(fact_9532_floor__real__def,axiom,
    ( archim6058952711729229775r_real
    = ( ^ [X: real] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_real @ ( ring_1_of_int_real @ Z5 ) @ X )
              & ( ord_less_real @ X @ ( ring_1_of_int_real @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_real_def
thf(fact_9533_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_9534_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_9535_or__nat__rec,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N: nat] :
          ( plus_plus_nat
          @ ( zero_n2687167440665602831ol_nat
            @ ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ M6 )
              | ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( bit_se1412395901928357646or_nat @ ( divide_divide_nat @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ).

% or_nat_rec
thf(fact_9536_or__not__num__neg_Opelims,axiom,
    ! [X4: num,Xa: num,Y: num] :
      ( ( ( bit_or_not_num_neg @ X4 @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
       => ( ( ( X4 = one )
           => ( ( Xa = one )
             => ( ( Y = one )
               => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [M5: num] :
                  ( ( Xa
                    = ( bit0 @ M5 ) )
                 => ( ( Y
                      = ( bit1 @ M5 ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit0 @ M5 ) ) ) ) ) )
           => ( ( ( X4 = one )
               => ! [M5: num] :
                    ( ( Xa
                      = ( bit1 @ M5 ) )
                   => ( ( Y
                        = ( bit1 @ M5 ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ one @ ( bit1 @ M5 ) ) ) ) ) )
             => ( ! [N3: num] :
                    ( ( X4
                      = ( bit0 @ N3 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( bit0 @ one ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ one ) ) ) ) )
               => ( ! [N3: num] :
                      ( ( X4
                        = ( bit0 @ N3 ) )
                     => ! [M5: num] :
                          ( ( Xa
                            = ( bit0 @ M5 ) )
                         => ( ( Y
                              = ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
                 => ( ! [N3: num] :
                        ( ( X4
                          = ( bit0 @ N3 ) )
                       => ! [M5: num] :
                            ( ( Xa
                              = ( bit1 @ M5 ) )
                           => ( ( Y
                                = ( bit0 @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit0 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) )
                   => ( ! [N3: num] :
                          ( ( X4
                            = ( bit1 @ N3 ) )
                         => ( ( Xa = one )
                           => ( ( Y = one )
                             => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ one ) ) ) ) )
                     => ( ! [N3: num] :
                            ( ( X4
                              = ( bit1 @ N3 ) )
                           => ! [M5: num] :
                                ( ( Xa
                                  = ( bit0 @ M5 ) )
                               => ( ( Y
                                    = ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit0 @ M5 ) ) ) ) ) )
                       => ~ ! [N3: num] :
                              ( ( X4
                                = ( bit1 @ N3 ) )
                             => ! [M5: num] :
                                  ( ( Xa
                                    = ( bit1 @ M5 ) )
                                 => ( ( Y
                                      = ( bitM @ ( bit_or_not_num_neg @ N3 @ M5 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_or3848514188828904588eg_rel @ ( product_Pair_num_num @ ( bit1 @ N3 ) @ ( bit1 @ M5 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% or_not_num_neg.pelims
thf(fact_9537_or__int__unfold,axiom,
    ( bit_se1409905431419307370or_int
    = ( ^ [K3: int,L2: int] :
          ( if_int
          @ ( ( K3
              = ( uminus_uminus_int @ one_one_int ) )
            | ( L2
              = ( uminus_uminus_int @ one_one_int ) ) )
          @ ( uminus_uminus_int @ one_one_int )
          @ ( if_int @ ( K3 = zero_zero_int ) @ L2 @ ( if_int @ ( L2 = zero_zero_int ) @ K3 @ ( plus_plus_int @ ( ord_max_int @ ( modulo_modulo_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( modulo_modulo_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se1409905431419307370or_int @ ( divide_divide_int @ K3 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ L2 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% or_int_unfold
thf(fact_9538_max__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ Q3 @ zero_z5237406670263579293d_enat )
      = Q3 ) ).

% max_enat_simps(2)
thf(fact_9539_max__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_ma741700101516333627d_enat @ zero_z5237406670263579293d_enat @ Q3 )
      = Q3 ) ).

% max_enat_simps(3)
thf(fact_9540_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_9541_max__0R,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ N2 @ zero_zero_nat )
      = N2 ) ).

% max_0R
thf(fact_9542_max__0L,axiom,
    ! [N2: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ N2 )
      = N2 ) ).

% max_0L
thf(fact_9543_max__nat_Oright__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ A @ zero_zero_nat )
      = A ) ).

% max_nat.right_neutral
thf(fact_9544_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_9545_max__nat_Oleft__neutral,axiom,
    ! [A: nat] :
      ( ( ord_max_nat @ zero_zero_nat @ A )
      = A ) ).

% max_nat.left_neutral
thf(fact_9546_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_9547_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_9548_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_9549_nat__add__max__left,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( plus_plus_nat @ ( ord_max_nat @ M @ N2 ) @ Q3 )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ Q3 ) @ ( plus_plus_nat @ N2 @ Q3 ) ) ) ).

% nat_add_max_left
thf(fact_9550_nat__add__max__right,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( plus_plus_nat @ M @ ( ord_max_nat @ N2 @ Q3 ) )
      = ( ord_max_nat @ ( plus_plus_nat @ M @ N2 ) @ ( plus_plus_nat @ M @ Q3 ) ) ) ).

% nat_add_max_right
thf(fact_9551_nat__mult__max__right,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_max_nat @ N2 @ Q3 ) )
      = ( ord_max_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_max_right
thf(fact_9552_nat__mult__max__left,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_max_nat @ M @ N2 ) @ Q3 )
      = ( ord_max_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N2 @ Q3 ) ) ) ).

% nat_mult_max_left
thf(fact_9553_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_9554_max__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_max_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ M3 @ N2 ) )
        @ M ) ) ).

% max_Suc2
thf(fact_9555_max__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_max_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ ( suc @ N2 )
        @ ^ [M3: nat] : ( suc @ ( ord_max_nat @ N2 @ M3 ) )
        @ M ) ) ).

% max_Suc1
thf(fact_9556_or__nat__unfold,axiom,
    ( bit_se1412395901928357646or_nat
    = ( ^ [M6: nat,N: nat] : ( if_nat @ ( M6 = zero_zero_nat ) @ N @ ( if_nat @ ( N = zero_zero_nat ) @ M6 @ ( plus_plus_nat @ ( ord_max_nat @ ( modulo_modulo_nat @ M6 @ ( 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 @ M6 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ ( divide_divide_nat @ N @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% or_nat_unfold
thf(fact_9557_floor__rat__def,axiom,
    ( archim3151403230148437115or_rat
    = ( ^ [X: rat] :
          ( the_int
          @ ^ [Z5: int] :
              ( ( ord_less_eq_rat @ ( ring_1_of_int_rat @ Z5 ) @ X )
              & ( ord_less_rat @ X @ ( ring_1_of_int_rat @ ( plus_plus_int @ Z5 @ one_one_int ) ) ) ) ) ) ) ).

% floor_rat_def
thf(fact_9558_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_9559_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_9560_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_9561_obtain__pos__sum,axiom,
    ! [R3: rat] :
      ( ( ord_less_rat @ zero_zero_rat @ R3 )
     => ~ ! [S3: rat] :
            ( ( ord_less_rat @ zero_zero_rat @ S3 )
           => ! [T3: rat] :
                ( ( ord_less_rat @ zero_zero_rat @ T3 )
               => ( R3
                 != ( plus_plus_rat @ S3 @ T3 ) ) ) ) ) ).

% obtain_pos_sum
thf(fact_9562_less__eq__rat__def,axiom,
    ( ord_less_eq_rat
    = ( ^ [X: rat,Y5: rat] :
          ( ( ord_less_rat @ X @ Y5 )
          | ( X = Y5 ) ) ) ) ).

% less_eq_rat_def
thf(fact_9563_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_9564_pred__def,axiom,
    ( pred
    = ( case_nat_nat @ zero_zero_nat
      @ ^ [X24: nat] : X24 ) ) ).

% pred_def
thf(fact_9565_bit__cut__integer__code,axiom,
    ( code_bit_cut_integer
    = ( ^ [K3: code_integer] :
          ( if_Pro5737122678794959658eger_o @ ( K3 = zero_z3403309356797280102nteger ) @ ( produc6677183202524767010eger_o @ zero_z3403309356797280102nteger @ $false )
          @ ( produc9125791028180074456eger_o
            @ ^ [R5: code_integer,S4: code_integer] : ( produc6677183202524767010eger_o @ ( if_Code_integer @ ( ord_le6747313008572928689nteger @ zero_z3403309356797280102nteger @ K3 ) @ R5 @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ S4 ) ) @ ( S4 = one_one_Code_integer ) )
            @ ( code_divmod_abs @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% bit_cut_integer_code
thf(fact_9566_normalize__negative,axiom,
    ! [Q3: int,P2: int] :
      ( ( ord_less_int @ Q3 @ zero_zero_int )
     => ( ( normalize @ ( product_Pair_int_int @ P2 @ Q3 ) )
        = ( normalize @ ( product_Pair_int_int @ ( uminus_uminus_int @ P2 ) @ ( uminus_uminus_int @ Q3 ) ) ) ) ) ).

% normalize_negative
thf(fact_9567_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_9568_prod__decode__aux_Oelims,axiom,
    ! [X4: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X4 @ Xa )
        = Y )
     => ( ( ( ord_less_eq_nat @ Xa @ X4 )
         => ( Y
            = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
         => ( Y
            = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) ) ) ).

% prod_decode_aux.elims
thf(fact_9569_normalize__denom__pos,axiom,
    ! [R3: product_prod_int_int,P2: int,Q3: int] :
      ( ( ( normalize @ R3 )
        = ( product_Pair_int_int @ P2 @ Q3 ) )
     => ( ord_less_int @ zero_zero_int @ Q3 ) ) ).

% normalize_denom_pos
thf(fact_9570_prod__decode__aux_Osimps,axiom,
    ( nat_prod_decode_aux
    = ( ^ [K3: nat,M6: nat] : ( if_Pro6206227464963214023at_nat @ ( ord_less_eq_nat @ M6 @ K3 ) @ ( product_Pair_nat_nat @ M6 @ ( minus_minus_nat @ K3 @ M6 ) ) @ ( nat_prod_decode_aux @ ( suc @ K3 ) @ ( minus_minus_nat @ M6 @ ( suc @ K3 ) ) ) ) ) ) ).

% prod_decode_aux.simps
thf(fact_9571_prod__decode__aux_Opelims,axiom,
    ! [X4: nat,Xa: nat,Y: product_prod_nat_nat] :
      ( ( ( nat_prod_decode_aux @ X4 @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_nat @ Xa @ X4 )
               => ( Y
                  = ( product_Pair_nat_nat @ Xa @ ( minus_minus_nat @ X4 @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_nat @ Xa @ X4 )
               => ( Y
                  = ( nat_prod_decode_aux @ ( suc @ X4 ) @ ( minus_minus_nat @ Xa @ ( suc @ X4 ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ nat_pr5047031295181774490ux_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).

% prod_decode_aux.pelims
thf(fact_9572_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,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ L2 @ S4 ) ) )
                @ ( 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,S4: code_integer] : ( if_Pro6119634080678213985nteger @ ( S4 = zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( uminus1351360451143612070nteger @ R5 ) @ zero_z3403309356797280102nteger ) @ ( produc1086072967326762835nteger @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ R5 ) @ one_one_Code_integer ) @ ( minus_8373710615458151222nteger @ ( uminus1351360451143612070nteger @ L2 ) @ S4 ) ) )
                    @ ( code_divmod_abs @ K3 @ L2 ) ) ) ) ) ) ) ) ) ).

% divmod_integer_code
thf(fact_9573_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_9574_finite__atLeastAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or1269000886237332187st_nat @ L @ U ) ) ).

% finite_atLeastAtMost
thf(fact_9575_finite__atLeastLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or4665077453230672383an_nat @ L @ U ) ) ).

% finite_atLeastLessThan
thf(fact_9576_finite__lessThan,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_lessThan_nat @ K ) ) ).

% finite_lessThan
thf(fact_9577_finite__atMost,axiom,
    ! [K: nat] : ( finite_finite_nat @ ( set_ord_atMost_nat @ K ) ) ).

% finite_atMost
thf(fact_9578_fst__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_fst_nat_nat @ ( divmod_nat @ M @ N2 ) )
      = ( divide_divide_nat @ M @ N2 ) ) ).

% fst_divmod_nat
thf(fact_9579_finite__nat__set__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [N9: set_nat] :
        ? [M6: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N9 )
         => ( ord_less_eq_nat @ X @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded_le
thf(fact_9580_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_9581_finite__M__bounded__by__nat,axiom,
    ! [P: nat > $o,I2: nat] :
      ( finite_finite_nat
      @ ( collect_nat
        @ ^ [K3: nat] :
            ( ( P @ K3 )
            & ( ord_less_nat @ K3 @ I2 ) ) ) ) ).

% finite_M_bounded_by_nat
thf(fact_9582_bounded__nat__set__is__finite,axiom,
    ! [N4: set_nat,N2: nat] :
      ( ! [X5: nat] :
          ( ( member_nat @ X5 @ N4 )
         => ( ord_less_nat @ X5 @ N2 ) )
     => ( finite_finite_nat @ N4 ) ) ).

% bounded_nat_set_is_finite
thf(fact_9583_finite__nat__set__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [N9: set_nat] :
        ? [M6: nat] :
        ! [X: nat] :
          ( ( member_nat @ X @ N9 )
         => ( ord_less_nat @ X @ M6 ) ) ) ) ).

% finite_nat_set_iff_bounded
thf(fact_9584_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_9585_subset__eq__atLeast0__atMost__finite,axiom,
    ! [N4: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N4 @ ( set_or1269000886237332187st_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N4 ) ) ).

% subset_eq_atLeast0_atMost_finite
thf(fact_9586_subset__eq__atLeast0__lessThan__finite,axiom,
    ! [N4: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( finite_finite_nat @ N4 ) ) ).

% subset_eq_atLeast0_lessThan_finite
thf(fact_9587_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_9588_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_9589_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_9590_finite__atLeastAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or1266510415728281911st_int @ L @ U ) ) ).

% finite_atLeastAtMost_int
thf(fact_9591_finite__atLeastLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ L @ U ) ) ).

% finite_atLeastLessThan_int
thf(fact_9592_finite__interval__int1,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int1
thf(fact_9593_finite__interval__int4,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B ) ) ) ) ).

% finite_interval_int4
thf(fact_9594_finite__interval__int2,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_eq_int @ A @ I3 )
            & ( ord_less_int @ I3 @ B ) ) ) ) ).

% finite_interval_int2
thf(fact_9595_finite__interval__int3,axiom,
    ! [A: int,B: int] :
      ( finite_finite_int
      @ ( collect_int
        @ ^ [I3: int] :
            ( ( ord_less_int @ A @ I3 )
            & ( ord_less_eq_int @ I3 @ B ) ) ) ) ).

% finite_interval_int3
thf(fact_9596_finite__nth__roots,axiom,
    ! [N2: nat,C: complex] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( finite3207457112153483333omplex
        @ ( collect_complex
          @ ^ [Z5: complex] :
              ( ( power_power_complex @ Z5 @ N2 )
              = C ) ) ) ) ).

% finite_nth_roots
thf(fact_9597_finite__atLeastZeroLessThan__int,axiom,
    ! [U: int] : ( finite_finite_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) ) ).

% finite_atLeastZeroLessThan_int
thf(fact_9598_finite__nat__iff__bounded__le,axiom,
    ( finite_finite_nat
    = ( ^ [S5: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_atMost_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded_le
thf(fact_9599_finite__nat__iff__bounded,axiom,
    ( finite_finite_nat
    = ( ^ [S5: set_nat] :
        ? [K3: nat] : ( ord_less_eq_set_nat @ S5 @ ( set_ord_lessThan_nat @ K3 ) ) ) ) ).

% finite_nat_iff_bounded
thf(fact_9600_finite__nat__bounded,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [K2: nat] : ( ord_less_eq_set_nat @ S2 @ ( set_ord_lessThan_nat @ K2 ) ) ) ).

% finite_nat_bounded
thf(fact_9601_infinite__int__iff__unbounded__le,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M6: int] :
          ? [N: int] :
            ( ( ord_less_eq_int @ M6 @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded_le
thf(fact_9602_infinite__int__iff__unbounded,axiom,
    ! [S2: set_int] :
      ( ( ~ ( finite_finite_int @ S2 ) )
      = ( ! [M6: int] :
          ? [N: int] :
            ( ( ord_less_int @ M6 @ ( abs_abs_int @ N ) )
            & ( member_int @ N @ S2 ) ) ) ) ).

% infinite_int_iff_unbounded
thf(fact_9603_unbounded__k__infinite,axiom,
    ! [K: nat,S2: set_nat] :
      ( ! [M5: nat] :
          ( ( ord_less_nat @ K @ M5 )
         => ? [N6: nat] :
              ( ( ord_less_nat @ M5 @ N6 )
              & ( member_nat @ N6 @ S2 ) ) )
     => ~ ( finite_finite_nat @ S2 ) ) ).

% unbounded_k_infinite
thf(fact_9604_infinite__nat__iff__unbounded,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M6: nat] :
          ? [N: nat] :
            ( ( ord_less_nat @ M6 @ N )
            & ( member_nat @ N @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded
thf(fact_9605_infinite__nat__iff__unbounded__le,axiom,
    ! [S2: set_nat] :
      ( ( ~ ( finite_finite_nat @ S2 ) )
      = ( ! [M6: nat] :
          ? [N: nat] :
            ( ( ord_less_eq_nat @ M6 @ N )
            & ( member_nat @ N @ S2 ) ) ) ) ).

% infinite_nat_iff_unbounded_le
thf(fact_9606_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
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = one_one_complex ) )
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = C ) ) ) ) ) ).

% bij_betw_nth_root_unity
thf(fact_9607_real__root__Suc__0,axiom,
    ! [X4: real] :
      ( ( root @ ( suc @ zero_zero_nat ) @ X4 )
      = X4 ) ).

% real_root_Suc_0
thf(fact_9608_real__root__eq__iff,axiom,
    ! [N2: nat,X4: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X4 )
          = ( root @ N2 @ Y ) )
        = ( X4 = Y ) ) ) ).

% real_root_eq_iff
thf(fact_9609_real__root__eq__0__iff,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X4 )
          = zero_zero_real )
        = ( X4 = zero_zero_real ) ) ) ).

% real_root_eq_0_iff
thf(fact_9610_real__root__less__iff,axiom,
    ! [N2: nat,X4: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X4 ) @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ X4 @ Y ) ) ) ).

% real_root_less_iff
thf(fact_9611_real__root__le__iff,axiom,
    ! [N2: nat,X4: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X4 ) @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ X4 @ Y ) ) ) ).

% real_root_le_iff
thf(fact_9612_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_9613_real__root__eq__1__iff,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( root @ N2 @ X4 )
          = one_one_real )
        = ( X4 = one_one_real ) ) ) ).

% real_root_eq_1_iff
thf(fact_9614_real__root__gt__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ zero_zero_real @ Y ) ) ) ).

% real_root_gt_0_iff
thf(fact_9615_real__root__lt__0__iff,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X4 ) @ zero_zero_real )
        = ( ord_less_real @ X4 @ zero_zero_real ) ) ) ).

% real_root_lt_0_iff
thf(fact_9616_real__root__ge__0__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ zero_zero_real @ Y ) ) ) ).

% real_root_ge_0_iff
thf(fact_9617_real__root__le__0__iff,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X4 ) @ zero_zero_real )
        = ( ord_less_eq_real @ X4 @ zero_zero_real ) ) ) ).

% real_root_le_0_iff
thf(fact_9618_real__root__gt__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_real @ one_one_real @ Y ) ) ) ).

% real_root_gt_1_iff
thf(fact_9619_real__root__lt__1__iff,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ ( root @ N2 @ X4 ) @ one_one_real )
        = ( ord_less_real @ X4 @ one_one_real ) ) ) ).

% real_root_lt_1_iff
thf(fact_9620_real__root__ge__1__iff,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ one_one_real @ ( root @ N2 @ Y ) )
        = ( ord_less_eq_real @ one_one_real @ Y ) ) ) ).

% real_root_ge_1_iff
thf(fact_9621_real__root__le__1__iff,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ ( root @ N2 @ X4 ) @ one_one_real )
        = ( ord_less_eq_real @ X4 @ one_one_real ) ) ) ).

% real_root_le_1_iff
thf(fact_9622_real__root__pow__pos2,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( power_power_real @ ( root @ N2 @ X4 ) @ N2 )
          = X4 ) ) ) ).

% real_root_pow_pos2
thf(fact_9623_real__root__pos__pos__le,axiom,
    ! [X4: real,N2: nat] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X4 ) ) ) ).

% real_root_pos_pos_le
thf(fact_9624_real__root__less__mono,axiom,
    ! [N2: nat,X4: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ X4 @ Y )
       => ( ord_less_real @ ( root @ N2 @ X4 ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_less_mono
thf(fact_9625_real__root__le__mono,axiom,
    ! [N2: nat,X4: real,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ X4 @ Y )
       => ( ord_less_eq_real @ ( root @ N2 @ X4 ) @ ( root @ N2 @ Y ) ) ) ) ).

% real_root_le_mono
thf(fact_9626_real__root__power,axiom,
    ! [N2: nat,X4: real,K: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X4 @ K ) )
        = ( power_power_real @ ( root @ N2 @ X4 ) @ K ) ) ) ).

% real_root_power
thf(fact_9627_real__root__abs,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( abs_abs_real @ X4 ) )
        = ( abs_abs_real @ ( root @ N2 @ X4 ) ) ) ) ).

% real_root_abs
thf(fact_9628_sgn__root,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( sgn_sgn_real @ ( root @ N2 @ X4 ) )
        = ( sgn_sgn_real @ X4 ) ) ) ).

% sgn_root
thf(fact_9629_real__root__gt__zero,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_real @ zero_zero_real @ ( root @ N2 @ X4 ) ) ) ) ).

% real_root_gt_zero
thf(fact_9630_real__root__strict__decreasing,axiom,
    ! [N2: nat,N4: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ( ord_less_real @ one_one_real @ X4 )
         => ( ord_less_real @ ( root @ N4 @ X4 ) @ ( root @ N2 @ X4 ) ) ) ) ) ).

% real_root_strict_decreasing
thf(fact_9631_sqrt__def,axiom,
    ( sqrt
    = ( root @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ).

% sqrt_def
thf(fact_9632_root__abs__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( abs_abs_real @ ( root @ N2 @ ( power_power_real @ Y @ N2 ) ) )
        = ( abs_abs_real @ Y ) ) ) ).

% root_abs_power
thf(fact_9633_real__root__pos__pos,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ord_less_eq_real @ zero_zero_real @ ( root @ N2 @ X4 ) ) ) ) ).

% real_root_pos_pos
thf(fact_9634_real__root__strict__increasing,axiom,
    ! [N2: nat,N4: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_nat @ N2 @ N4 )
       => ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( ( ord_less_real @ X4 @ one_one_real )
           => ( ord_less_real @ ( root @ N2 @ X4 ) @ ( root @ N4 @ X4 ) ) ) ) ) ) ).

% real_root_strict_increasing
thf(fact_9635_real__root__decreasing,axiom,
    ! [N2: nat,N4: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ( ord_less_eq_real @ one_one_real @ X4 )
         => ( ord_less_eq_real @ ( root @ N4 @ X4 ) @ ( root @ N2 @ X4 ) ) ) ) ) ).

% real_root_decreasing
thf(fact_9636_real__root__pow__pos,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( power_power_real @ ( root @ N2 @ X4 ) @ N2 )
          = X4 ) ) ) ).

% real_root_pow_pos
thf(fact_9637_real__root__pos__unique,axiom,
    ! [N2: nat,Y: real,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ Y )
       => ( ( ( power_power_real @ Y @ N2 )
            = X4 )
         => ( ( root @ N2 @ X4 )
            = Y ) ) ) ) ).

% real_root_pos_unique
thf(fact_9638_real__root__power__cancel,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
       => ( ( root @ N2 @ ( power_power_real @ X4 @ N2 ) )
          = X4 ) ) ) ).

% real_root_power_cancel
thf(fact_9639_odd__real__root__power__cancel,axiom,
    ! [N2: nat,X4: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( root @ N2 @ ( power_power_real @ X4 @ N2 ) )
        = X4 ) ) ).

% odd_real_root_power_cancel
thf(fact_9640_odd__real__root__unique,axiom,
    ! [N2: nat,Y: real,X4: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( ( power_power_real @ Y @ N2 )
          = X4 )
       => ( ( root @ N2 @ X4 )
          = Y ) ) ) ).

% odd_real_root_unique
thf(fact_9641_odd__real__root__pow,axiom,
    ! [N2: nat,X4: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( power_power_real @ ( root @ N2 @ X4 ) @ N2 )
        = X4 ) ) ).

% odd_real_root_pow
thf(fact_9642_real__root__increasing,axiom,
    ! [N2: nat,N4: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_eq_nat @ N2 @ N4 )
       => ( ( ord_less_eq_real @ zero_zero_real @ X4 )
         => ( ( ord_less_eq_real @ X4 @ one_one_real )
           => ( ord_less_eq_real @ ( root @ N2 @ X4 ) @ ( root @ N4 @ X4 ) ) ) ) ) ) ).

% real_root_increasing
thf(fact_9643_root__sgn__power,axiom,
    ! [N2: nat,Y: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( root @ N2 @ ( times_times_real @ ( sgn_sgn_real @ Y ) @ ( power_power_real @ ( abs_abs_real @ Y ) @ N2 ) ) )
        = Y ) ) ).

% root_sgn_power
thf(fact_9644_sgn__power__root,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( times_times_real @ ( sgn_sgn_real @ ( root @ N2 @ X4 ) ) @ ( power_power_real @ ( abs_abs_real @ ( root @ N2 @ X4 ) ) @ N2 ) )
        = X4 ) ) ).

% sgn_power_root
thf(fact_9645_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_9646_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_9647_log__base__root,axiom,
    ! [N2: nat,B: real,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ B )
       => ( ( log @ ( root @ N2 @ B ) @ X4 )
          = ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( log @ B @ X4 ) ) ) ) ) ).

% log_base_root
thf(fact_9648_split__root,axiom,
    ! [P: real > $o,N2: nat,X4: real] :
      ( ( P @ ( root @ N2 @ X4 ) )
      = ( ( ( N2 = zero_zero_nat )
         => ( P @ zero_zero_real ) )
        & ( ( ord_less_nat @ zero_zero_nat @ N2 )
         => ! [Y5: real] :
              ( ( ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N2 ) )
                = X4 )
             => ( P @ Y5 ) ) ) ) ) ).

% split_root
thf(fact_9649_root__powr__inverse,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( ( root @ N2 @ X4 )
          = ( powr_real @ X4 @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N2 ) ) ) ) ) ) ).

% root_powr_inverse
thf(fact_9650_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_9651_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_9652_card__lessThan,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_lessThan_nat @ U ) )
      = U ) ).

% card_lessThan
thf(fact_9653_card__Collect__less__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_nat @ I3 @ N2 ) ) )
      = N2 ) ).

% card_Collect_less_nat
thf(fact_9654_card__atMost,axiom,
    ! [U: nat] :
      ( ( finite_card_nat @ ( set_ord_atMost_nat @ U ) )
      = ( suc @ U ) ) ).

% card_atMost
thf(fact_9655_card__atLeastLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or4665077453230672383an_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_atLeastLessThan
thf(fact_9656_card__Collect__le__nat,axiom,
    ! [N2: nat] :
      ( ( finite_card_nat
        @ ( collect_nat
          @ ^ [I3: nat] : ( ord_less_eq_nat @ I3 @ N2 ) ) )
      = ( suc @ N2 ) ) ).

% card_Collect_le_nat
thf(fact_9657_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_9658_card__atLeastLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_atLeastLessThan_int
thf(fact_9659_snd__divmod__nat,axiom,
    ! [M: nat,N2: nat] :
      ( ( product_snd_nat_nat @ ( divmod_nat @ M @ N2 ) )
      = ( modulo_modulo_nat @ M @ N2 ) ) ).

% snd_divmod_nat
thf(fact_9660_card__atLeastAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or1266510415728281911st_int @ L @ U ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ U @ L ) @ one_one_int ) ) ) ).

% card_atLeastAtMost_int
thf(fact_9661_lessThan__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_lessThan_nat @ ( suc @ K ) )
      = ( insert_nat @ K @ ( set_ord_lessThan_nat @ K ) ) ) ).

% lessThan_Suc
thf(fact_9662_atMost__Suc,axiom,
    ! [K: nat] :
      ( ( set_ord_atMost_nat @ ( suc @ K ) )
      = ( insert_nat @ ( suc @ K ) @ ( set_ord_atMost_nat @ K ) ) ) ).

% atMost_Suc
thf(fact_9663_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_9664_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_9665_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_9666_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_9667_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_9668_lessThan__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_lessThan_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( pred_numeral @ K ) @ ( set_ord_lessThan_nat @ ( pred_numeral @ K ) ) ) ) ).

% lessThan_nat_numeral
thf(fact_9669_card__less,axiom,
    ! [M7: set_nat,I2: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) )
       != zero_zero_nat ) ) ).

% card_less
thf(fact_9670_card__less__Suc,axiom,
    ! [M7: set_nat,I2: nat] :
      ( ( member_nat @ zero_zero_nat @ M7 )
     => ( ( suc
          @ ( finite_card_nat
            @ ( collect_nat
              @ ^ [K3: nat] :
                  ( ( member_nat @ ( suc @ K3 ) @ M7 )
                  & ( ord_less_nat @ K3 @ I2 ) ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).

% card_less_Suc
thf(fact_9671_card__less__Suc2,axiom,
    ! [M7: set_nat,I2: nat] :
      ( ~ ( member_nat @ zero_zero_nat @ M7 )
     => ( ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ ( suc @ K3 ) @ M7 )
                & ( ord_less_nat @ K3 @ I2 ) ) ) )
        = ( finite_card_nat
          @ ( collect_nat
            @ ^ [K3: nat] :
                ( ( member_nat @ K3 @ M7 )
                & ( ord_less_nat @ K3 @ ( suc @ I2 ) ) ) ) ) ) ) ).

% card_less_Suc2
thf(fact_9672_atMost__nat__numeral,axiom,
    ! [K: num] :
      ( ( set_ord_atMost_nat @ ( numeral_numeral_nat @ K ) )
      = ( insert_nat @ ( numeral_numeral_nat @ K ) @ ( set_ord_atMost_nat @ ( pred_numeral @ K ) ) ) ) ).

% atMost_nat_numeral
thf(fact_9673_card__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( finite_card_int @ ( set_or4662586982721622107an_int @ zero_zero_int @ U ) )
      = ( nat2 @ U ) ) ).

% card_atLeastZeroLessThan_int
thf(fact_9674_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_9675_subset__eq__atLeast0__lessThan__card,axiom,
    ! [N4: set_nat,N2: nat] :
      ( ( ord_less_eq_set_nat @ N4 @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) )
     => ( ord_less_eq_nat @ ( finite_card_nat @ N4 ) @ N2 ) ) ).

% subset_eq_atLeast0_lessThan_card
thf(fact_9676_card__sum__le__nat__sum,axiom,
    ! [S2: set_nat] :
      ( ord_less_eq_nat
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( finite_card_nat @ S2 ) ) )
      @ ( groups3542108847815614940at_nat
        @ ^ [X: nat] : X
        @ S2 ) ) ).

% card_sum_le_nat_sum
thf(fact_9677_card__nth__roots,axiom,
    ! [C: complex,N2: nat] :
      ( ( C != zero_zero_complex )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( finite_card_complex
            @ ( collect_complex
              @ ^ [Z5: complex] :
                  ( ( power_power_complex @ Z5 @ N2 )
                  = C ) ) )
          = N2 ) ) ) ).

% card_nth_roots
thf(fact_9678_card__roots__unity__eq,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( finite_card_complex
          @ ( collect_complex
            @ ^ [Z5: complex] :
                ( ( power_power_complex @ Z5 @ N2 )
                = one_one_complex ) ) )
        = N2 ) ) ).

% card_roots_unity_eq
thf(fact_9679_set__decode__plus__power__2,axiom,
    ! [N2: nat,Z: nat] :
      ( ~ ( member_nat @ N2 @ ( nat_set_decode @ Z ) )
     => ( ( nat_set_decode @ ( plus_plus_nat @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ Z ) )
        = ( insert_nat @ N2 @ ( nat_set_decode @ Z ) ) ) ) ).

% set_decode_plus_power_2
thf(fact_9680_snd__divmod__integer,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_integer @ K @ L ) )
      = ( modulo364778990260209775nteger @ K @ L ) ) ).

% snd_divmod_integer
thf(fact_9681_snd__divmod__abs,axiom,
    ! [K: code_integer,L: code_integer] :
      ( ( produc6174133586879617921nteger @ ( code_divmod_abs @ K @ L ) )
      = ( modulo364778990260209775nteger @ ( abs_abs_Code_integer @ K ) @ ( abs_abs_Code_integer @ L ) ) ) ).

% snd_divmod_abs
thf(fact_9682_atLeastAtMostPlus1__int__conv,axiom,
    ! [M: int,N2: int] :
      ( ( ord_less_eq_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
     => ( ( set_or1266510415728281911st_int @ M @ ( plus_plus_int @ one_one_int @ N2 ) )
        = ( insert_int @ ( plus_plus_int @ one_one_int @ N2 ) @ ( set_or1266510415728281911st_int @ M @ N2 ) ) ) ) ).

% atLeastAtMostPlus1_int_conv
thf(fact_9683_minus__one__mod__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ one_one_int ) @ ( numeral_numeral_int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ).

% minus_one_mod_numeral
thf(fact_9684_one__mod__minus__numeral,axiom,
    ! [N2: num] :
      ( ( modulo_modulo_int @ one_one_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ one @ N2 ) ) ) ) ) ).

% one_mod_minus_numeral
thf(fact_9685_minus__numeral__mod__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ).

% minus_numeral_mod_numeral
thf(fact_9686_numeral__mod__minus__numeral,axiom,
    ! [M: num,N2: num] :
      ( ( modulo_modulo_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( uminus_uminus_int @ ( adjust_mod @ ( numeral_numeral_int @ N2 ) @ ( product_snd_int_int @ ( unique5052692396658037445od_int @ M @ N2 ) ) ) ) ) ).

% numeral_mod_minus_numeral
thf(fact_9687_Divides_Oadjust__mod__def,axiom,
    ( adjust_mod
    = ( ^ [L2: int,R5: int] : ( if_int @ ( R5 = zero_zero_int ) @ zero_zero_int @ ( minus_minus_int @ L2 @ R5 ) ) ) ) ).

% Divides.adjust_mod_def
thf(fact_9688_and__int_Oelims,axiom,
    ! [X4: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
        = Y )
     => ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
            & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( uminus_uminus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
        & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
              & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
         => ( Y
            = ( plus_plus_int
              @ ( zero_n2684676970156552555ol_int
                @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                  & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
              @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% and_int.elims
thf(fact_9689_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_9690_bezw_Oelims,axiom,
    ! [X4: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X4 @ Xa )
        = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y
            = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Xa ) ) ) ) ) ) ) ) ) ).

% bezw.elims
thf(fact_9691_simp__from__to,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( if_set_int @ ( ord_less_int @ J3 @ I3 ) @ bot_bot_set_int @ ( insert_int @ I3 @ ( set_or1266510415728281911st_int @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ) ).

% simp_from_to
thf(fact_9692_bezw__non__0,axiom,
    ! [Y: nat,X4: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ Y )
     => ( ( bezw @ X4 @ Y )
        = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X4 @ Y ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X4 @ Y ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y @ ( modulo_modulo_nat @ X4 @ Y ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Y ) ) ) ) ) ) ) ).

% bezw_non_0
thf(fact_9693_bezw_Osimps,axiom,
    ( bezw
    = ( ^ [X: nat,Y5: nat] : ( if_Pro3027730157355071871nt_int @ ( Y5 = zero_zero_nat ) @ ( product_Pair_int_int @ one_one_int @ zero_zero_int ) @ ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X @ Y5 ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X @ Y5 ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Y5 @ ( modulo_modulo_nat @ X @ Y5 ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X @ Y5 ) ) ) ) ) ) ) ) ).

% bezw.simps
thf(fact_9694_bezw_Opelims,axiom,
    ! [X4: nat,Xa: nat,Y: product_prod_int_int] :
      ( ( ( bezw @ X4 @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ one_one_int @ zero_zero_int ) ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y
                  = ( product_Pair_int_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( minus_minus_int @ ( product_fst_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( times_times_int @ ( product_snd_int_int @ ( bezw @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) @ ( semiri1314217659103216013at_int @ ( divide_divide_nat @ X4 @ Xa ) ) ) ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ bezw_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).

% bezw.pelims
thf(fact_9695_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_9696_and__int_Opelims,axiom,
    ! [X4: int,Xa: int,Y: int] :
      ( ( ( bit_se725231765392027082nd_int @ X4 @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
       => ~ ( ( ( ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                  & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( uminus_uminus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) ) ) ) )
              & ( ~ ( ( member_int @ X4 @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) )
                    & ( member_int @ Xa @ ( insert_int @ zero_zero_int @ ( insert_int @ ( uminus_uminus_int @ one_one_int ) @ bot_bot_set_int ) ) ) )
               => ( Y
                  = ( plus_plus_int
                    @ ( zero_n2684676970156552555ol_int
                      @ ( ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ X4 )
                        & ~ ( dvd_dvd_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ Xa ) ) )
                    @ ( times_times_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( bit_se725231765392027082nd_int @ ( divide_divide_int @ X4 @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) @ ( divide_divide_int @ Xa @ ( numeral_numeral_int @ ( bit0 @ one ) ) ) ) ) ) ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ bit_and_int_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).

% and_int.pelims
thf(fact_9697_lessThan__0,axiom,
    ( ( set_ord_lessThan_nat @ zero_zero_nat )
    = bot_bot_set_nat ) ).

% lessThan_0
thf(fact_9698_atLeastLessThan__singleton,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ ( suc @ M ) )
      = ( insert_nat @ M @ bot_bot_set_nat ) ) ).

% atLeastLessThan_singleton
thf(fact_9699_atMost__0,axiom,
    ( ( set_ord_atMost_nat @ zero_zero_nat )
    = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) ).

% atMost_0
thf(fact_9700_bot__enat__def,axiom,
    bot_bo4199563552545308370d_enat = zero_z5237406670263579293d_enat ).

% bot_enat_def
thf(fact_9701_bot__nat__def,axiom,
    bot_bot_nat = zero_zero_nat ).

% bot_nat_def
thf(fact_9702_atLeastLessThan0,axiom,
    ! [M: nat] :
      ( ( set_or4665077453230672383an_nat @ M @ zero_zero_nat )
      = bot_bot_set_nat ) ).

% atLeastLessThan0
thf(fact_9703_lessThan__empty__iff,axiom,
    ! [N2: nat] :
      ( ( ( set_ord_lessThan_nat @ N2 )
        = bot_bot_set_nat )
      = ( N2 = zero_zero_nat ) ) ).

% lessThan_empty_iff
thf(fact_9704_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_9705_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_9706_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_9707_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_9708_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_9709_normalize__def,axiom,
    ( normalize
    = ( ^ [P5: product_prod_int_int] :
          ( if_Pro3027730157355071871nt_int @ ( ord_less_int @ zero_zero_int @ ( product_snd_int_int @ P5 ) ) @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) )
          @ ( if_Pro3027730157355071871nt_int
            @ ( ( product_snd_int_int @ P5 )
              = zero_zero_int )
            @ ( product_Pair_int_int @ zero_zero_int @ one_one_int )
            @ ( product_Pair_int_int @ ( divide_divide_int @ ( product_fst_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) @ ( divide_divide_int @ ( product_snd_int_int @ P5 ) @ ( uminus_uminus_int @ ( gcd_gcd_int @ ( product_fst_int_int @ P5 ) @ ( product_snd_int_int @ P5 ) ) ) ) ) ) ) ) ) ).

% normalize_def
thf(fact_9710_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_9711_gcd__red__int,axiom,
    ( gcd_gcd_int
    = ( ^ [X: int,Y5: int] : ( gcd_gcd_int @ Y5 @ ( modulo_modulo_int @ X @ Y5 ) ) ) ) ).

% gcd_red_int
thf(fact_9712_gcd__ge__0__int,axiom,
    ! [X4: int,Y: int] : ( ord_less_eq_int @ zero_zero_int @ ( gcd_gcd_int @ X4 @ Y ) ) ).

% gcd_ge_0_int
thf(fact_9713_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_9714_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_9715_gcd__cases__int,axiom,
    ! [X4: int,Y: int,P: int > $o] :
      ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
       => ( ( ord_less_eq_int @ zero_zero_int @ Y )
         => ( P @ ( gcd_gcd_int @ X4 @ Y ) ) ) )
     => ( ( ( ord_less_eq_int @ zero_zero_int @ X4 )
         => ( ( ord_less_eq_int @ Y @ zero_zero_int )
           => ( P @ ( gcd_gcd_int @ X4 @ ( uminus_uminus_int @ Y ) ) ) ) )
       => ( ( ( ord_less_eq_int @ X4 @ zero_zero_int )
           => ( ( ord_less_eq_int @ zero_zero_int @ Y )
             => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X4 ) @ Y ) ) ) )
         => ( ( ( ord_less_eq_int @ X4 @ zero_zero_int )
             => ( ( ord_less_eq_int @ Y @ zero_zero_int )
               => ( P @ ( gcd_gcd_int @ ( uminus_uminus_int @ X4 ) @ ( uminus_uminus_int @ Y ) ) ) ) )
           => ( P @ ( gcd_gcd_int @ X4 @ Y ) ) ) ) ) ) ).

% gcd_cases_int
thf(fact_9716_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_9717_gcd__non__0__int,axiom,
    ! [Y: int,X4: int] :
      ( ( ord_less_int @ zero_zero_int @ Y )
     => ( ( gcd_gcd_int @ X4 @ Y )
        = ( gcd_gcd_int @ Y @ ( modulo_modulo_int @ X4 @ Y ) ) ) ) ).

% gcd_non_0_int
thf(fact_9718_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_9719_gcd__1__nat,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ one_one_nat )
      = one_one_nat ) ).

% gcd_1_nat
thf(fact_9720_gcd__Suc__0,axiom,
    ! [M: nat] :
      ( ( gcd_gcd_nat @ M @ ( suc @ zero_zero_nat ) )
      = ( suc @ zero_zero_nat ) ) ).

% gcd_Suc_0
thf(fact_9721_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_9722_gcd__red__nat,axiom,
    ( gcd_gcd_nat
    = ( ^ [X: nat,Y5: nat] : ( gcd_gcd_nat @ Y5 @ ( modulo_modulo_nat @ X @ Y5 ) ) ) ) ).

% gcd_red_nat
thf(fact_9723_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_9724_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_9725_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_9726_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_9727_gcd__non__0__nat,axiom,
    ! [Y: nat,X4: nat] :
      ( ( Y != zero_zero_nat )
     => ( ( gcd_gcd_nat @ X4 @ Y )
        = ( gcd_gcd_nat @ Y @ ( modulo_modulo_nat @ X4 @ Y ) ) ) ) ).

% gcd_non_0_nat
thf(fact_9728_gcd__nat_Osimps,axiom,
    ( gcd_gcd_nat
    = ( ^ [X: nat,Y5: nat] : ( if_nat @ ( Y5 = zero_zero_nat ) @ X @ ( gcd_gcd_nat @ Y5 @ ( modulo_modulo_nat @ X @ Y5 ) ) ) ) ) ).

% gcd_nat.simps
thf(fact_9729_gcd__nat_Oelims,axiom,
    ! [X4: nat,Xa: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X4 @ Xa )
        = Y )
     => ( ( ( Xa = zero_zero_nat )
         => ( Y = X4 ) )
        & ( ( Xa != zero_zero_nat )
         => ( Y
            = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) ) ) ) ).

% gcd_nat.elims
thf(fact_9730_bezout__nat,axiom,
    ! [A: nat,B: nat] :
      ( ( A != zero_zero_nat )
     => ? [X5: nat,Y3: nat] :
          ( ( times_times_nat @ A @ X5 )
          = ( plus_plus_nat @ ( times_times_nat @ B @ Y3 ) @ ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_nat
thf(fact_9731_bezout__gcd__nat_H,axiom,
    ! [B: nat,A: nat] :
    ? [X5: nat,Y3: nat] :
      ( ( ( ord_less_eq_nat @ ( times_times_nat @ B @ Y3 ) @ ( times_times_nat @ A @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ A @ X5 ) @ ( times_times_nat @ B @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) )
      | ( ( ord_less_eq_nat @ ( times_times_nat @ A @ Y3 ) @ ( times_times_nat @ B @ X5 ) )
        & ( ( minus_minus_nat @ ( times_times_nat @ B @ X5 ) @ ( times_times_nat @ A @ Y3 ) )
          = ( gcd_gcd_nat @ A @ B ) ) ) ) ).

% bezout_gcd_nat'
thf(fact_9732_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_9733_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 )
           => ( ! [I: nat] :
                  ( ( ord_less_nat @ K2 @ I )
                 => ( P @ I ) )
             => ( P @ K2 ) ) )
       => ( P @ M ) ) ) ).

% nat_descend_induct
thf(fact_9734_gcd__nat_Opelims,axiom,
    ! [X4: nat,Xa: nat,Y: nat] :
      ( ( ( gcd_gcd_nat @ X4 @ Xa )
        = Y )
     => ( ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) )
       => ~ ( ( ( ( Xa = zero_zero_nat )
               => ( Y = X4 ) )
              & ( ( Xa != zero_zero_nat )
               => ( Y
                  = ( gcd_gcd_nat @ Xa @ ( modulo_modulo_nat @ X4 @ Xa ) ) ) ) )
           => ~ ( accp_P4275260045618599050at_nat @ gcd_nat_rel @ ( product_Pair_nat_nat @ X4 @ Xa ) ) ) ) ) ).

% gcd_nat.pelims
thf(fact_9735_drop__bit__numeral__minus__bit1,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( inc @ K ) ) ) ) ) ).

% drop_bit_numeral_minus_bit1
thf(fact_9736_finite__enumerate,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ? [R2: nat > nat] :
          ( ( strict1292158309912662752at_nat @ R2 @ ( set_ord_lessThan_nat @ ( finite_card_nat @ S2 ) ) )
          & ! [N6: nat] :
              ( ( ord_less_nat @ N6 @ ( finite_card_nat @ S2 ) )
             => ( member_nat @ ( R2 @ N6 ) @ S2 ) ) ) ) ).

% finite_enumerate
thf(fact_9737_drop__bit__nonnegative__int__iff,axiom,
    ! [N2: nat,K: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ ( bit_se8568078237143864401it_int @ N2 @ K ) )
      = ( ord_less_eq_int @ zero_zero_int @ K ) ) ).

% drop_bit_nonnegative_int_iff
thf(fact_9738_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_9739_drop__bit__minus__one,axiom,
    ! [N2: nat] :
      ( ( bit_se8568078237143864401it_int @ N2 @ ( uminus_uminus_int @ one_one_int ) )
      = ( uminus_uminus_int @ one_one_int ) ) ).

% drop_bit_minus_one
thf(fact_9740_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_9741_drop__bit__numeral__minus__bit0,axiom,
    ! [L: num,K: num] :
      ( ( bit_se8568078237143864401it_int @ ( numeral_numeral_nat @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ K ) ) ) )
      = ( bit_se8568078237143864401it_int @ ( pred_numeral @ L ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ K ) ) ) ) ).

% drop_bit_numeral_minus_bit0
thf(fact_9742_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_9743_drop__bit__push__bit__int,axiom,
    ! [M: nat,N2: nat,K: int] :
      ( ( bit_se8568078237143864401it_int @ M @ ( bit_se545348938243370406it_int @ N2 @ K ) )
      = ( bit_se8568078237143864401it_int @ ( minus_minus_nat @ M @ N2 ) @ ( bit_se545348938243370406it_int @ ( minus_minus_nat @ N2 @ M ) @ K ) ) ) ).

% drop_bit_push_bit_int
thf(fact_9744_drop__bit__int__def,axiom,
    ( bit_se8568078237143864401it_int
    = ( ^ [N: nat,K3: int] : ( divide_divide_int @ K3 @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% drop_bit_int_def
thf(fact_9745_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_9746_drop__bit__nat__eq,axiom,
    ! [N2: nat,K: int] :
      ( ( bit_se8570568707652914677it_nat @ N2 @ ( nat2 @ K ) )
      = ( nat2 @ ( bit_se8568078237143864401it_int @ N2 @ K ) ) ) ).

% drop_bit_nat_eq
thf(fact_9747_drop__bit__nat__def,axiom,
    ( bit_se8570568707652914677it_nat
    = ( ^ [N: nat,M6: nat] : ( divide_divide_nat @ M6 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) ) ) ).

% drop_bit_nat_def
thf(fact_9748_card__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or5832277885323065728an_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ ( plus_plus_int @ L @ one_one_int ) ) ) ) ).

% card_greaterThanLessThan_int
thf(fact_9749_finite__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or5832277885323065728an_int @ L @ U ) ) ).

% finite_greaterThanLessThan_int
thf(fact_9750_atLeastPlusOneLessThan__greaterThanLessThan__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or4662586982721622107an_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or5832277885323065728an_int @ L @ U ) ) ).

% atLeastPlusOneLessThan_greaterThanLessThan_int
thf(fact_9751_xor__minus__numerals_I1_J,axiom,
    ! [N2: num,K: int] :
      ( ( bit_se6526347334894502574or_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) @ K )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ ( neg_numeral_sub_int @ N2 @ one ) @ K ) ) ) ).

% xor_minus_numerals(1)
thf(fact_9752_xor__minus__numerals_I2_J,axiom,
    ! [K: int,N2: num] :
      ( ( bit_se6526347334894502574or_int @ K @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( bit_ri7919022796975470100ot_int @ ( bit_se6526347334894502574or_int @ K @ ( neg_numeral_sub_int @ N2 @ one ) ) ) ) ).

% xor_minus_numerals(2)
thf(fact_9753_finite__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% finite_greaterThanLessThan
thf(fact_9754_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_9755_atLeastSucLessThan__greaterThanLessThan,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or4665077453230672383an_nat @ ( suc @ L ) @ U )
      = ( set_or5834768355832116004an_nat @ L @ U ) ) ).

% atLeastSucLessThan_greaterThanLessThan
thf(fact_9756_tanh__real__bounds,axiom,
    ! [X4: real] : ( member_real @ ( tanh_real @ X4 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) ).

% tanh_real_bounds
thf(fact_9757_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_9758_Suc__funpow,axiom,
    ! [N2: nat] :
      ( ( compow_nat_nat @ N2 @ suc )
      = ( plus_plus_nat @ N2 ) ) ).

% Suc_funpow
thf(fact_9759_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_9760_max__nat_Osemilattice__neutr__order__axioms,axiom,
    ( semila1623282765462674594er_nat @ ord_max_nat @ zero_zero_nat
    @ ^ [X: nat,Y5: nat] : ( ord_less_eq_nat @ Y5 @ X )
    @ ^ [X: nat,Y5: nat] : ( ord_less_nat @ Y5 @ X ) ) ).

% max_nat.semilattice_neutr_order_axioms
thf(fact_9761_nat__of__integer__code__post_I3_J,axiom,
    ! [K: num] :
      ( ( code_nat_of_integer @ ( numera6620942414471956472nteger @ K ) )
      = ( numeral_numeral_nat @ K ) ) ).

% nat_of_integer_code_post(3)
thf(fact_9762_nat__of__integer__code__post_I2_J,axiom,
    ( ( code_nat_of_integer @ one_one_Code_integer )
    = one_one_nat ) ).

% nat_of_integer_code_post(2)
thf(fact_9763_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_9764_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_9765_modulo__integer_Orep__eq,axiom,
    ! [X4: code_integer,Xa: code_integer] :
      ( ( code_int_of_integer @ ( modulo364778990260209775nteger @ X4 @ Xa ) )
      = ( modulo_modulo_int @ ( code_int_of_integer @ X4 ) @ ( code_int_of_integer @ Xa ) ) ) ).

% modulo_integer.rep_eq
thf(fact_9766_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_9767_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_9768_less__eq__integer_Orep__eq,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [X: code_integer,Xa4: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ X ) @ ( code_int_of_integer @ Xa4 ) ) ) ) ).

% less_eq_integer.rep_eq
thf(fact_9769_integer__less__eq__iff,axiom,
    ( ord_le3102999989581377725nteger
    = ( ^ [K3: code_integer,L2: code_integer] : ( ord_less_eq_int @ ( code_int_of_integer @ K3 ) @ ( code_int_of_integer @ L2 ) ) ) ) ).

% integer_less_eq_iff
thf(fact_9770_times__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( times_times_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y5: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ ( times_times_nat @ X @ U2 ) @ ( times_times_nat @ Y5 @ V4 ) ) @ ( plus_plus_nat @ ( times_times_nat @ X @ V4 ) @ ( times_times_nat @ Y5 @ U2 ) ) ) )
          @ Xa
          @ X4 ) ) ) ).

% times_int.abs_eq
thf(fact_9771_one__int__def,axiom,
    ( one_one_int
    = ( abs_Integ @ ( product_Pair_nat_nat @ one_one_nat @ zero_zero_nat ) ) ) ).

% one_int_def
thf(fact_9772_less__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( ord_less_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y5: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
        @ Xa
        @ X4 ) ) ).

% less_int.abs_eq
thf(fact_9773_less__eq__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( ord_less_eq_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( produc8739625826339149834_nat_o
        @ ^ [X: nat,Y5: nat] :
            ( produc6081775807080527818_nat_o
            @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ U2 @ Y5 ) ) )
        @ Xa
        @ X4 ) ) ).

% less_eq_int.abs_eq
thf(fact_9774_plus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( plus_plus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y5: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ U2 ) @ ( plus_plus_nat @ Y5 @ V4 ) ) )
          @ Xa
          @ X4 ) ) ) ).

% plus_int.abs_eq
thf(fact_9775_minus__int_Oabs__eq,axiom,
    ! [Xa: product_prod_nat_nat,X4: product_prod_nat_nat] :
      ( ( minus_minus_int @ ( abs_Integ @ Xa ) @ ( abs_Integ @ X4 ) )
      = ( abs_Integ
        @ ( produc27273713700761075at_nat
          @ ^ [X: nat,Y5: nat] :
              ( produc2626176000494625587at_nat
              @ ^ [U2: nat,V4: nat] : ( product_Pair_nat_nat @ ( plus_plus_nat @ X @ V4 ) @ ( plus_plus_nat @ Y5 @ U2 ) ) )
          @ Xa
          @ X4 ) ) ) ).

% minus_int.abs_eq
thf(fact_9776_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_9777_pred__nat__def,axiom,
    ( pred_nat
    = ( collec3392354462482085612at_nat
      @ ( produc6081775807080527818_nat_o
        @ ^ [M6: nat,N: nat] :
            ( N
            = ( suc @ M6 ) ) ) ) ) ).

% pred_nat_def
thf(fact_9778_num__of__nat__numeral__eq,axiom,
    ! [Q3: num] :
      ( ( num_of_nat @ ( numeral_numeral_nat @ Q3 ) )
      = Q3 ) ).

% num_of_nat_numeral_eq
thf(fact_9779_num__of__nat_Osimps_I1_J,axiom,
    ( ( num_of_nat @ zero_zero_nat )
    = one ) ).

% num_of_nat.simps(1)
thf(fact_9780_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_9781_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_9782_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_9783_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_9784_less__eq__int_Orep__eq,axiom,
    ( ord_less_eq_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y5: nat,Z5: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_eq_int.rep_eq
thf(fact_9785_less__int_Orep__eq,axiom,
    ( ord_less_int
    = ( ^ [X: int,Xa4: int] :
          ( produc8739625826339149834_nat_o
          @ ^ [Y5: nat,Z5: nat] :
              ( produc6081775807080527818_nat_o
              @ ^ [U2: nat,V4: nat] : ( ord_less_nat @ ( plus_plus_nat @ Y5 @ V4 ) @ ( plus_plus_nat @ U2 @ Z5 ) ) )
          @ ( rep_Integ @ X )
          @ ( rep_Integ @ Xa4 ) ) ) ) ).

% less_int.rep_eq
thf(fact_9786_VEBT__internal_Ovalid_H_Oelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_valid @ X4 @ Xa )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa = one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
             => ( ( Deg2 = Xa )
                & ! [X5: vEBT_VEBT] :
                    ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                   => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                & ( vEBT_VEBT_valid @ Summary3 @ ( 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
                  @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
                    & ! [X: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                  @ ( produc6081775807080527818_nat_o
                    @ ^ [Mi2: nat,Ma2: nat] :
                        ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                        & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                        & ! [I3: nat] :
                            ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                           => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
                              = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
                        & ( ( Mi2 = Ma2 )
                         => ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                        & ( ( Mi2 != Ma2 )
                         => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
                            & ! [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 @ Mi2 @ X )
                                    & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) )
                  @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(3)
thf(fact_9787_VEBT__internal_Ovalid_H_Oelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_valid @ X4 @ Xa )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Xa != one_one_nat ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
             => ~ ( ( Deg2 = Xa )
                  & ! [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 @ Summary3 @ ( 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
                    @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                    @ ( produc6081775807080527818_nat_o
                      @ ^ [Mi2: nat,Ma2: nat] :
                          ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                          & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                          & ! [I3: nat] :
                              ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                             => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
                                = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
                          & ( ( Mi2 = Ma2 )
                           => ! [X: vEBT_VEBT] :
                                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                               => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                          & ( ( Mi2 != Ma2 )
                           => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
                              & ! [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 @ Mi2 @ X )
                                      & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) )
                    @ Mima ) ) ) ) ) ).

% VEBT_internal.valid'.elims(2)
thf(fact_9788_VEBT__internal_Ovalid_H_Oelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X4 @ Xa )
        = Y )
     => ( ( ? [Uu2: $o,Uv2: $o] :
              ( X4
              = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
         => ( Y
            = ( Xa != one_one_nat ) ) )
       => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
              ( ( X4
                = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
             => ( Y
                = ( ~ ( ( Deg2 = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary3 @ ( 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
                        @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi2: nat,Ma2: nat] :
                              ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                              & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
                              & ( ( Mi2 = Ma2 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                              & ( ( Mi2 != Ma2 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
                                  & ! [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 @ Mi2 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.elims(1)
thf(fact_9789_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
          @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary @ X3 )
            & ! [X: vEBT_VEBT] :
                ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
               => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
          @ ( produc6081775807080527818_nat_o
            @ ^ [Mi2: nat,Ma2: nat] :
                ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg ) )
                & ! [I3: nat] :
                    ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                   => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList2 @ I3 ) @ X3 ) )
                      = ( vEBT_V8194947554948674370ptions @ Summary @ I3 ) ) )
                & ( ( Mi2 = Ma2 )
                 => ! [X: vEBT_VEBT] :
                      ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList2 ) )
                     => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                & ( ( Mi2 != Ma2 )
                 => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList2 @ Ma2 )
                    & ! [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 @ Mi2 @ X )
                            & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) )
          @ Mima2 ) ) ) ).

% VEBT_internal.valid'.simps(2)
thf(fact_9790_VEBT__internal_Ovalid_H_Opelims_I1_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat,Y: $o] :
      ( ( ( vEBT_VEBT_valid @ X4 @ Xa )
        = Y )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( Y
                  = ( Xa = one_one_nat ) )
               => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
               => ( ( Y
                    = ( ( Deg2 = Xa )
                      & ! [X: vEBT_VEBT] :
                          ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                         => ( vEBT_VEBT_valid @ X @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                      & ( vEBT_VEBT_valid @ Summary3 @ ( 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
                        @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi2: nat,Ma2: nat] :
                              ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                              & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
                              & ( ( Mi2 = Ma2 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                              & ( ( Mi2 != Ma2 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
                                  & ! [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 @ Mi2 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) )
                        @ Mima ) ) )
                 => ~ ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Xa ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(1)
thf(fact_9791_VEBT__internal_Ovalid_H_Opelims_I2_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ( vEBT_VEBT_valid @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
               => ( Xa != one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Xa ) )
                 => ~ ( ( Deg2 = Xa )
                      & ! [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 @ Summary3 @ ( 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
                        @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
                          & ! [X: vEBT_VEBT] :
                              ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                             => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                        @ ( produc6081775807080527818_nat_o
                          @ ^ [Mi2: nat,Ma2: nat] :
                              ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                              & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                              & ! [I3: nat] :
                                  ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                                 => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
                                    = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
                              & ( ( Mi2 = Ma2 )
                               => ! [X: vEBT_VEBT] :
                                    ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                   => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                              & ( ( Mi2 != Ma2 )
                               => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
                                  & ! [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 @ Mi2 @ X )
                                          & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) )
                        @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(2)
thf(fact_9792_VEBT__internal_Ovalid_H_Opelims_I3_J,axiom,
    ! [X4: vEBT_VEBT,Xa: nat] :
      ( ~ ( vEBT_VEBT_valid @ X4 @ Xa )
     => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ X4 @ Xa ) )
       => ( ! [Uu2: $o,Uv2: $o] :
              ( ( X4
                = ( vEBT_Leaf @ Uu2 @ Uv2 ) )
             => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Leaf @ Uu2 @ Uv2 ) @ Xa ) )
               => ( Xa = one_one_nat ) ) )
         => ~ ! [Mima: option4927543243414619207at_nat,Deg2: nat,TreeList3: list_VEBT_VEBT,Summary3: vEBT_VEBT] :
                ( ( X4
                  = ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) )
               => ( ( accp_P2887432264394892906BT_nat @ vEBT_VEBT_valid_rel @ ( produc738532404422230701BT_nat @ ( vEBT_Node @ Mima @ Deg2 @ TreeList3 @ Summary3 ) @ Xa ) )
                 => ( ( Deg2 = Xa )
                    & ! [X5: vEBT_VEBT] :
                        ( ( member_VEBT_VEBT @ X5 @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                       => ( vEBT_VEBT_valid @ X5 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) )
                    & ( vEBT_VEBT_valid @ Summary3 @ ( 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
                      @ ( ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ Summary3 @ X3 )
                        & ! [X: vEBT_VEBT] :
                            ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                           => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                      @ ( produc6081775807080527818_nat_o
                        @ ^ [Mi2: nat,Ma2: nat] :
                            ( ( ord_less_eq_nat @ Mi2 @ Ma2 )
                            & ( ord_less_nat @ Ma2 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ Deg2 ) )
                            & ! [I3: nat] :
                                ( ( ord_less_nat @ I3 @ ( power_power_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ ( minus_minus_nat @ Deg2 @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
                               => ( ( ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ ( nth_VEBT_VEBT @ TreeList3 @ I3 ) @ X3 ) )
                                  = ( vEBT_V8194947554948674370ptions @ Summary3 @ I3 ) ) )
                            & ( ( Mi2 = Ma2 )
                             => ! [X: vEBT_VEBT] :
                                  ( ( member_VEBT_VEBT @ X @ ( set_VEBT_VEBT2 @ TreeList3 ) )
                                 => ~ ? [X3: nat] : ( vEBT_V8194947554948674370ptions @ X @ X3 ) ) )
                            & ( ( Mi2 != Ma2 )
                             => ( ( vEBT_V5917875025757280293ildren @ ( divide_divide_nat @ Deg2 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ TreeList3 @ Ma2 )
                                & ! [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 @ Mi2 @ X )
                                        & ( ord_less_eq_nat @ X @ Ma2 ) ) ) ) ) ) ) )
                      @ Mima ) ) ) ) ) ) ) ).

% VEBT_internal.valid'.pelims(3)
thf(fact_9793_Sup__int__def,axiom,
    ( complete_Sup_Sup_int
    = ( ^ [X3: set_int] :
          ( the_int
          @ ^ [X: int] :
              ( ( member_int @ X @ X3 )
              & ! [Y5: int] :
                  ( ( member_int @ Y5 @ X3 )
                 => ( ord_less_eq_int @ Y5 @ X ) ) ) ) ) ) ).

% Sup_int_def
thf(fact_9794_take__bit__numeral__minus__numeral__int,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int
        @ ^ [Q5: num] : ( bit_se2923211474154528505it_int @ ( numeral_numeral_nat @ M ) @ ( minus_minus_int @ ( power_power_int @ ( numeral_numeral_int @ ( bit0 @ one ) ) @ ( numeral_numeral_nat @ M ) ) @ ( numeral_numeral_int @ Q5 ) ) )
        @ ( bit_take_bit_num @ ( numeral_numeral_nat @ M ) @ N2 ) ) ) ).

% take_bit_numeral_minus_numeral_int
thf(fact_9795_and__minus__numerals_I7_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(7)
thf(fact_9796_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_9797_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_9798_take__bit__num__simps_I5_J,axiom,
    ! [R3: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ one )
      = ( some_num @ one ) ) ).

% take_bit_num_simps(5)
thf(fact_9799_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_9800_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_9801_take__bit__num__simps_I6_J,axiom,
    ! [R3: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit0 @ M ) )
      = ( case_o6005452278849405969um_num @ none_num
        @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
        @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ).

% take_bit_num_simps(6)
thf(fact_9802_take__bit__num__simps_I7_J,axiom,
    ! [R3: num,M: num] :
      ( ( bit_take_bit_num @ ( numeral_numeral_nat @ R3 ) @ ( bit1 @ M ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ ( pred_numeral @ R3 ) @ M ) ) ) ) ).

% take_bit_num_simps(7)
thf(fact_9803_and__minus__numerals_I8_J,axiom,
    ! [N2: num,M: num] :
      ( ( bit_se725231765392027082nd_int @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) @ ( numeral_numeral_int @ M ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(8)
thf(fact_9804_and__minus__numerals_I4_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit1 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bit0 @ N2 ) ) ) ) ).

% and_minus_numerals(4)
thf(fact_9805_and__minus__numerals_I3_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( uminus_uminus_int @ ( numeral_numeral_int @ ( bit0 @ N2 ) ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ ( bitM @ N2 ) ) ) ) ).

% and_minus_numerals(3)
thf(fact_9806_and__not__num_Osimps_I1_J,axiom,
    ( ( bit_and_not_num @ one @ one )
    = none_num ) ).

% and_not_num.simps(1)
thf(fact_9807_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_9808_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_9809_and__not__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_and_not_num @ one @ ( bit0 @ N2 ) )
      = ( some_num @ one ) ) ).

% and_not_num.simps(2)
thf(fact_9810_and__not__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(4)
thf(fact_9811_GreatestI__ex__nat,axiom,
    ! [P: nat > $o,B: nat] :
      ( ? [X_1: nat] : ( P @ X_1 )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_ex_nat
thf(fact_9812_Greatest__le__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( ord_less_eq_nat @ K @ ( order_Greatest_nat @ P ) ) ) ) ).

% Greatest_le_nat
thf(fact_9813_GreatestI__nat,axiom,
    ! [P: nat > $o,K: nat,B: nat] :
      ( ( P @ K )
     => ( ! [Y3: nat] :
            ( ( P @ Y3 )
           => ( ord_less_eq_nat @ Y3 @ B ) )
       => ( P @ ( order_Greatest_nat @ P ) ) ) ) ).

% GreatestI_nat
thf(fact_9814_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_9815_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_9816_and__not__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% and_not_num.simps(7)
thf(fact_9817_and__not__num__eq__Some__iff,axiom,
    ! [M: num,N2: num,Q3: num] :
      ( ( ( bit_and_not_num @ M @ N2 )
        = ( some_num @ Q3 ) )
      = ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
        = ( numeral_numeral_int @ Q3 ) ) ) ).

% and_not_num_eq_Some_iff
thf(fact_9818_and__not__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(8)
thf(fact_9819_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_9820_int__numeral__not__and__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ M ) ) @ ( numeral_numeral_int @ N2 ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ N2 @ M ) ) ) ).

% int_numeral_not_and_num
thf(fact_9821_int__numeral__and__not__num,axiom,
    ! [M: num,N2: num] :
      ( ( bit_se725231765392027082nd_int @ ( numeral_numeral_int @ M ) @ ( bit_ri7919022796975470100ot_int @ ( numeral_numeral_int @ N2 ) ) )
      = ( case_option_int_num @ zero_zero_int @ numeral_numeral_int @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% int_numeral_and_not_num
thf(fact_9822_take__bit__num__def,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M6: num] :
          ( if_option_num
          @ ( ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M6 ) )
            = zero_zero_nat )
          @ none_num
          @ ( some_num @ ( num_of_nat @ ( bit_se2925701944663578781it_nat @ N @ ( numeral_numeral_nat @ M6 ) ) ) ) ) ) ) ).

% take_bit_num_def
thf(fact_9823_nth__sorted__list__of__set__greaterThanLessThan,axiom,
    ! [N2: nat,J: nat,I2: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ ( suc @ I2 ) ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I2 @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanLessThan
thf(fact_9824_and__not__num_Oelims,axiom,
    ! [X4: num,Xa: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X4 @ Xa )
        = Y )
     => ( ( ( X4 = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X4 = one )
           => ( ? [N3: num] :
                  ( Xa
                  = ( bit0 @ N3 ) )
             => ( Y
               != ( some_num @ one ) ) ) )
         => ( ( ( X4 = one )
             => ( ? [N3: num] :
                    ( Xa
                    = ( bit1 @ N3 ) )
               => ( Y != none_num ) ) )
           => ( ! [M5: num] :
                  ( ( X4
                    = ( bit0 @ M5 ) )
                 => ( ( Xa = one )
                   => ( Y
                     != ( some_num @ ( bit0 @ M5 ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X4
                      = ( bit0 @ M5 ) )
                   => ! [N3: num] :
                        ( ( Xa
                          = ( bit0 @ N3 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X4
                        = ( bit0 @ M5 ) )
                     => ! [N3: num] :
                          ( ( Xa
                            = ( bit1 @ N3 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X4
                          = ( bit1 @ M5 ) )
                       => ( ( Xa = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M5 ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X4
                            = ( bit1 @ M5 ) )
                         => ! [N3: num] :
                              ( ( Xa
                                = ( bit0 @ N3 ) )
                             => ( Y
                               != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                  @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                  @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X4
                              = ( bit1 @ M5 ) )
                           => ! [N3: num] :
                                ( ( Xa
                                  = ( bit1 @ N3 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.elims
thf(fact_9825_Bit__Operations_Otake__bit__num__code,axiom,
    ( bit_take_bit_num
    = ( ^ [N: nat,M6: num] :
          ( produc478579273971653890on_num
          @ ^ [A3: nat,X: num] :
              ( case_nat_option_num @ none_num
              @ ^ [O: nat] :
                  ( case_num_option_num @ ( some_num @ one )
                  @ ^ [P5: num] :
                      ( case_o6005452278849405969um_num @ none_num
                      @ ^ [Q5: num] : ( some_num @ ( bit0 @ Q5 ) )
                      @ ( bit_take_bit_num @ O @ P5 ) )
                  @ ^ [P5: num] : ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_take_bit_num @ O @ P5 ) ) )
                  @ X )
              @ A3 )
          @ ( product_Pair_nat_num @ N @ M6 ) ) ) ) ).

% Bit_Operations.take_bit_num_code
thf(fact_9826_and__not__num_Osimps_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(5)
thf(fact_9827_and__not__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(9)
thf(fact_9828_and__not__num_Osimps_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_and_not_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M @ N2 ) ) ) ).

% and_not_num.simps(6)
thf(fact_9829_and__not__num_Opelims,axiom,
    ! [X4: num,Xa: num,Y: option_num] :
      ( ( ( bit_and_not_num @ X4 @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
       => ( ( ( X4 = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [N3: num] :
                  ( ( Xa
                    = ( bit0 @ N3 ) )
                 => ( ( Y
                      = ( some_num @ one ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
           => ( ( ( X4 = one )
               => ! [N3: num] :
                    ( ( Xa
                      = ( bit1 @ N3 ) )
                   => ( ( Y = none_num )
                     => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X4
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( some_num @ ( bit0 @ M5 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X4
                        = ( bit0 @ M5 ) )
                     => ! [N3: num] :
                          ( ( Xa
                            = ( bit0 @ N3 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X4
                          = ( bit0 @ M5 ) )
                       => ! [N3: num] :
                            ( ( Xa
                              = ( bit1 @ N3 ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X4
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X4
                              = ( bit1 @ M5 ) )
                           => ! [N3: num] :
                                ( ( Xa
                                  = ( bit0 @ N3 ) )
                               => ( ( Y
                                    = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                      @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                      @ ( bit_and_not_num @ M5 @ N3 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X4
                                = ( bit1 @ M5 ) )
                             => ! [N3: num] :
                                  ( ( Xa
                                    = ( bit1 @ N3 ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_and_not_num @ M5 @ N3 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_and_not_num_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_not_num.pelims
thf(fact_9830_and__num_Oelims,axiom,
    ! [X4: num,Xa: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X4 @ Xa )
        = Y )
     => ( ( ( X4 = one )
         => ( ( Xa = one )
           => ( Y
             != ( some_num @ one ) ) ) )
       => ( ( ( X4 = one )
           => ( ? [N3: num] :
                  ( Xa
                  = ( bit0 @ N3 ) )
             => ( Y != none_num ) ) )
         => ( ( ( X4 = one )
             => ( ? [N3: num] :
                    ( Xa
                    = ( bit1 @ N3 ) )
               => ( Y
                 != ( some_num @ one ) ) ) )
           => ( ( ? [M5: num] :
                    ( X4
                    = ( bit0 @ M5 ) )
               => ( ( Xa = one )
                 => ( Y != none_num ) ) )
             => ( ! [M5: num] :
                    ( ( X4
                      = ( bit0 @ M5 ) )
                   => ! [N3: num] :
                        ( ( Xa
                          = ( bit0 @ N3 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X4
                        = ( bit0 @ M5 ) )
                     => ! [N3: num] :
                          ( ( Xa
                            = ( bit1 @ N3 ) )
                         => ( Y
                           != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
                 => ( ( ? [M5: num] :
                          ( X4
                          = ( bit1 @ M5 ) )
                     => ( ( Xa = one )
                       => ( Y
                         != ( some_num @ one ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X4
                            = ( bit1 @ M5 ) )
                         => ! [N3: num] :
                              ( ( Xa
                                = ( bit0 @ N3 ) )
                             => ( Y
                               != ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X4
                              = ( bit1 @ M5 ) )
                           => ! [N3: num] :
                                ( ( Xa
                                  = ( bit1 @ N3 ) )
                               => ( Y
                                 != ( case_o6005452278849405969um_num @ ( some_num @ one )
                                    @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                    @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.elims
thf(fact_9831_xor__num_Oelims,axiom,
    ! [X4: num,Xa: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X4 @ Xa )
        = Y )
     => ( ( ( X4 = one )
         => ( ( Xa = one )
           => ( Y != none_num ) ) )
       => ( ( ( X4 = one )
           => ! [N3: num] :
                ( ( Xa
                  = ( bit0 @ N3 ) )
               => ( Y
                 != ( some_num @ ( bit1 @ N3 ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [N3: num] :
                  ( ( Xa
                    = ( bit1 @ N3 ) )
                 => ( Y
                   != ( some_num @ ( bit0 @ N3 ) ) ) ) )
           => ( ! [M5: num] :
                  ( ( X4
                    = ( bit0 @ M5 ) )
                 => ( ( Xa = one )
                   => ( Y
                     != ( some_num @ ( bit1 @ M5 ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X4
                      = ( bit0 @ M5 ) )
                   => ! [N3: num] :
                        ( ( Xa
                          = ( bit0 @ N3 ) )
                       => ( Y
                         != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X4
                        = ( bit0 @ M5 ) )
                     => ! [N3: num] :
                          ( ( Xa
                            = ( bit1 @ N3 ) )
                         => ( Y
                           != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X4
                          = ( bit1 @ M5 ) )
                       => ( ( Xa = one )
                         => ( Y
                           != ( some_num @ ( bit0 @ M5 ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X4
                            = ( bit1 @ M5 ) )
                         => ! [N3: num] :
                              ( ( Xa
                                = ( bit0 @ N3 ) )
                             => ( Y
                               != ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) )
                     => ~ ! [M5: num] :
                            ( ( X4
                              = ( bit1 @ M5 ) )
                           => ! [N3: num] :
                                ( ( Xa
                                  = ( bit1 @ N3 ) )
                               => ( Y
                                 != ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.elims
thf(fact_9832_and__num_Osimps_I1_J,axiom,
    ( ( bit_un7362597486090784418nd_num @ one @ one )
    = ( some_num @ one ) ) ).

% and_num.simps(1)
thf(fact_9833_xor__num_Osimps_I1_J,axiom,
    ( ( bit_un2480387367778600638or_num @ one @ one )
    = none_num ) ).

% xor_num.simps(1)
thf(fact_9834_and__num_Osimps_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(5)
thf(fact_9835_xor__num_Osimps_I5_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit0 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).

% xor_num.simps(5)
thf(fact_9836_and__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ one )
      = ( some_num @ one ) ) ).

% and_num.simps(7)
thf(fact_9837_and__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit1 @ N2 ) )
      = ( some_num @ one ) ) ).

% and_num.simps(3)
thf(fact_9838_and__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ one )
      = none_num ) ).

% and_num.simps(4)
thf(fact_9839_and__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_un7362597486090784418nd_num @ one @ ( bit0 @ N2 ) )
      = none_num ) ).

% and_num.simps(2)
thf(fact_9840_xor__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ).

% xor_num.simps(9)
thf(fact_9841_and__num_Osimps_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(6)
thf(fact_9842_and__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(8)
thf(fact_9843_xor__num_Osimps_I7_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ one )
      = ( some_num @ ( bit0 @ M ) ) ) ).

% xor_num.simps(7)
thf(fact_9844_xor__num_Osimps_I4_J,axiom,
    ! [M: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ one )
      = ( some_num @ ( bit1 @ M ) ) ) ).

% xor_num.simps(4)
thf(fact_9845_xor__num_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit1 @ N2 ) )
      = ( some_num @ ( bit0 @ N2 ) ) ) ).

% xor_num.simps(3)
thf(fact_9846_xor__num_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( bit_un2480387367778600638or_num @ one @ ( bit0 @ N2 ) )
      = ( some_num @ ( bit1 @ N2 ) ) ) ).

% xor_num.simps(2)
thf(fact_9847_and__num_Osimps_I9_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un7362597486090784418nd_num @ ( bit1 @ M ) @ ( bit1 @ N2 ) )
      = ( case_o6005452278849405969um_num @ ( some_num @ one )
        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
        @ ( bit_un7362597486090784418nd_num @ M @ N2 ) ) ) ).

% and_num.simps(9)
thf(fact_9848_xor__num_Osimps_I8_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit1 @ M ) @ ( bit0 @ N2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).

% xor_num.simps(8)
thf(fact_9849_xor__num_Osimps_I6_J,axiom,
    ! [M: num,N2: num] :
      ( ( bit_un2480387367778600638or_num @ ( bit0 @ M ) @ ( bit1 @ N2 ) )
      = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M @ N2 ) ) ) ) ).

% xor_num.simps(6)
thf(fact_9850_and__num_Opelims,axiom,
    ! [X4: num,Xa: num,Y: option_num] :
      ( ( ( bit_un7362597486090784418nd_num @ X4 @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
       => ( ( ( X4 = one )
           => ( ( Xa = one )
             => ( ( Y
                  = ( some_num @ one ) )
               => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [N3: num] :
                  ( ( Xa
                    = ( bit0 @ N3 ) )
                 => ( ( Y = none_num )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
           => ( ( ( X4 = one )
               => ! [N3: num] :
                    ( ( Xa
                      = ( bit1 @ N3 ) )
                   => ( ( Y
                        = ( some_num @ one ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X4
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y = none_num )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X4
                        = ( bit0 @ M5 ) )
                     => ! [N3: num] :
                          ( ( Xa
                            = ( bit0 @ N3 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X4
                          = ( bit0 @ M5 ) )
                       => ! [N3: num] :
                            ( ( Xa
                              = ( bit1 @ N3 ) )
                           => ( ( Y
                                = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X4
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ one ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X4
                              = ( bit1 @ M5 ) )
                           => ! [N3: num] :
                                ( ( Xa
                                  = ( bit0 @ N3 ) )
                               => ( ( Y
                                    = ( map_option_num_num @ bit0 @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X4
                                = ( bit1 @ M5 ) )
                             => ! [N3: num] :
                                  ( ( Xa
                                    = ( bit1 @ N3 ) )
                                 => ( ( Y
                                      = ( case_o6005452278849405969um_num @ ( some_num @ one )
                                        @ ^ [N10: num] : ( some_num @ ( bit1 @ N10 ) )
                                        @ ( bit_un7362597486090784418nd_num @ M5 @ N3 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un4731106466462545111um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% and_num.pelims
thf(fact_9851_xor__num_Opelims,axiom,
    ! [X4: num,Xa: num,Y: option_num] :
      ( ( ( bit_un2480387367778600638or_num @ X4 @ Xa )
        = Y )
     => ( ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ X4 @ Xa ) )
       => ( ( ( X4 = one )
           => ( ( Xa = one )
             => ( ( Y = none_num )
               => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ one ) ) ) ) )
         => ( ( ( X4 = one )
             => ! [N3: num] :
                  ( ( Xa
                    = ( bit0 @ N3 ) )
                 => ( ( Y
                      = ( some_num @ ( bit1 @ N3 ) ) )
                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit0 @ N3 ) ) ) ) ) )
           => ( ( ( X4 = one )
               => ! [N3: num] :
                    ( ( Xa
                      = ( bit1 @ N3 ) )
                   => ( ( Y
                        = ( some_num @ ( bit0 @ N3 ) ) )
                     => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ one @ ( bit1 @ N3 ) ) ) ) ) )
             => ( ! [M5: num] :
                    ( ( X4
                      = ( bit0 @ M5 ) )
                   => ( ( Xa = one )
                     => ( ( Y
                          = ( some_num @ ( bit1 @ M5 ) ) )
                       => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ one ) ) ) ) )
               => ( ! [M5: num] :
                      ( ( X4
                        = ( bit0 @ M5 ) )
                     => ! [N3: num] :
                          ( ( Xa
                            = ( bit0 @ N3 ) )
                         => ( ( Y
                              = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
                           => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
                 => ( ! [M5: num] :
                        ( ( X4
                          = ( bit0 @ M5 ) )
                       => ! [N3: num] :
                            ( ( Xa
                              = ( bit1 @ N3 ) )
                           => ( ( Y
                                = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit0 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) )
                   => ( ! [M5: num] :
                          ( ( X4
                            = ( bit1 @ M5 ) )
                         => ( ( Xa = one )
                           => ( ( Y
                                = ( some_num @ ( bit0 @ M5 ) ) )
                             => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ one ) ) ) ) )
                     => ( ! [M5: num] :
                            ( ( X4
                              = ( bit1 @ M5 ) )
                           => ! [N3: num] :
                                ( ( Xa
                                  = ( bit0 @ N3 ) )
                               => ( ( Y
                                    = ( some_num @ ( case_option_num_num @ one @ bit1 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) ) )
                                 => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit0 @ N3 ) ) ) ) ) )
                       => ~ ! [M5: num] :
                              ( ( X4
                                = ( bit1 @ M5 ) )
                             => ! [N3: num] :
                                  ( ( Xa
                                    = ( bit1 @ N3 ) )
                                 => ( ( Y
                                      = ( map_option_num_num @ bit0 @ ( bit_un2480387367778600638or_num @ M5 @ N3 ) ) )
                                   => ~ ( accp_P3113834385874906142um_num @ bit_un2901131394128224187um_rel @ ( product_Pair_num_num @ ( bit1 @ M5 ) @ ( bit1 @ N3 ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% xor_num.pelims
thf(fact_9852_xor__num__dict,axiom,
    bit_un2480387367778600638or_num = bit_un6178654185764691216or_num ).

% xor_num_dict
thf(fact_9853_and__num__rel__dict,axiom,
    bit_un4731106466462545111um_rel = bit_un5425074673868309765um_rel ).

% and_num_rel_dict
thf(fact_9854_xor__num__rel__dict,axiom,
    bit_un2901131394128224187um_rel = bit_un3595099601533988841um_rel ).

% xor_num_rel_dict
thf(fact_9855_and__num__dict,axiom,
    bit_un7362597486090784418nd_num = bit_un1837492267222099188nd_num ).

% and_num_dict
thf(fact_9856_nth__sorted__list__of__set__greaterThanAtMost,axiom,
    ! [N2: nat,J: nat,I2: nat] :
      ( ( ord_less_nat @ N2 @ ( minus_minus_nat @ J @ I2 ) )
     => ( ( nth_nat @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) ) @ N2 )
        = ( suc @ ( plus_plus_nat @ I2 @ N2 ) ) ) ) ).

% nth_sorted_list_of_set_greaterThanAtMost
thf(fact_9857_pow_Osimps_I3_J,axiom,
    ! [X4: num,Y: num] :
      ( ( pow @ X4 @ ( bit1 @ Y ) )
      = ( times_times_num @ ( sqr @ ( pow @ X4 @ Y ) ) @ X4 ) ) ).

% pow.simps(3)
thf(fact_9858_finite__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] : ( finite_finite_nat @ ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% finite_greaterThanAtMost
thf(fact_9859_card__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( finite_card_nat @ ( set_or6659071591806873216st_nat @ L @ U ) )
      = ( minus_minus_nat @ U @ L ) ) ).

% card_greaterThanAtMost
thf(fact_9860_sqr_Osimps_I2_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit0 @ N2 ) )
      = ( bit0 @ ( bit0 @ ( sqr @ N2 ) ) ) ) ).

% sqr.simps(2)
thf(fact_9861_sqr_Osimps_I1_J,axiom,
    ( ( sqr @ one )
    = one ) ).

% sqr.simps(1)
thf(fact_9862_atLeastSucAtMost__greaterThanAtMost,axiom,
    ! [L: nat,U: nat] :
      ( ( set_or1269000886237332187st_nat @ ( suc @ L ) @ U )
      = ( set_or6659071591806873216st_nat @ L @ U ) ) ).

% atLeastSucAtMost_greaterThanAtMost
thf(fact_9863_sqr__conv__mult,axiom,
    ( sqr
    = ( ^ [X: num] : ( times_times_num @ X @ X ) ) ) ).

% sqr_conv_mult
thf(fact_9864_pow_Osimps_I2_J,axiom,
    ! [X4: num,Y: num] :
      ( ( pow @ X4 @ ( bit0 @ Y ) )
      = ( sqr @ ( pow @ X4 @ Y ) ) ) ).

% pow.simps(2)
thf(fact_9865_sqr_Osimps_I3_J,axiom,
    ! [N2: num] :
      ( ( sqr @ ( bit1 @ N2 ) )
      = ( bit1 @ ( bit0 @ ( plus_plus_num @ ( sqr @ N2 ) @ N2 ) ) ) ) ).

% sqr.simps(3)
thf(fact_9866_integer__of__num__triv_I2_J,axiom,
    ( ( code_integer_of_num @ ( bit0 @ one ) )
    = ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ).

% integer_of_num_triv(2)
thf(fact_9867_finite__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] : ( finite_finite_int @ ( set_or6656581121297822940st_int @ L @ U ) ) ).

% finite_greaterThanAtMost_int
thf(fact_9868_card__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( finite_card_int @ ( set_or6656581121297822940st_int @ L @ U ) )
      = ( nat2 @ ( minus_minus_int @ U @ L ) ) ) ).

% card_greaterThanAtMost_int
thf(fact_9869_atLeastPlusOneAtMost__greaterThanAtMost__int,axiom,
    ! [L: int,U: int] :
      ( ( set_or1266510415728281911st_int @ ( plus_plus_int @ L @ one_one_int ) @ U )
      = ( set_or6656581121297822940st_int @ L @ U ) ) ).

% atLeastPlusOneAtMost_greaterThanAtMost_int
thf(fact_9870_integer__of__num__triv_I1_J,axiom,
    ( ( code_integer_of_num @ one )
    = one_one_Code_integer ) ).

% integer_of_num_triv(1)
thf(fact_9871_integer__of__num_I2_J,axiom,
    ! [N2: num] :
      ( ( code_integer_of_num @ ( bit0 @ N2 ) )
      = ( plus_p5714425477246183910nteger @ ( code_integer_of_num @ N2 ) @ ( code_integer_of_num @ N2 ) ) ) ).

% integer_of_num(2)
thf(fact_9872_Rats__eq__int__div__nat,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I3: int,N: nat] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( semiri5074537144036343181t_real @ N ) ) )
          & ( N != zero_zero_nat ) ) ) ) ).

% Rats_eq_int_div_nat
thf(fact_9873_image__minus__const__atLeastLessThan__nat,axiom,
    ! [C: nat,Y: nat,X4: nat] :
      ( ( ( ord_less_nat @ C @ Y )
       => ( ( image_nat_nat
            @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
            @ ( set_or4665077453230672383an_nat @ X4 @ Y ) )
          = ( set_or4665077453230672383an_nat @ ( minus_minus_nat @ X4 @ C ) @ ( minus_minus_nat @ Y @ C ) ) ) )
      & ( ~ ( ord_less_nat @ C @ Y )
       => ( ( ( ord_less_nat @ X4 @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X4 @ Y ) )
              = ( insert_nat @ zero_zero_nat @ bot_bot_set_nat ) ) )
          & ( ~ ( ord_less_nat @ X4 @ Y )
           => ( ( image_nat_nat
                @ ^ [I3: nat] : ( minus_minus_nat @ I3 @ C )
                @ ( set_or4665077453230672383an_nat @ X4 @ Y ) )
              = bot_bot_set_nat ) ) ) ) ) ).

% image_minus_const_atLeastLessThan_nat
thf(fact_9874_bij__betw__Suc,axiom,
    ! [M7: set_nat,N4: set_nat] :
      ( ( bij_betw_nat_nat @ suc @ M7 @ N4 )
      = ( ( image_nat_nat @ suc @ M7 )
        = N4 ) ) ).

% bij_betw_Suc
thf(fact_9875_Rats__abs__iff,axiom,
    ! [X4: real] :
      ( ( member_real @ ( abs_abs_real @ X4 ) @ field_5140801741446780682s_real )
      = ( member_real @ X4 @ field_5140801741446780682s_real ) ) ).

% Rats_abs_iff
thf(fact_9876_image__Suc__atLeastAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or1269000886237332187st_nat @ I2 @ J ) )
      = ( set_or1269000886237332187st_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastAtMost
thf(fact_9877_image__Suc__atLeastLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( ( image_nat_nat @ suc @ ( set_or4665077453230672383an_nat @ I2 @ J ) )
      = ( set_or4665077453230672383an_nat @ ( suc @ I2 ) @ ( suc @ J ) ) ) ).

% image_Suc_atLeastLessThan
thf(fact_9878_Rats__dense__in__real,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ? [X5: real] :
          ( ( member_real @ X5 @ field_5140801741446780682s_real )
          & ( ord_less_real @ X4 @ X5 )
          & ( ord_less_real @ X5 @ Y ) ) ) ).

% Rats_dense_in_real
thf(fact_9879_Rats__no__bot__less,axiom,
    ! [X4: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_real @ X5 @ X4 ) ) ).

% Rats_no_bot_less
thf(fact_9880_Rats__no__top__le,axiom,
    ! [X4: real] :
    ? [X5: real] :
      ( ( member_real @ X5 @ field_5140801741446780682s_real )
      & ( ord_less_eq_real @ X4 @ X5 ) ) ).

% Rats_no_top_le
thf(fact_9881_zero__notin__Suc__image,axiom,
    ! [A2: set_nat] :
      ~ ( member_nat @ zero_zero_nat @ ( image_nat_nat @ suc @ A2 ) ) ).

% zero_notin_Suc_image
thf(fact_9882_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_9883_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_9884_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_9885_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_9886_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_9887_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_9888_Rats__eq__int__div__int,axiom,
    ( field_5140801741446780682s_real
    = ( collect_real
      @ ^ [Uu3: real] :
        ? [I3: int,J3: int] :
          ( ( Uu3
            = ( divide_divide_real @ ( ring_1_of_int_real @ I3 ) @ ( ring_1_of_int_real @ J3 ) ) )
          & ( J3 != zero_zero_int ) ) ) ) ).

% Rats_eq_int_div_int
thf(fact_9889_Inf__real__def,axiom,
    ( comple4887499456419720421f_real
    = ( ^ [X3: set_real] : ( uminus_uminus_real @ ( comple1385675409528146559p_real @ ( image_real_real @ uminus_uminus_real @ X3 ) ) ) ) ) ).

% Inf_real_def
thf(fact_9890_finite__int__iff__bounded__le,axiom,
    ( finite_finite_int
    = ( ^ [S5: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_atMost_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded_le
thf(fact_9891_finite__int__iff__bounded,axiom,
    ( finite_finite_int
    = ( ^ [S5: set_int] :
        ? [K3: int] : ( ord_less_eq_set_int @ ( image_int_int @ abs_abs_int @ S5 ) @ ( set_ord_lessThan_int @ K3 ) ) ) ) ).

% finite_int_iff_bounded
thf(fact_9892_image__int__atLeastAtMost,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or1269000886237332187st_nat @ A @ B ) )
      = ( set_or1266510415728281911st_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastAtMost
thf(fact_9893_image__int__atLeastLessThan,axiom,
    ! [A: nat,B: nat] :
      ( ( image_nat_int @ semiri1314217659103216013at_int @ ( set_or4665077453230672383an_nat @ A @ B ) )
      = ( set_or4662586982721622107an_int @ ( semiri1314217659103216013at_int @ A ) @ ( semiri1314217659103216013at_int @ B ) ) ) ).

% image_int_atLeastLessThan
thf(fact_9894_image__add__int__atLeastLessThan,axiom,
    ! [L: int,U: int] :
      ( ( image_int_int
        @ ^ [X: int] : ( plus_plus_int @ X @ L )
        @ ( set_or4662586982721622107an_int @ zero_zero_int @ ( minus_minus_int @ U @ L ) ) )
      = ( set_or4662586982721622107an_int @ L @ U ) ) ).

% image_add_int_atLeastLessThan
thf(fact_9895_image__atLeastZeroLessThan__int,axiom,
    ! [U: int] :
      ( ( ord_less_eq_int @ zero_zero_int @ U )
     => ( ( set_or4662586982721622107an_int @ zero_zero_int @ U )
        = ( image_nat_int @ semiri1314217659103216013at_int @ ( set_ord_lessThan_nat @ ( nat2 @ U ) ) ) ) ) ).

% image_atLeastZeroLessThan_int
thf(fact_9896_suminf__eq__SUP__real,axiom,
    ! [X8: nat > real] :
      ( ( summable_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ zero_zero_real @ ( X8 @ I4 ) )
       => ( ( suminf_real @ X8 )
          = ( comple1385675409528146559p_real
            @ ( image_nat_real
              @ ^ [I3: nat] : ( groups6591440286371151544t_real @ X8 @ ( set_ord_lessThan_nat @ I3 ) )
              @ top_top_set_nat ) ) ) ) ) ).

% suminf_eq_SUP_real
thf(fact_9897_UN__atMost__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atMost_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atMost_UNIV
thf(fact_9898_UN__lessThan__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_lessThan_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_lessThan_UNIV
thf(fact_9899_range__mod,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( image_nat_nat
          @ ^ [M6: nat] : ( modulo_modulo_nat @ M6 @ N2 )
          @ top_top_set_nat )
        = ( set_or4665077453230672383an_nat @ zero_zero_nat @ N2 ) ) ) ).

% range_mod
thf(fact_9900_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_9901_card__UNIV__unit,axiom,
    ( ( finite410649719033368117t_unit @ top_to1996260823553986621t_unit )
    = one_one_nat ) ).

% card_UNIV_unit
thf(fact_9902_card__UNIV__bool,axiom,
    ( ( finite_card_o @ top_top_set_o )
    = ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ).

% card_UNIV_bool
thf(fact_9903_root__def,axiom,
    ( root
    = ( ^ [N: nat,X: real] :
          ( if_real @ ( N = zero_zero_nat ) @ zero_zero_real
          @ ( the_in5290026491893676941l_real @ top_top_set_real
            @ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N ) )
            @ X ) ) ) ) ).

% root_def
thf(fact_9904_card__UNIV__char,axiom,
    ( ( finite_card_char @ top_top_set_char )
    = ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% card_UNIV_char
thf(fact_9905_DERIV__even__real__root,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
       => ( ( ord_less_real @ X4 @ 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 @ X4 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).

% DERIV_even_real_root
thf(fact_9906_MVT2,axiom,
    ! [A: real,B: real,F: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B )
             => ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
       => ? [Z2: real] :
            ( ( ord_less_real @ A @ Z2 )
            & ( ord_less_real @ Z2 @ B )
            & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
              = ( times_times_real @ ( minus_minus_real @ B @ A ) @ ( F4 @ Z2 ) ) ) ) ) ) ).

% MVT2
thf(fact_9907_DERIV__nonneg__imp__nondecreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_eq_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_nonneg_imp_nondecreasing
thf(fact_9908_DERIV__nonpos__imp__nonincreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_eq_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_eq_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_nonpos_imp_nonincreasing
thf(fact_9909_DERIV__neg__imp__decreasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ zero_zero_real ) ) ) )
       => ( ord_less_real @ ( F @ B ) @ ( F @ A ) ) ) ) ).

% DERIV_neg_imp_decreasing
thf(fact_9910_DERIV__pos__imp__increasing,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ord_less_eq_real @ A @ X5 )
           => ( ( ord_less_eq_real @ X5 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing
thf(fact_9911_deriv__nonneg__imp__mono,axiom,
    ! [A: real,B: real,G: real > real,G2: real > real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B ) )
         => ( has_fi5821293074295781190e_real @ G @ ( G2 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1222579329274155063t_real @ A @ B ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( G2 @ X5 ) ) )
       => ( ( ord_less_eq_real @ A @ B )
         => ( ord_less_eq_real @ ( G @ A ) @ ( G @ B ) ) ) ) ) ).

% deriv_nonneg_imp_mono
thf(fact_9912_DERIV__isconst3,axiom,
    ! [A: real,B: real,X4: real,Y: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( member_real @ X4 @ ( set_or1633881224788618240n_real @ A @ B ) )
       => ( ( member_real @ Y @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ! [X5: real] :
                ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
           => ( ( F @ X4 )
              = ( F @ Y ) ) ) ) ) ) ).

% DERIV_isconst3
thf(fact_9913_DERIV__local__const,axiom,
    ! [F: real > real,L: real,X4: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ D )
             => ( ( F @ X4 )
                = ( F @ Y3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_const
thf(fact_9914_DERIV__neg__dec__left,axiom,
    ! [F: real > real,L: real,X4: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_neg_dec_left
thf(fact_9915_DERIV__pos__inc__left,axiom,
    ! [F: real > real,L: real,X4: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ).

% DERIV_pos_inc_left
thf(fact_9916_DERIV__neg__dec__right,axiom,
    ! [F: real > real,L: real,X4: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ).

% DERIV_neg_dec_right
thf(fact_9917_DERIV__pos__inc__right,axiom,
    ! [F: real > real,L: real,X4: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( ord_less_real @ H4 @ D3 )
                 => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ).

% DERIV_pos_inc_right
thf(fact_9918_DERIV__ln,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( inverse_inverse_real @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_ln
thf(fact_9919_has__real__derivative__neg__dec__right,axiom,
    ! [F: real > real,L: real,X4: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X4 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_right
thf(fact_9920_has__real__derivative__pos__inc__right,axiom,
    ! [F: real > real,L: real,X4: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( plus_plus_real @ X4 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( plus_plus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_right
thf(fact_9921_has__real__derivative__pos__inc__left,axiom,
    ! [F: real > real,L: real,X4: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
     => ( ( ord_less_real @ zero_zero_real @ L )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X4 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) @ ( F @ X4 ) ) ) ) ) ) ) ) ).

% has_real_derivative_pos_inc_left
thf(fact_9922_has__real__derivative__neg__dec__left,axiom,
    ! [F: real > real,L: real,X4: real,S2: set_real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ S2 ) )
     => ( ( ord_less_real @ L @ zero_zero_real )
       => ? [D3: real] :
            ( ( ord_less_real @ zero_zero_real @ D3 )
            & ! [H4: real] :
                ( ( ord_less_real @ zero_zero_real @ H4 )
               => ( ( member_real @ ( minus_minus_real @ X4 @ H4 ) @ S2 )
                 => ( ( ord_less_real @ H4 @ D3 )
                   => ( ord_less_real @ ( F @ X4 ) @ ( F @ ( minus_minus_real @ X4 @ H4 ) ) ) ) ) ) ) ) ) ).

% has_real_derivative_neg_dec_left
thf(fact_9923_DERIV__const__average,axiom,
    ! [A: real,B: real,V: real > real,K: real] :
      ( ( A != B )
     => ( ! [X5: real] : ( has_fi5821293074295781190e_real @ V @ K @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ( ( V @ ( divide_divide_real @ ( plus_plus_real @ A @ B ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) )
          = ( divide_divide_real @ ( plus_plus_real @ ( V @ A ) @ ( V @ B ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% DERIV_const_average
thf(fact_9924_DERIV__local__max,axiom,
    ! [F: real > real,L: real,X4: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ Y3 ) @ ( F @ X4 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_max
thf(fact_9925_DERIV__local__min,axiom,
    ! [F: real > real,L: real,X4: real,D: real] :
      ( ( has_fi5821293074295781190e_real @ F @ L @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ D )
       => ( ! [Y3: real] :
              ( ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ X4 @ Y3 ) ) @ D )
             => ( ord_less_eq_real @ ( F @ X4 ) @ ( F @ Y3 ) ) )
         => ( L = zero_zero_real ) ) ) ) ).

% DERIV_local_min
thf(fact_9926_DERIV__ln__divide,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( has_fi5821293074295781190e_real @ ln_ln_real @ ( divide_divide_real @ one_one_real @ X4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_ln_divide
thf(fact_9927_DERIV__fun__pow,axiom,
    ! [G: real > real,M: real,X4: real,N2: nat] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( has_fi5821293074295781190e_real
        @ ^ [X: real] : ( power_power_real @ ( G @ X ) @ N2 )
        @ ( times_times_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( G @ X4 ) @ ( minus_minus_nat @ N2 @ one_one_nat ) ) ) @ M )
        @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_fun_pow
thf(fact_9928_DERIV__pow,axiom,
    ! [N2: nat,X4: real,S: set_real] :
      ( has_fi5821293074295781190e_real
      @ ^ [X: real] : ( power_power_real @ X @ N2 )
      @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ X4 @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) )
      @ ( topolo2177554685111907308n_real @ X4 @ S ) ) ).

% DERIV_pow
thf(fact_9929_has__real__derivative__powr,axiom,
    ! [Z: real,R3: real] :
      ( ( ord_less_real @ zero_zero_real @ Z )
     => ( has_fi5821293074295781190e_real
        @ ^ [Z5: real] : ( powr_real @ Z5 @ R3 )
        @ ( times_times_real @ R3 @ ( powr_real @ Z @ ( minus_minus_real @ R3 @ one_one_real ) ) )
        @ ( topolo2177554685111907308n_real @ Z @ top_top_set_real ) ) ) ).

% has_real_derivative_powr
thf(fact_9930_DERIV__log,axiom,
    ! [X4: real,B: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( has_fi5821293074295781190e_real @ ( log @ B ) @ ( divide_divide_real @ one_one_real @ ( times_times_real @ ( ln_ln_real @ B ) @ X4 ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_log
thf(fact_9931_DERIV__fun__powr,axiom,
    ! [G: real > real,M: real,X4: real,R3: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X4 ) )
       => ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( powr_real @ ( G @ X ) @ R3 )
          @ ( times_times_real @ ( times_times_real @ R3 @ ( powr_real @ ( G @ X4 ) @ ( minus_minus_real @ R3 @ ( semiri5074537144036343181t_real @ one_one_nat ) ) ) ) @ M )
          @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_fun_powr
thf(fact_9932_DERIV__powr,axiom,
    ! [G: real > real,M: real,X4: real,F: real > real,R3: real] :
      ( ( has_fi5821293074295781190e_real @ G @ M @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
     => ( ( ord_less_real @ zero_zero_real @ ( G @ X4 ) )
       => ( ( has_fi5821293074295781190e_real @ F @ R3 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) )
         => ( has_fi5821293074295781190e_real
            @ ^ [X: real] : ( powr_real @ ( G @ X ) @ ( F @ X ) )
            @ ( times_times_real @ ( powr_real @ ( G @ X4 ) @ ( F @ X4 ) ) @ ( plus_plus_real @ ( times_times_real @ R3 @ ( ln_ln_real @ ( G @ X4 ) ) ) @ ( divide_divide_real @ ( times_times_real @ M @ ( F @ X4 ) ) @ ( G @ X4 ) ) ) )
            @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).

% DERIV_powr
thf(fact_9933_DERIV__real__sqrt,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ zero_zero_real @ X4 )
     => ( has_fi5821293074295781190e_real @ sqrt @ ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_real_sqrt
thf(fact_9934_DERIV__series_H,axiom,
    ! [F: real > nat > real,F4: real > nat > real,X0: real,A: real,B: real,L5: nat > real] :
      ( ! [N3: nat] :
          ( has_fi5821293074295781190e_real
          @ ^ [X: real] : ( F @ X @ N3 )
          @ ( F4 @ X0 @ N3 )
          @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
           => ( summable_real @ ( F @ X5 ) ) )
       => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ A @ B ) )
         => ( ( summable_real @ ( F4 @ X0 ) )
           => ( ( summable_real @ L5 )
             => ( ! [N3: nat,X5: real,Y3: real] :
                    ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ A @ B ) )
                   => ( ( member_real @ Y3 @ ( set_or1633881224788618240n_real @ A @ B ) )
                     => ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ ( F @ X5 @ N3 ) @ ( F @ Y3 @ N3 ) ) ) @ ( times_times_real @ ( L5 @ N3 ) @ ( abs_abs_real @ ( minus_minus_real @ X5 @ Y3 ) ) ) ) ) )
               => ( has_fi5821293074295781190e_real
                  @ ^ [X: real] : ( suminf_real @ ( F @ X ) )
                  @ ( suminf_real @ ( F4 @ X0 ) )
                  @ ( topolo2177554685111907308n_real @ X0 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_series'
thf(fact_9935_DERIV__arctan,axiom,
    ! [X4: real] : ( has_fi5821293074295781190e_real @ arctan @ ( inverse_inverse_real @ ( plus_plus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ).

% DERIV_arctan
thf(fact_9936_DERIV__real__sqrt__generic,axiom,
    ! [X4: real,D4: real] :
      ( ( X4 != zero_zero_real )
     => ( ( ( ord_less_real @ zero_zero_real @ X4 )
         => ( D4
            = ( divide_divide_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
       => ( ( ( ord_less_real @ X4 @ zero_zero_real )
           => ( D4
              = ( divide_divide_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( sqrt @ X4 ) ) ) @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) )
         => ( has_fi5821293074295781190e_real @ sqrt @ D4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ).

% DERIV_real_sqrt_generic
thf(fact_9937_arsinh__real__has__field__derivative,axiom,
    ! [X4: real,A2: set_real] : ( has_fi5821293074295781190e_real @ arsinh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( plus_plus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A2 ) ) ).

% arsinh_real_has_field_derivative
thf(fact_9938_arcosh__real__has__field__derivative,axiom,
    ! [X4: real,A2: set_real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( has_fi5821293074295781190e_real @ arcosh_real @ ( divide_divide_real @ one_one_real @ ( sqrt @ ( minus_minus_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_real ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A2 ) ) ) ).

% arcosh_real_has_field_derivative
thf(fact_9939_artanh__real__has__field__derivative,axiom,
    ! [X4: real,A2: set_real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real @ artanh_real @ ( divide_divide_real @ one_one_real @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ A2 ) ) ) ).

% artanh_real_has_field_derivative
thf(fact_9940_DERIV__power__series_H,axiom,
    ! [R: real,F: nat > real,X0: real] :
      ( ! [X5: real] :
          ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
         => ( summable_real
            @ ^ [N: nat] : ( times_times_real @ ( times_times_real @ ( F @ N ) @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) @ ( power_power_real @ X5 @ N ) ) ) )
     => ( ( member_real @ X0 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ R ) @ R ) )
       => ( ( ord_less_real @ zero_zero_real @ R )
         => ( has_fi5821293074295781190e_real
            @ ^ [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_9941_DERIV__real__root,axiom,
    ! [N2: nat,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ord_less_real @ zero_zero_real @ X4 )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X4 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_real_root
thf(fact_9942_DERIV__arccos,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arccos @ ( inverse_inverse_real @ ( uminus_uminus_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_arccos
thf(fact_9943_DERIV__arcsin,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( has_fi5821293074295781190e_real @ arcsin @ ( inverse_inverse_real @ ( sqrt @ ( minus_minus_real @ one_one_real @ ( power_power_real @ X4 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_arcsin
thf(fact_9944_Maclaurin__all__le__objl,axiom,
    ! [Diff: nat > real > real,F: real > real,X4: real,N2: nat] :
      ( ( ( ( Diff @ zero_zero_nat )
          = F )
        & ! [M5: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
     => ? [T3: real] :
          ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X4 ) )
          & ( ( F @ X4 )
            = ( plus_plus_real
              @ ( groups6591440286371151544t_real
                @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
                @ ( set_ord_lessThan_nat @ N2 ) )
              @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X4 @ N2 ) ) ) ) ) ) ).

% Maclaurin_all_le_objl
thf(fact_9945_Maclaurin__all__le,axiom,
    ! [Diff: nat > real > real,F: real > real,X4: real,N2: nat] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M5: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X4 ) )
            & ( ( F @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X4 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_all_le
thf(fact_9946_DERIV__odd__real__root,axiom,
    ! [N2: nat,X4: real] :
      ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
     => ( ( X4 != zero_zero_real )
       => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X4 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ).

% DERIV_odd_real_root
thf(fact_9947_Maclaurin__minus,axiom,
    ! [H: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ H @ zero_zero_real )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M5: nat,T3: real] :
                ( ( ( ord_less_nat @ M5 @ N2 )
                  & ( ord_less_eq_real @ H @ T3 )
                  & ( ord_less_eq_real @ T3 @ zero_zero_real ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ H @ T3 )
                & ( ord_less_real @ T3 @ zero_zero_real )
                & ( ( F @ H )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_minus
thf(fact_9948_Maclaurin2,axiom,
    ! [H: real,Diff: nat > real > real,F: real > real,N2: nat] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M5: nat,T3: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ zero_zero_real @ T3 )
                & ( ord_less_eq_real @ T3 @ H ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ? [T3: real] :
              ( ( ord_less_real @ zero_zero_real @ T3 )
              & ( ord_less_eq_real @ T3 @ H )
              & ( ( F @ H )
                = ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
                    @ ( set_ord_lessThan_nat @ N2 ) )
                  @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H @ N2 ) ) ) ) ) ) ) ) ).

% Maclaurin2
thf(fact_9949_Maclaurin,axiom,
    ! [H: real,N2: nat,Diff: nat > real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ H )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( ( Diff @ zero_zero_nat )
            = F )
         => ( ! [M5: nat,T3: real] :
                ( ( ( ord_less_nat @ M5 @ N2 )
                  & ( ord_less_eq_real @ zero_zero_real @ T3 )
                  & ( ord_less_eq_real @ T3 @ H ) )
               => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ T3 )
                & ( ord_less_real @ T3 @ H )
                & ( ( F @ H )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ H @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ H @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin
thf(fact_9950_Maclaurin__all__lt,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X4: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ( ord_less_nat @ zero_zero_nat @ N2 )
       => ( ( X4 != zero_zero_real )
         => ( ! [M5: nat,X5: real] : ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
           => ? [T3: real] :
                ( ( ord_less_real @ zero_zero_real @ ( abs_abs_real @ T3 ) )
                & ( ord_less_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X4 ) )
                & ( ( F @ X4 )
                  = ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
                      @ ( set_ord_lessThan_nat @ N2 ) )
                    @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X4 @ N2 ) ) ) ) ) ) ) ) ) ).

% Maclaurin_all_lt
thf(fact_9951_Maclaurin__bi__le,axiom,
    ! [Diff: nat > real > real,F: real > real,N2: nat,X4: real] :
      ( ( ( Diff @ zero_zero_nat )
        = F )
     => ( ! [M5: nat,T3: real] :
            ( ( ( ord_less_nat @ M5 @ N2 )
              & ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X4 ) ) )
           => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
       => ? [T3: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ T3 ) @ ( abs_abs_real @ X4 ) )
            & ( ( F @ X4 )
              = ( plus_plus_real
                @ ( groups6591440286371151544t_real
                  @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ X4 @ M6 ) )
                  @ ( set_ord_lessThan_nat @ N2 ) )
                @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ X4 @ N2 ) ) ) ) ) ) ) ).

% Maclaurin_bi_le
thf(fact_9952_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 )
       => ( ! [M5: nat,T3: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ A @ T3 )
                & ( ord_less_eq_real @ T3 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ? [T3: real] :
                  ( ( ord_less_real @ A @ T3 )
                  & ( ord_less_real @ T3 @ C )
                  & ( ( F @ A )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ A @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_down
thf(fact_9953_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 )
       => ( ! [M5: nat,T3: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ A @ T3 )
                & ( ord_less_eq_real @ T3 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_real @ C @ B )
             => ? [T3: real] :
                  ( ( ord_less_real @ C @ T3 )
                  & ( ord_less_real @ T3 @ B )
                  & ( ( F @ B )
                    = ( plus_plus_real
                      @ ( groups6591440286371151544t_real
                        @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ M6 ) )
                        @ ( set_ord_lessThan_nat @ N2 ) )
                      @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ B @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ).

% Taylor_up
thf(fact_9954_Taylor,axiom,
    ! [N2: nat,Diff: nat > real > real,F: real > real,A: real,B: real,C: real,X4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( ( Diff @ zero_zero_nat )
          = F )
       => ( ! [M5: nat,T3: real] :
              ( ( ( ord_less_nat @ M5 @ N2 )
                & ( ord_less_eq_real @ A @ T3 )
                & ( ord_less_eq_real @ T3 @ B ) )
             => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
         => ( ( ord_less_eq_real @ A @ C )
           => ( ( ord_less_eq_real @ C @ B )
             => ( ( ord_less_eq_real @ A @ X4 )
               => ( ( ord_less_eq_real @ X4 @ B )
                 => ( ( X4 != C )
                   => ? [T3: real] :
                        ( ( ( ord_less_real @ X4 @ C )
                         => ( ( ord_less_real @ X4 @ T3 )
                            & ( ord_less_real @ T3 @ C ) ) )
                        & ( ~ ( ord_less_real @ X4 @ C )
                         => ( ( ord_less_real @ C @ T3 )
                            & ( ord_less_real @ T3 @ X4 ) ) )
                        & ( ( F @ X4 )
                          = ( plus_plus_real
                            @ ( groups6591440286371151544t_real
                              @ ^ [M6: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ M6 @ C ) @ ( semiri2265585572941072030t_real @ M6 ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ C ) @ M6 ) )
                              @ ( set_ord_lessThan_nat @ N2 ) )
                            @ ( times_times_real @ ( divide_divide_real @ ( Diff @ N2 @ T3 ) @ ( semiri2265585572941072030t_real @ N2 ) ) @ ( power_power_real @ ( minus_minus_real @ X4 @ C ) @ N2 ) ) ) ) ) ) ) ) ) ) ) ) ) ).

% Taylor
thf(fact_9955_Maclaurin__lemma2,axiom,
    ! [N2: nat,H: real,Diff: nat > real > real,K: nat,B3: real] :
      ( ! [M5: nat,T3: real] :
          ( ( ( ord_less_nat @ M5 @ N2 )
            & ( ord_less_eq_real @ zero_zero_real @ T3 )
            & ( ord_less_eq_real @ T3 @ H ) )
         => ( has_fi5821293074295781190e_real @ ( Diff @ M5 ) @ ( Diff @ ( suc @ M5 ) @ T3 ) @ ( topolo2177554685111907308n_real @ T3 @ top_top_set_real ) ) )
     => ( ( N2
          = ( suc @ K ) )
       => ! [M2: nat,T4: real] :
            ( ( ( ord_less_nat @ M2 @ N2 )
              & ( ord_less_eq_real @ zero_zero_real @ T4 )
              & ( ord_less_eq_real @ T4 @ H ) )
           => ( has_fi5821293074295781190e_real
              @ ^ [U2: real] :
                  ( minus_minus_real @ ( Diff @ M2 @ U2 )
                  @ ( plus_plus_real
                    @ ( groups6591440286371151544t_real
                      @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ M2 @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ U2 @ P5 ) )
                      @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ M2 ) ) )
                    @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ U2 @ ( minus_minus_nat @ N2 @ M2 ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ M2 ) ) ) ) ) )
              @ ( minus_minus_real @ ( Diff @ ( suc @ M2 ) @ T4 )
                @ ( plus_plus_real
                  @ ( groups6591440286371151544t_real
                    @ ^ [P5: nat] : ( times_times_real @ ( divide_divide_real @ ( Diff @ ( plus_plus_nat @ ( suc @ M2 ) @ P5 ) @ zero_zero_real ) @ ( semiri2265585572941072030t_real @ P5 ) ) @ ( power_power_real @ T4 @ P5 ) )
                    @ ( set_ord_lessThan_nat @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) )
                  @ ( times_times_real @ B3 @ ( divide_divide_real @ ( power_power_real @ T4 @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) @ ( semiri2265585572941072030t_real @ ( minus_minus_nat @ N2 @ ( suc @ M2 ) ) ) ) ) ) )
              @ ( topolo2177554685111907308n_real @ T4 @ top_top_set_real ) ) ) ) ) ).

% Maclaurin_lemma2
thf(fact_9956_DERIV__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( abs_abs_real @ X4 ) @ one_one_real )
     => ( has_fi5821293074295781190e_real
        @ ^ [X9: real] :
            ( suminf_real
            @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( times_times_real @ ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) @ ( power_power_real @ X9 @ ( plus_plus_nat @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) @ one_one_nat ) ) ) ) )
        @ ( suminf_real
          @ ^ [K3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ K3 ) @ ( power_power_real @ X4 @ ( times_times_nat @ K3 @ ( numeral_numeral_nat @ ( bit0 @ one ) ) ) ) ) )
        @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ).

% DERIV_arctan_series
thf(fact_9957_DERIV__real__root__generic,axiom,
    ! [N2: nat,X4: real,D4: real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( X4 != zero_zero_real )
       => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
           => ( ( ord_less_real @ zero_zero_real @ X4 )
             => ( D4
                = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X4 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) )
         => ( ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
             => ( ( ord_less_real @ X4 @ zero_zero_real )
               => ( D4
                  = ( uminus_uminus_real @ ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X4 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) ) ) )
           => ( ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
               => ( D4
                  = ( inverse_inverse_real @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ ( power_power_real @ ( root @ N2 @ X4 ) @ ( minus_minus_nat @ N2 @ ( suc @ zero_zero_nat ) ) ) ) ) ) )
             => ( has_fi5821293074295781190e_real @ ( root @ N2 ) @ D4 @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ).

% DERIV_real_root_generic
thf(fact_9958_UNIV__char__of__nat,axiom,
    ( top_top_set_char
    = ( image_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ) ).

% UNIV_char_of_nat
thf(fact_9959_nat__of__char__less__256,axiom,
    ! [C: char] : ( ord_less_nat @ ( comm_s629917340098488124ar_nat @ C ) @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% nat_of_char_less_256
thf(fact_9960_range__nat__of__char,axiom,
    ( ( image_char_nat @ comm_s629917340098488124ar_nat @ top_top_set_char )
    = ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ) ).

% range_nat_of_char
thf(fact_9961_integer__of__char__code,axiom,
    ! [B0: $o,B1: $o,B22: $o,B32: $o,B42: $o,B52: $o,B62: $o,B72: $o] :
      ( ( integer_of_char @ ( char2 @ B0 @ B1 @ B22 @ B32 @ B42 @ B52 @ B62 @ B72 ) )
      = ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( plus_p5714425477246183910nteger @ ( times_3573771949741848930nteger @ ( zero_n356916108424825756nteger @ B72 ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B62 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B52 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B42 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B32 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B22 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B1 ) ) @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) @ ( zero_n356916108424825756nteger @ B0 ) ) ) ).

% integer_of_char_code
thf(fact_9962_String_Ochar__of__ascii__of,axiom,
    ! [C: char] :
      ( ( comm_s629917340098488124ar_nat @ ( ascii_of @ C ) )
      = ( bit_se2925701944663578781it_nat @ ( numeral_numeral_nat @ ( bit1 @ ( bit1 @ one ) ) ) @ ( comm_s629917340098488124ar_nat @ C ) ) ) ).

% String.char_of_ascii_of
thf(fact_9963_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_9964_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_9965_sorted__list__of__set__greaterThanAtMost,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ ( suc @ I2 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ I2 @ J ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or6659071591806873216st_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanAtMost
thf(fact_9966_sorted__list__of__set__greaterThanLessThan,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ ( suc @ I2 ) @ J )
     => ( ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ I2 @ J ) )
        = ( cons_nat @ ( suc @ I2 ) @ ( linord2614967742042102400et_nat @ ( set_or5834768355832116004an_nat @ ( suc @ I2 ) @ J ) ) ) ) ) ).

% sorted_list_of_set_greaterThanLessThan
thf(fact_9967_upto__aux__rec,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int,Js: list_int] : ( if_list_int @ ( ord_less_int @ J3 @ I3 ) @ Js @ ( upto_aux @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) @ ( cons_int @ J3 @ Js ) ) ) ) ) ).

% upto_aux_rec
thf(fact_9968_upto_Opelims,axiom,
    ! [X4: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X4 @ Xa )
        = Y )
     => ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) )
       => ~ ( ( ( ( ord_less_eq_int @ X4 @ Xa )
               => ( Y
                  = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
              & ( ~ ( ord_less_eq_int @ X4 @ Xa )
               => ( Y = nil_int ) ) )
           => ~ ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ X4 @ Xa ) ) ) ) ) ).

% upto.pelims
thf(fact_9969_upto__Nil,axiom,
    ! [I2: int,J: int] :
      ( ( ( upto @ I2 @ J )
        = nil_int )
      = ( ord_less_int @ J @ I2 ) ) ).

% upto_Nil
thf(fact_9970_upto__Nil2,axiom,
    ! [I2: int,J: int] :
      ( ( nil_int
        = ( upto @ I2 @ J ) )
      = ( ord_less_int @ J @ I2 ) ) ).

% upto_Nil2
thf(fact_9971_upto__empty,axiom,
    ! [J: int,I2: int] :
      ( ( ord_less_int @ J @ I2 )
     => ( ( upto @ I2 @ J )
        = nil_int ) ) ).

% upto_empty
thf(fact_9972_upto__single,axiom,
    ! [I2: int] :
      ( ( upto @ I2 @ I2 )
      = ( cons_int @ I2 @ nil_int ) ) ).

% upto_single
thf(fact_9973_nth__upto,axiom,
    ! [I2: int,K: nat,J: int] :
      ( ( ord_less_eq_int @ ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) @ J )
     => ( ( nth_int @ ( upto @ I2 @ J ) @ K )
        = ( plus_plus_int @ I2 @ ( semiri1314217659103216013at_int @ K ) ) ) ) ).

% nth_upto
thf(fact_9974_length__upto,axiom,
    ! [I2: int,J: int] :
      ( ( size_size_list_int @ ( upto @ I2 @ J ) )
      = ( nat2 @ ( plus_plus_int @ ( minus_minus_int @ J @ I2 ) @ one_one_int ) ) ) ).

% length_upto
thf(fact_9975_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_9976_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_9977_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_9978_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_9979_upto__aux__def,axiom,
    ( upto_aux
    = ( ^ [I3: int,J3: int] : ( append_int @ ( upto @ I3 @ J3 ) ) ) ) ).

% upto_aux_def
thf(fact_9980_sup__enat__def,axiom,
    sup_su3973961784419623482d_enat = ord_ma741700101516333627d_enat ).

% sup_enat_def
thf(fact_9981_sup__nat__def,axiom,
    sup_sup_nat = ord_max_nat ).

% sup_nat_def
thf(fact_9982_upto__code,axiom,
    ( upto
    = ( ^ [I3: int,J3: int] : ( upto_aux @ I3 @ J3 @ nil_int ) ) ) ).

% upto_code
thf(fact_9983_atLeastAtMost__upto,axiom,
    ( set_or1266510415728281911st_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ J3 ) ) ) ) ).

% atLeastAtMost_upto
thf(fact_9984_distinct__upto,axiom,
    ! [I2: int,J: int] : ( distinct_int @ ( upto @ I2 @ J ) ) ).

% distinct_upto
thf(fact_9985_atLeastLessThan__add__Un,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( set_or4665077453230672383an_nat @ I2 @ ( plus_plus_nat @ J @ K ) )
        = ( sup_sup_set_nat @ ( set_or4665077453230672383an_nat @ I2 @ J ) @ ( set_or4665077453230672383an_nat @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% atLeastLessThan_add_Un
thf(fact_9986_upto__split2,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ J ) @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ).

% upto_split2
thf(fact_9987_upto__split1,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( upto @ J @ K ) ) ) ) ) ).

% upto_split1
thf(fact_9988_atLeastLessThan__upto,axiom,
    ( set_or4662586982721622107an_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ I3 @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% atLeastLessThan_upto
thf(fact_9989_greaterThanAtMost__upto,axiom,
    ( set_or6656581121297822940st_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) ) ) ).

% greaterThanAtMost_upto
thf(fact_9990_upto_Osimps,axiom,
    ( upto
    = ( ^ [I3: int,J3: int] : ( if_list_int @ ( ord_less_eq_int @ I3 @ J3 ) @ ( cons_int @ I3 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ J3 ) ) @ nil_int ) ) ) ).

% upto.simps
thf(fact_9991_upto_Oelims,axiom,
    ! [X4: int,Xa: int,Y: list_int] :
      ( ( ( upto @ X4 @ Xa )
        = Y )
     => ( ( ( ord_less_eq_int @ X4 @ Xa )
         => ( Y
            = ( cons_int @ X4 @ ( upto @ ( plus_plus_int @ X4 @ one_one_int ) @ Xa ) ) ) )
        & ( ~ ( ord_less_eq_int @ X4 @ Xa )
         => ( Y = nil_int ) ) ) ) ).

% upto.elims
thf(fact_9992_upto__rec1,axiom,
    ! [I2: int,J: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( upto @ I2 @ J )
        = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) ) ).

% upto_rec1
thf(fact_9993_upto__rec2,axiom,
    ! [I2: int,J: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( upto @ I2 @ J )
        = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ nil_int ) ) ) ) ).

% upto_rec2
thf(fact_9994_greaterThanLessThan__upto,axiom,
    ( set_or5832277885323065728an_int
    = ( ^ [I3: int,J3: int] : ( set_int2 @ ( upto @ ( plus_plus_int @ I3 @ one_one_int ) @ ( minus_minus_int @ J3 @ one_one_int ) ) ) ) ) ).

% greaterThanLessThan_upto
thf(fact_9995_upto__split3,axiom,
    ! [I2: int,J: int,K: int] :
      ( ( ord_less_eq_int @ I2 @ J )
     => ( ( ord_less_eq_int @ J @ K )
       => ( ( upto @ I2 @ K )
          = ( append_int @ ( upto @ I2 @ ( minus_minus_int @ J @ one_one_int ) ) @ ( cons_int @ J @ ( upto @ ( plus_plus_int @ J @ one_one_int ) @ K ) ) ) ) ) ) ).

% upto_split3
thf(fact_9996_upto_Opsimps,axiom,
    ! [I2: int,J: int] :
      ( ( accp_P1096762738010456898nt_int @ upto_rel @ ( product_Pair_int_int @ I2 @ J ) )
     => ( ( ( ord_less_eq_int @ I2 @ J )
         => ( ( upto @ I2 @ J )
            = ( cons_int @ I2 @ ( upto @ ( plus_plus_int @ I2 @ one_one_int ) @ J ) ) ) )
        & ( ~ ( ord_less_eq_int @ I2 @ J )
         => ( ( upto @ I2 @ J )
            = nil_int ) ) ) ) ).

% upto.psimps
thf(fact_9997_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 )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R2 ) )
               => ( ord_less_real @ zero_zero_real @ ( F @ X2 ) ) ) ) ) ) ).

% LIM_fun_gt_zero
thf(fact_9998_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 )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R2 ) )
               => ( ( F @ X2 )
                 != zero_zero_real ) ) ) ) ) ).

% LIM_fun_not_zero
thf(fact_9999_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 )
       => ? [R2: real] :
            ( ( ord_less_real @ zero_zero_real @ R2 )
            & ! [X2: real] :
                ( ( ( X2 != C )
                  & ( ord_less_real @ ( abs_abs_real @ ( minus_minus_real @ C @ X2 ) ) @ R2 ) )
               => ( ord_less_real @ ( F @ X2 ) @ zero_zero_real ) ) ) ) ) ).

% LIM_fun_less_zero
thf(fact_10000_LIM__cos__div__sin,axiom,
    ( filterlim_real_real
    @ ^ [X: real] : ( divide_divide_real @ ( cos_real @ X ) @ ( sin_real @ X ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ top_top_set_real ) ) ).

% LIM_cos_div_sin
thf(fact_10001_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
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) ) ) ) ) ) ) ).

% summable_Leibniz(3)
thf(fact_10002_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
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
              @ ( set_or1222579329274155063t_real
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                @ ( groups6591440286371151544t_real
                  @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                  @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) ) ) ) ) ) ).

% summable_Leibniz(2)
thf(fact_10003_filterlim__Suc,axiom,
    filterlim_nat_nat @ suc @ at_top_nat @ at_top_nat ).

% filterlim_Suc
thf(fact_10004_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_10005_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_10006_monoseq__convergent,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( topolo6980174941875973593q_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ ( abs_abs_real @ ( X8 @ I4 ) ) @ B3 )
       => ~ ! [L6: real] :
              ~ ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat ) ) ) ).

% monoseq_convergent
thf(fact_10007_LIMSEQ__root,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( root @ N @ ( semiri5074537144036343181t_real @ N ) )
    @ ( topolo2815343760600316023s_real @ one_one_real )
    @ at_top_nat ) ).

% LIMSEQ_root
thf(fact_10008_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_10009_LIMSEQ__inverse__zero,axiom,
    ! [X8: nat > real] :
      ( ! [R2: real] :
        ? [N7: nat] :
        ! [N3: nat] :
          ( ( ord_less_eq_nat @ N7 @ N3 )
         => ( ord_less_real @ R2 @ ( X8 @ N3 ) ) )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( X8 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_zero
thf(fact_10010_lim__inverse__n_H,axiom,
    ( filterlim_nat_real
    @ ^ [N: nat] : ( divide_divide_real @ one_one_real @ ( semiri5074537144036343181t_real @ N ) )
    @ ( topolo2815343760600316023s_real @ zero_zero_real )
    @ at_top_nat ) ).

% lim_inverse_n'
thf(fact_10011_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_10012_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_10013_LIMSEQ__inverse__real__of__nat__add,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R3 @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add
thf(fact_10014_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_10015_LIMSEQ__realpow__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( filterlim_nat_real @ ( power_power_real @ X4 ) @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat ) ) ) ).

% LIMSEQ_realpow_zero
thf(fact_10016_LIMSEQ__divide__realpow__zero,axiom,
    ! [X4: real,A: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( divide_divide_real @ A @ ( power_power_real @ X4 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_divide_realpow_zero
thf(fact_10017_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_10018_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_10019_LIMSEQ__inverse__realpow__zero,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( filterlim_nat_real
        @ ^ [N: nat] : ( inverse_inverse_real @ ( power_power_real @ X4 @ N ) )
        @ ( topolo2815343760600316023s_real @ zero_zero_real )
        @ at_top_nat ) ) ).

% LIMSEQ_inverse_realpow_zero
thf(fact_10020_LIMSEQ__inverse__real__of__nat__add__minus,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( plus_plus_real @ R3 @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus
thf(fact_10021_tendsto__exp__limit__sequentially,axiom,
    ! [X4: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( power_power_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ ( semiri5074537144036343181t_real @ N ) ) ) @ N )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
      @ at_top_nat ) ).

% tendsto_exp_limit_sequentially
thf(fact_10022_LIMSEQ__inverse__real__of__nat__add__minus__mult,axiom,
    ! [R3: real] :
      ( filterlim_nat_real
      @ ^ [N: nat] : ( times_times_real @ R3 @ ( plus_plus_real @ one_one_real @ ( uminus_uminus_real @ ( inverse_inverse_real @ ( semiri5074537144036343181t_real @ ( suc @ N ) ) ) ) ) )
      @ ( topolo2815343760600316023s_real @ R3 )
      @ at_top_nat ) ).

% LIMSEQ_inverse_real_of_nat_add_minus_mult
thf(fact_10023_summable__Leibniz_I1_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( summable_real
          @ ^ [N: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ N ) @ ( A @ N ) ) ) ) ) ).

% summable_Leibniz(1)
thf(fact_10024_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_10025_cos__diff__limit__1,axiom,
    ! [Theta: nat > real,Theta2: real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( minus_minus_real @ ( Theta @ J3 ) @ Theta2 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ~ ! [K2: nat > int] :
            ~ ( filterlim_nat_real
              @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
              @ ( topolo2815343760600316023s_real @ Theta2 )
              @ at_top_nat ) ) ).

% cos_diff_limit_1
thf(fact_10026_cos__limit__1,axiom,
    ! [Theta: nat > real] :
      ( ( filterlim_nat_real
        @ ^ [J3: nat] : ( cos_real @ ( Theta @ J3 ) )
        @ ( topolo2815343760600316023s_real @ one_one_real )
        @ at_top_nat )
     => ? [K2: nat > int] :
          ( filterlim_nat_real
          @ ^ [J3: nat] : ( minus_minus_real @ ( Theta @ J3 ) @ ( times_times_real @ ( ring_1_of_int_real @ ( K2 @ J3 ) ) @ ( times_times_real @ ( numeral_numeral_real @ ( bit0 @ one ) ) @ pi ) ) )
          @ ( topolo2815343760600316023s_real @ zero_zero_real )
          @ at_top_nat ) ) ).

% cos_limit_1
thf(fact_10027_summable__Leibniz_I4_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(4)
thf(fact_10028_zeroseq__arctan__series,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ ( abs_abs_real @ X4 ) @ 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 @ X4 @ ( 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_10029_summable__Leibniz_H_I2_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) ) )
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) ) ) ) ) ).

% summable_Leibniz'(2)
thf(fact_10030_summable__Leibniz_H_I3_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(3)
thf(fact_10031_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
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) ) )
                  @ L4 )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat )
              & ! [N6: nat] :
                  ( ord_less_eq_real @ L4
                  @ ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N6 ) @ one_one_nat ) ) ) )
              & ( filterlim_nat_real
                @ ^ [N: nat] :
                    ( groups6591440286371151544t_real
                    @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                    @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
                @ ( topolo2815343760600316023s_real @ L4 )
                @ at_top_nat ) ) ) ) ) ).

% sums_alternating_upper_lower
thf(fact_10032_summable__Leibniz_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ( topolo6980174941875973593q_real @ A )
       => ( filterlim_nat_real
          @ ^ [N: nat] :
              ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
          @ ( topolo2815343760600316023s_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
          @ at_top_nat ) ) ) ).

% summable_Leibniz(5)
thf(fact_10033_summable__Leibniz_H_I5_J,axiom,
    ! [A: nat > real] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( filterlim_nat_real
            @ ^ [N: nat] :
                ( groups6591440286371151544t_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
                @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N ) @ one_one_nat ) ) )
            @ ( topolo2815343760600316023s_real
              @ ( suminf_real
                @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) ) )
            @ at_top_nat ) ) ) ) ).

% summable_Leibniz'(5)
thf(fact_10034_summable__Leibniz_H_I4_J,axiom,
    ! [A: nat > real,N2: nat] :
      ( ( filterlim_nat_real @ A @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ at_top_nat )
     => ( ! [N3: nat] : ( ord_less_eq_real @ zero_zero_real @ ( A @ N3 ) )
       => ( ! [N3: nat] : ( ord_less_eq_real @ ( A @ ( suc @ N3 ) ) @ ( A @ N3 ) )
         => ( ord_less_eq_real
            @ ( suminf_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) ) )
            @ ( groups6591440286371151544t_real
              @ ^ [I3: nat] : ( times_times_real @ ( power_power_real @ ( uminus_uminus_real @ one_one_real ) @ I3 ) @ ( A @ I3 ) )
              @ ( set_ord_lessThan_nat @ ( plus_plus_nat @ ( times_times_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 ) @ one_one_nat ) ) ) ) ) ) ) ).

% summable_Leibniz'(4)
thf(fact_10035_eventually__sequentially__Suc,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( suc @ I3 ) )
        @ at_top_nat )
      = ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentially_Suc
thf(fact_10036_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_10037_le__sequentially,axiom,
    ! [F5: filter_nat] :
      ( ( ord_le2510731241096832064er_nat @ F5 @ at_top_nat )
      = ( ! [N9: nat] : ( eventually_nat @ ( ord_less_eq_nat @ N9 ) @ F5 ) ) ) ).

% le_sequentially
thf(fact_10038_eventually__sequentially,axiom,
    ! [P: nat > $o] :
      ( ( eventually_nat @ P @ at_top_nat )
      = ( ? [N9: nat] :
          ! [N: nat] :
            ( ( ord_less_eq_nat @ N9 @ N )
           => ( P @ N ) ) ) ) ).

% eventually_sequentially
thf(fact_10039_eventually__sequentiallyI,axiom,
    ! [C: nat,P: nat > $o] :
      ( ! [X5: nat] :
          ( ( ord_less_eq_nat @ C @ X5 )
         => ( P @ X5 ) )
     => ( eventually_nat @ P @ at_top_nat ) ) ).

% eventually_sequentiallyI
thf(fact_10040_sequentially__offset,axiom,
    ! [P: nat > $o,K: nat] :
      ( ( eventually_nat @ P @ at_top_nat )
     => ( eventually_nat
        @ ^ [I3: nat] : ( P @ ( plus_plus_nat @ I3 @ K ) )
        @ at_top_nat ) ) ).

% sequentially_offset
thf(fact_10041_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_10042_tanh__real__at__top,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ one_one_real ) @ at_top_real ).

% tanh_real_at_top
thf(fact_10043_artanh__real__at__left__1,axiom,
    filterlim_real_real @ artanh_real @ at_top_real @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5984915006950818249n_real @ one_one_real ) ) ).

% artanh_real_at_left_1
thf(fact_10044_tendsto__power__div__exp__0,axiom,
    ! [K: nat] :
      ( filterlim_real_real
      @ ^ [X: real] : ( divide_divide_real @ ( power_power_real @ X @ K ) @ ( exp_real @ X ) )
      @ ( topolo2815343760600316023s_real @ zero_zero_real )
      @ at_top_real ) ).

% tendsto_power_div_exp_0
thf(fact_10045_tendsto__exp__limit__at__top,axiom,
    ! [X4: real] :
      ( filterlim_real_real
      @ ^ [Y5: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( divide_divide_real @ X4 @ Y5 ) ) @ Y5 )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
      @ at_top_real ) ).

% tendsto_exp_limit_at_top
thf(fact_10046_filterlim__tan__at__left,axiom,
    filterlim_real_real @ tan_real @ at_top_real @ ( topolo2177554685111907308n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) @ ( set_or5984915006950818249n_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ).

% filterlim_tan_at_left
thf(fact_10047_DERIV__neg__imp__decreasing__at__top,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ B @ X5 )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
              & ( ord_less_real @ Y4 @ zero_zero_real ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_top_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_neg_imp_decreasing_at_top
thf(fact_10048_tendsto__arctan__at__top,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ at_top_real ).

% tendsto_arctan_at_top
thf(fact_10049_at__top__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_top_real @ at_infinity_real ).

% at_top_le_at_infinity
thf(fact_10050_Bseq__eq__bounded,axiom,
    ! [F: nat > real,A: real,B: real] :
      ( ( ord_less_eq_set_real @ ( image_nat_real @ F @ top_top_set_nat ) @ ( set_or1222579329274155063t_real @ A @ B ) )
     => ( bfun_nat_real @ F @ at_top_nat ) ) ).

% Bseq_eq_bounded
thf(fact_10051_Bseq__realpow,axiom,
    ! [X4: real] :
      ( ( ord_less_eq_real @ zero_zero_real @ X4 )
     => ( ( ord_less_eq_real @ X4 @ one_one_real )
       => ( bfun_nat_real @ ( power_power_real @ X4 ) @ at_top_nat ) ) ) ).

% Bseq_realpow
thf(fact_10052_filterlim__pow__at__bot__even,axiom,
    ! [N2: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
            @ at_top_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_even
thf(fact_10053_at__bot__le__at__infinity,axiom,
    ord_le4104064031414453916r_real @ at_bot_real @ at_infinity_real ).

% at_bot_le_at_infinity
thf(fact_10054_tanh__real__at__bot,axiom,
    filterlim_real_real @ tanh_real @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ one_one_real ) ) @ at_bot_real ).

% tanh_real_at_bot
thf(fact_10055_DERIV__pos__imp__increasing__at__bot,axiom,
    ! [B: real,F: real > real,Flim: real] :
      ( ! [X5: real] :
          ( ( ord_less_eq_real @ X5 @ B )
         => ? [Y4: real] :
              ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
              & ( ord_less_real @ zero_zero_real @ Y4 ) ) )
     => ( ( filterlim_real_real @ F @ ( topolo2815343760600316023s_real @ Flim ) @ at_bot_real )
       => ( ord_less_real @ Flim @ ( F @ B ) ) ) ) ).

% DERIV_pos_imp_increasing_at_bot
thf(fact_10056_filterlim__pow__at__bot__odd,axiom,
    ! [N2: nat,F: real > real,F5: filter_real] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( ( filterlim_real_real @ F @ at_bot_real @ F5 )
       => ( ~ ( dvd_dvd_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ N2 )
         => ( filterlim_real_real
            @ ^ [X: real] : ( power_power_real @ ( F @ X ) @ N2 )
            @ at_bot_real
            @ F5 ) ) ) ) ).

% filterlim_pow_at_bot_odd
thf(fact_10057_tendsto__arctan__at__bot,axiom,
    filterlim_real_real @ arctan @ ( topolo2815343760600316023s_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) @ at_bot_real ).

% tendsto_arctan_at_bot
thf(fact_10058_tendsto__exp__limit__at__right,axiom,
    ! [X4: real] :
      ( filterlim_real_real
      @ ^ [Y5: real] : ( powr_real @ ( plus_plus_real @ one_one_real @ ( times_times_real @ X4 @ Y5 ) ) @ ( divide_divide_real @ one_one_real @ Y5 ) )
      @ ( topolo2815343760600316023s_real @ ( exp_real @ X4 ) )
      @ ( topolo2177554685111907308n_real @ zero_zero_real @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ).

% tendsto_exp_limit_at_right
thf(fact_10059_filterlim__tan__at__right,axiom,
    filterlim_real_real @ tan_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ ( divide_divide_real @ pi @ ( numeral_numeral_real @ ( bit0 @ one ) ) ) ) ) ) ).

% filterlim_tan_at_right
thf(fact_10060_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_10061_tendsto__arcosh__at__left__1,axiom,
    filterlim_real_real @ arcosh_real @ ( topolo2815343760600316023s_real @ zero_zero_real ) @ ( topolo2177554685111907308n_real @ one_one_real @ ( set_or5849166863359141190n_real @ one_one_real ) ) ).

% tendsto_arcosh_at_left_1
thf(fact_10062_artanh__real__at__right__1,axiom,
    filterlim_real_real @ artanh_real @ at_bot_real @ ( topolo2177554685111907308n_real @ ( uminus_uminus_real @ one_one_real ) @ ( set_or5849166863359141190n_real @ ( uminus_uminus_real @ one_one_real ) ) ) ).

% artanh_real_at_right_1
thf(fact_10063_atLeast__0,axiom,
    ( ( set_ord_atLeast_nat @ zero_zero_nat )
    = top_top_set_nat ) ).

% atLeast_0
thf(fact_10064_atLeast__Suc__greaterThan,axiom,
    ! [K: nat] :
      ( ( set_ord_atLeast_nat @ ( suc @ K ) )
      = ( set_or1210151606488870762an_nat @ K ) ) ).

% atLeast_Suc_greaterThan
thf(fact_10065_decseq__bounded,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I4 ) )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% decseq_bounded
thf(fact_10066_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_10067_greaterThan__0,axiom,
    ( ( set_or1210151606488870762an_nat @ zero_zero_nat )
    = ( image_nat_nat @ suc @ top_top_set_nat ) ) ).

% greaterThan_0
thf(fact_10068_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_10069_decseq__convergent,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_9091379641038594480t_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ B3 @ ( X8 @ I4 ) )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I: nat] : ( ord_less_eq_real @ L6 @ ( X8 @ I ) ) ) ) ) ).

% decseq_convergent
thf(fact_10070_UN__atLeast__UNIV,axiom,
    ( ( comple7399068483239264473et_nat @ ( image_nat_set_nat @ set_ord_atLeast_nat @ top_top_set_nat ) )
    = top_top_set_nat ) ).

% UN_atLeast_UNIV
thf(fact_10071_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_10072_isCont__Lb__Ub,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ( ord_less_eq_real @ A @ X5 )
              & ( ord_less_eq_real @ X5 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
       => ? [L6: real,M9: real] :
            ( ! [X2: real] :
                ( ( ( ord_less_eq_real @ A @ X2 )
                  & ( ord_less_eq_real @ X2 @ B ) )
               => ( ( ord_less_eq_real @ L6 @ ( F @ X2 ) )
                  & ( ord_less_eq_real @ ( F @ X2 ) @ M9 ) ) )
            & ! [Y4: real] :
                ( ( ( ord_less_eq_real @ L6 @ Y4 )
                  & ( ord_less_eq_real @ Y4 @ M9 ) )
               => ? [X5: real] :
                    ( ( ord_less_eq_real @ A @ X5 )
                    & ( ord_less_eq_real @ X5 @ B )
                    & ( ( F @ X5 )
                      = Y4 ) ) ) ) ) ) ).

% isCont_Lb_Ub
thf(fact_10073_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_10074_isCont__inverse__function2,axiom,
    ! [A: real,X4: real,B: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ A @ X4 )
     => ( ( ord_less_real @ X4 @ B )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B )
               => ( ( G @ ( F @ Z2 ) )
                  = Z2 ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_eq_real @ A @ Z2 )
               => ( ( ord_less_eq_real @ Z2 @ B )
                 => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X4 ) @ top_top_set_real ) @ G ) ) ) ) ) ).

% isCont_inverse_function2
thf(fact_10075_isCont__arcosh,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ one_one_real @ X4 )
     => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arcosh_real ) ) ).

% isCont_arcosh
thf(fact_10076_DERIV__inverse__function,axiom,
    ! [F: real > real,D4: real,G: real > real,X4: real,A: real,B: real] :
      ( ( has_fi5821293074295781190e_real @ F @ D4 @ ( topolo2177554685111907308n_real @ ( G @ X4 ) @ top_top_set_real ) )
     => ( ( D4 != zero_zero_real )
       => ( ( ord_less_real @ A @ X4 )
         => ( ( ord_less_real @ X4 @ B )
           => ( ! [Y3: real] :
                  ( ( ord_less_real @ A @ Y3 )
                 => ( ( ord_less_real @ Y3 @ B )
                   => ( ( F @ ( G @ Y3 ) )
                      = Y3 ) ) )
             => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ G )
               => ( has_fi5821293074295781190e_real @ G @ ( inverse_inverse_real @ D4 ) @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) ) ) ) ) ) ) ) ).

% DERIV_inverse_function
thf(fact_10077_isCont__arccos,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arccos ) ) ) ).

% isCont_arccos
thf(fact_10078_isCont__arcsin,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ arcsin ) ) ) ).

% isCont_arcsin
thf(fact_10079_LIM__less__bound,axiom,
    ! [B: real,X4: real,F: real > real] :
      ( ( ord_less_real @ B @ X4 )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ ( set_or1633881224788618240n_real @ B @ X4 ) )
           => ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) ) )
       => ( ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ F )
         => ( ord_less_eq_real @ zero_zero_real @ ( F @ X4 ) ) ) ) ) ).

% LIM_less_bound
thf(fact_10080_isCont__artanh,axiom,
    ! [X4: real] :
      ( ( ord_less_real @ ( uminus_uminus_real @ one_one_real ) @ X4 )
     => ( ( ord_less_real @ X4 @ one_one_real )
       => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X4 @ top_top_set_real ) @ artanh_real ) ) ) ).

% isCont_artanh
thf(fact_10081_isCont__inverse__function,axiom,
    ! [D: real,X4: real,G: real > real,F: real > real] :
      ( ( ord_less_real @ zero_zero_real @ D )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X4 ) ) @ D )
           => ( ( G @ ( F @ Z2 ) )
              = Z2 ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ ( abs_abs_real @ ( minus_minus_real @ Z2 @ X4 ) ) @ D )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) )
         => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ ( F @ X4 ) @ top_top_set_real ) @ G ) ) ) ) ).

% isCont_inverse_function
thf(fact_10082_GMVT_H,axiom,
    ! [A: real,B: real,F: real > real,G: real > real,G2: real > real,F4: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [Z2: real] :
            ( ( ord_less_eq_real @ A @ Z2 )
           => ( ( ord_less_eq_real @ Z2 @ B )
             => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ F ) ) )
       => ( ! [Z2: real] :
              ( ( ord_less_eq_real @ A @ Z2 )
             => ( ( ord_less_eq_real @ Z2 @ B )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) @ G ) ) )
         => ( ! [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
               => ( ( ord_less_real @ Z2 @ B )
                 => ( has_fi5821293074295781190e_real @ G @ ( G2 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
           => ( ! [Z2: real] :
                  ( ( ord_less_real @ A @ Z2 )
                 => ( ( ord_less_real @ Z2 @ B )
                   => ( has_fi5821293074295781190e_real @ F @ ( F4 @ Z2 ) @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) )
             => ? [C3: real] :
                  ( ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ B )
                  & ( ( times_times_real @ ( minus_minus_real @ ( F @ B ) @ ( F @ A ) ) @ ( G2 @ C3 ) )
                    = ( times_times_real @ ( minus_minus_real @ ( G @ B ) @ ( G @ A ) ) @ ( F4 @ C3 ) ) ) ) ) ) ) ) ) ).

% GMVT'
thf(fact_10083_GMVT,axiom,
    ! [A: real,B: real,F: real > real,G: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ( ord_less_eq_real @ A @ X5 )
              & ( ord_less_eq_real @ X5 @ B ) )
           => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ F ) )
       => ( ! [X5: real] :
              ( ( ( ord_less_real @ A @ X5 )
                & ( ord_less_real @ X5 @ B ) )
             => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
         => ( ! [X5: real] :
                ( ( ( ord_less_eq_real @ A @ X5 )
                  & ( ord_less_eq_real @ X5 @ B ) )
               => ( topolo4422821103128117721l_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) @ G ) )
           => ( ! [X5: real] :
                  ( ( ( ord_less_real @ A @ X5 )
                    & ( ord_less_real @ X5 @ B ) )
                 => ( differ6690327859849518006l_real @ G @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) )
             => ? [G_c: real,F_c: real,C3: real] :
                  ( ( has_fi5821293074295781190e_real @ G @ G_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( has_fi5821293074295781190e_real @ F @ F_c @ ( topolo2177554685111907308n_real @ C3 @ top_top_set_real ) )
                  & ( ord_less_real @ A @ C3 )
                  & ( ord_less_real @ C3 @ 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_10084_MVT,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B )
               => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ? [L4: real,Z2: real] :
              ( ( ord_less_real @ A @ Z2 )
              & ( ord_less_real @ Z2 @ B )
              & ( has_fi5821293074295781190e_real @ F @ L4 @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) )
              & ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                = ( times_times_real @ ( minus_minus_real @ B @ A ) @ L4 ) ) ) ) ) ) ).

% MVT
thf(fact_10085_continuous__on__arcosh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A2 )
           => ( ord_less_eq_real @ one_one_real @ ( F @ X5 ) ) )
       => ( topolo5044208981011980120l_real @ A2
          @ ^ [X: real] : ( arcosh_real @ ( F @ X ) ) ) ) ) ).

% continuous_on_arcosh'
thf(fact_10086_continuous__image__closed__interval,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_eq_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ? [C3: real,D3: real] :
            ( ( ( image_real_real @ F @ ( set_or1222579329274155063t_real @ A @ B ) )
              = ( set_or1222579329274155063t_real @ C3 @ D3 ) )
            & ( ord_less_eq_real @ C3 @ D3 ) ) ) ) ).

% continuous_image_closed_interval
thf(fact_10087_continuous__on__arcosh,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( set_ord_atLeast_real @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A2 @ arcosh_real ) ) ).

% continuous_on_arcosh
thf(fact_10088_continuous__on__arccos_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arccos ).

% continuous_on_arccos'
thf(fact_10089_continuous__on__arcsin_H,axiom,
    topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) @ arcsin ).

% continuous_on_arcsin'
thf(fact_10090_continuous__on__artanh_H,axiom,
    ! [A2: set_real,F: real > real] :
      ( ( topolo5044208981011980120l_real @ A2 @ F )
     => ( ! [X5: real] :
            ( ( member_real @ X5 @ A2 )
           => ( member_real @ ( F @ X5 ) @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) ) )
       => ( topolo5044208981011980120l_real @ A2
          @ ^ [X: real] : ( artanh_real @ ( F @ X ) ) ) ) ) ).

% continuous_on_artanh'
thf(fact_10091_Rolle__deriv,axiom,
    ! [A: real,B: real,F: real > real,F4: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( ( F @ A )
          = ( F @ B ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ! [X5: real] :
                ( ( ord_less_real @ A @ X5 )
               => ( ( ord_less_real @ X5 @ B )
                 => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
                & ( ord_less_real @ Z2 @ B )
                & ( ( F4 @ Z2 )
                  = ( ^ [V4: real] : zero_zero_real ) ) ) ) ) ) ) ).

% Rolle_deriv
thf(fact_10092_mvt,axiom,
    ! [A: real,B: real,F: real > real,F4: real > real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B )
               => ( has_de1759254742604945161l_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ~ ! [Xi: real] :
                ( ( ord_less_real @ A @ Xi )
               => ( ( ord_less_real @ Xi @ B )
                 => ( ( minus_minus_real @ ( F @ B ) @ ( F @ A ) )
                   != ( F4 @ Xi @ ( minus_minus_real @ B @ A ) ) ) ) ) ) ) ) ).

% mvt
thf(fact_10093_DERIV__pos__imp__increasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ord_less_real @ A @ X5 )
           => ( ( ord_less_real @ X5 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ zero_zero_real @ Y4 ) ) ) )
       => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
         => ( ord_less_real @ ( F @ A ) @ ( F @ B ) ) ) ) ) ).

% DERIV_pos_imp_increasing_open
thf(fact_10094_DERIV__neg__imp__decreasing__open,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ! [X5: real] :
            ( ( ord_less_real @ A @ X5 )
           => ( ( ord_less_real @ X5 @ B )
             => ? [Y4: real] :
                  ( ( has_fi5821293074295781190e_real @ F @ Y4 @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
                  & ( ord_less_real @ Y4 @ 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_10095_DERIV__isconst__end,axiom,
    ! [A: real,B: real,F: real > real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ( ( F @ B )
            = ( F @ A ) ) ) ) ) ).

% DERIV_isconst_end
thf(fact_10096_continuous__on__artanh,axiom,
    ! [A2: set_real] :
      ( ( ord_less_eq_set_real @ A2 @ ( set_or1633881224788618240n_real @ ( uminus_uminus_real @ one_one_real ) @ one_one_real ) )
     => ( topolo5044208981011980120l_real @ A2 @ artanh_real ) ) ).

% continuous_on_artanh
thf(fact_10097_DERIV__isconst2,axiom,
    ! [A: real,B: real,F: real > real,X4: real] :
      ( ( ord_less_real @ A @ B )
     => ( ( topolo5044208981011980120l_real @ ( set_or1222579329274155063t_real @ A @ B ) @ F )
       => ( ! [X5: real] :
              ( ( ord_less_real @ A @ X5 )
             => ( ( ord_less_real @ X5 @ B )
               => ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
         => ( ( ord_less_eq_real @ A @ X4 )
           => ( ( ord_less_eq_real @ X4 @ B )
             => ( ( F @ X4 )
                = ( F @ A ) ) ) ) ) ) ) ).

% DERIV_isconst2
thf(fact_10098_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 )
         => ( ! [X5: real] :
                ( ( ord_less_real @ A @ X5 )
               => ( ( ord_less_real @ X5 @ B )
                 => ( differ6690327859849518006l_real @ F @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) ) ) )
           => ? [Z2: real] :
                ( ( ord_less_real @ A @ Z2 )
                & ( ord_less_real @ Z2 @ B )
                & ( has_fi5821293074295781190e_real @ F @ zero_zero_real @ ( topolo2177554685111907308n_real @ Z2 @ top_top_set_real ) ) ) ) ) ) ) ).

% Rolle
thf(fact_10099_mono__Suc,axiom,
    order_mono_nat_nat @ suc ).

% mono_Suc
thf(fact_10100_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_10101_incseq__bounded,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B3 )
       => ( bfun_nat_real @ X8 @ at_top_nat ) ) ) ).

% incseq_bounded
thf(fact_10102_incseq__convergent,axiom,
    ! [X8: nat > real,B3: real] :
      ( ( order_mono_nat_real @ X8 )
     => ( ! [I4: nat] : ( ord_less_eq_real @ ( X8 @ I4 ) @ B3 )
       => ~ ! [L6: real] :
              ( ( filterlim_nat_real @ X8 @ ( topolo2815343760600316023s_real @ L6 ) @ at_top_nat )
             => ~ ! [I: nat] : ( ord_less_eq_real @ ( X8 @ I ) @ L6 ) ) ) ) ).

% incseq_convergent
thf(fact_10103_mono__ge2__power__minus__self,axiom,
    ! [K: nat] :
      ( ( ord_less_eq_nat @ ( numeral_numeral_nat @ ( bit0 @ one ) ) @ K )
     => ( order_mono_nat_nat
        @ ^ [M6: nat] : ( minus_minus_nat @ ( power_power_nat @ K @ M6 ) @ M6 ) ) ) ).

% mono_ge2_power_minus_self
thf(fact_10104_nonneg__incseq__Bseq__subseq__iff,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
     => ( ( order_mono_nat_real @ F )
       => ( ( order_5726023648592871131at_nat @ G )
         => ( ( bfun_nat_real
              @ ^ [X: nat] : ( F @ ( G @ X ) )
              @ at_top_nat )
            = ( bfun_nat_real @ F @ at_top_nat ) ) ) ) ) ).

% nonneg_incseq_Bseq_subseq_iff
thf(fact_10105_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_10106_inj__sgn__power,axiom,
    ! [N2: nat] :
      ( ( ord_less_nat @ zero_zero_nat @ N2 )
     => ( inj_on_real_real
        @ ^ [Y5: real] : ( times_times_real @ ( sgn_sgn_real @ Y5 ) @ ( power_power_real @ ( abs_abs_real @ Y5 ) @ N2 ) )
        @ top_top_set_real ) ) ).

% inj_sgn_power
thf(fact_10107_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_10108_pos__deriv__imp__strict__mono,axiom,
    ! [F: real > real,F4: real > real] :
      ( ! [X5: real] : ( has_fi5821293074295781190e_real @ F @ ( F4 @ X5 ) @ ( topolo2177554685111907308n_real @ X5 @ top_top_set_real ) )
     => ( ! [X5: real] : ( ord_less_real @ zero_zero_real @ ( F4 @ X5 ) )
       => ( order_7092887310737990675l_real @ F ) ) ) ).

% pos_deriv_imp_strict_mono
thf(fact_10109_inj__on__diff__nat,axiom,
    ! [N4: set_nat,K: nat] :
      ( ! [N3: nat] :
          ( ( member_nat @ N3 @ N4 )
         => ( ord_less_eq_nat @ K @ N3 ) )
     => ( inj_on_nat_nat
        @ ^ [N: nat] : ( minus_minus_nat @ N @ K )
        @ N4 ) ) ).

% inj_on_diff_nat
thf(fact_10110_inj__Suc,axiom,
    ! [N4: set_nat] : ( inj_on_nat_nat @ suc @ N4 ) ).

% inj_Suc
thf(fact_10111_inj__on__char__of__nat,axiom,
    inj_on_nat_char @ unique3096191561947761185of_nat @ ( set_or4665077453230672383an_nat @ zero_zero_nat @ ( numeral_numeral_nat @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ ( bit0 @ one ) ) ) ) ) ) ) ) ) ) ).

% inj_on_char_of_nat
thf(fact_10112_suminf__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
         => ( ! [X5: nat] :
                ( ~ ( member_nat @ X5 @ ( image_nat_nat @ G @ top_top_set_nat ) )
               => ( ( F @ X5 )
                  = zero_zero_real ) )
           => ( ( suminf_real @ ( comp_nat_real_nat @ F @ G ) )
              = ( suminf_real @ F ) ) ) ) ) ) ).

% suminf_reindex
thf(fact_10113_summable__reindex,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
         => ( summable_real @ ( comp_nat_real_nat @ F @ G ) ) ) ) ) ).

% summable_reindex
thf(fact_10114_suminf__reindex__mono,axiom,
    ! [F: nat > real,G: nat > nat] :
      ( ( summable_real @ F )
     => ( ( inj_on_nat_nat @ G @ top_top_set_nat )
       => ( ! [X5: nat] : ( ord_less_eq_real @ zero_zero_real @ ( F @ X5 ) )
         => ( ord_less_eq_real @ ( suminf_real @ ( comp_nat_real_nat @ F @ G ) ) @ ( suminf_real @ F ) ) ) ) ) ).

% suminf_reindex_mono
thf(fact_10115_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_10116_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_10117_uniformity__real__def,axiom,
    ( topolo1511823702728130853y_real
    = ( comple2936214249959783750l_real
      @ ( image_2178119161166701260l_real
        @ ^ [E3: real] :
            ( princi6114159922880469582l_real
            @ ( collec3799799289383736868l_real
              @ ( produc5414030515140494994real_o
                @ ^ [X: real,Y5: real] : ( ord_less_real @ ( real_V975177566351809787t_real @ X @ Y5 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_real_def
thf(fact_10118_uniformity__complex__def,axiom,
    ( topolo896644834953643431omplex
    = ( comple8358262395181532106omplex
      @ ( image_5971271580939081552omplex
        @ ^ [E3: real] :
            ( princi3496590319149328850omplex
            @ ( collec8663557070575231912omplex
              @ ( produc6771430404735790350plex_o
                @ ^ [X: complex,Y5: complex] : ( ord_less_real @ ( real_V3694042436643373181omplex @ X @ Y5 ) @ E3 ) ) ) )
        @ ( set_or5849166863359141190n_real @ zero_zero_real ) ) ) ) ).

% uniformity_complex_def
thf(fact_10119_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_10120_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_10121_le__rat,axiom,
    ! [B: int,D: int,A: int,C: int] :
      ( ( B != zero_zero_int )
     => ( ( D != zero_zero_int )
       => ( ( ord_less_eq_rat @ ( fract @ A @ B ) @ ( fract @ C @ D ) )
          = ( ord_less_eq_int @ ( times_times_int @ ( times_times_int @ A @ D ) @ ( times_times_int @ B @ D ) ) @ ( times_times_int @ ( times_times_int @ C @ B ) @ ( times_times_int @ B @ D ) ) ) ) ) ) ).

% le_rat
thf(fact_10122_Rat__induct__pos,axiom,
    ! [P: rat > $o,Q3: rat] :
      ( ! [A5: int,B5: int] :
          ( ( ord_less_int @ zero_zero_int @ B5 )
         => ( P @ ( fract @ A5 @ B5 ) ) )
     => ( P @ Q3 ) ) ).

% Rat_induct_pos
thf(fact_10123_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_10124_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_10125_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_10126_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_10127_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_10128_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_10129_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_10130_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_10131_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_10132_less__rat__def,axiom,
    ( ord_less_rat
    = ( ^ [X: rat,Y5: rat] : ( positive @ ( minus_minus_rat @ Y5 @ X ) ) ) ) ).

% less_rat_def
thf(fact_10133_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_10134_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_10135_min__0R,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ N2 @ zero_zero_nat )
      = zero_zero_nat ) ).

% min_0R
thf(fact_10136_min__0L,axiom,
    ! [N2: nat] :
      ( ( ord_min_nat @ zero_zero_nat @ N2 )
      = zero_zero_nat ) ).

% min_0L
thf(fact_10137_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_10138_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_10139_inf__nat__def,axiom,
    inf_inf_nat = ord_min_nat ).

% inf_nat_def
thf(fact_10140_nat__mult__min__left,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ ( ord_min_nat @ M @ N2 ) @ Q3 )
      = ( ord_min_nat @ ( times_times_nat @ M @ Q3 ) @ ( times_times_nat @ N2 @ Q3 ) ) ) ).

% nat_mult_min_left
thf(fact_10141_nat__mult__min__right,axiom,
    ! [M: nat,N2: nat,Q3: nat] :
      ( ( times_times_nat @ M @ ( ord_min_nat @ N2 @ Q3 ) )
      = ( ord_min_nat @ ( times_times_nat @ M @ N2 ) @ ( times_times_nat @ M @ Q3 ) ) ) ).

% nat_mult_min_right
thf(fact_10142_min__diff,axiom,
    ! [M: nat,I2: nat,N2: nat] :
      ( ( ord_min_nat @ ( minus_minus_nat @ M @ I2 ) @ ( minus_minus_nat @ N2 @ I2 ) )
      = ( minus_minus_nat @ ( ord_min_nat @ M @ N2 ) @ I2 ) ) ).

% min_diff
thf(fact_10143_concat__bit__assoc__sym,axiom,
    ! [M: nat,N2: nat,K: int,L: int,R3: int] :
      ( ( bit_concat_bit @ M @ ( bit_concat_bit @ N2 @ K @ L ) @ R3 )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_concat_bit @ ( minus_minus_nat @ M @ N2 ) @ L @ R3 ) ) ) ).

% concat_bit_assoc_sym
thf(fact_10144_take__bit__concat__bit__eq,axiom,
    ! [M: nat,N2: nat,K: int,L: int] :
      ( ( bit_se2923211474154528505it_int @ M @ ( bit_concat_bit @ N2 @ K @ L ) )
      = ( bit_concat_bit @ ( ord_min_nat @ M @ N2 ) @ K @ ( bit_se2923211474154528505it_int @ ( minus_minus_nat @ M @ N2 ) @ L ) ) ) ).

% take_bit_concat_bit_eq
thf(fact_10145_min__Suc1,axiom,
    ! [N2: nat,M: nat] :
      ( ( ord_min_nat @ ( suc @ N2 ) @ M )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M3: nat] : ( suc @ ( ord_min_nat @ N2 @ M3 ) )
        @ M ) ) ).

% min_Suc1
thf(fact_10146_min__Suc2,axiom,
    ! [M: nat,N2: nat] :
      ( ( ord_min_nat @ M @ ( suc @ N2 ) )
      = ( case_nat_nat @ zero_zero_nat
        @ ^ [M3: nat] : ( suc @ ( ord_min_nat @ M3 @ N2 ) )
        @ M ) ) ).

% min_Suc2
thf(fact_10147_min__enat__simps_I2_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ Q3 @ zero_z5237406670263579293d_enat )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(2)
thf(fact_10148_min__enat__simps_I3_J,axiom,
    ! [Q3: extended_enat] :
      ( ( ord_mi8085742599997312461d_enat @ zero_z5237406670263579293d_enat @ Q3 )
      = zero_z5237406670263579293d_enat ) ).

% min_enat_simps(3)
thf(fact_10149_inf__enat__def,axiom,
    inf_in1870772243966228564d_enat = ord_mi8085742599997312461d_enat ).

% inf_enat_def
thf(fact_10150_num__of__integer__code,axiom,
    ( code_num_of_integer
    = ( ^ [K3: code_integer] :
          ( if_num @ ( ord_le3102999989581377725nteger @ K3 @ one_one_Code_integer ) @ one
          @ ( produc7336495610019696514er_num
            @ ^ [L2: code_integer,J3: code_integer] : ( if_num @ ( J3 = zero_z3403309356797280102nteger ) @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ ( plus_plus_num @ ( plus_plus_num @ ( code_num_of_integer @ L2 ) @ ( code_num_of_integer @ L2 ) ) @ one ) )
            @ ( code_divmod_integer @ K3 @ ( numera6620942414471956472nteger @ ( bit0 @ one ) ) ) ) ) ) ) ).

% num_of_integer_code
thf(fact_10151_card__le__Suc__Max,axiom,
    ! [S2: set_nat] :
      ( ( finite_finite_nat @ S2 )
     => ( ord_less_eq_nat @ ( finite_card_nat @ S2 ) @ ( suc @ ( lattic8265883725875713057ax_nat @ S2 ) ) ) ) ).

% card_le_Suc_Max
thf(fact_10152_divide__nat__def,axiom,
    ( divide_divide_nat
    = ( ^ [M6: 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 ) @ M6 ) ) ) ) ) ) ).

% divide_nat_def
thf(fact_10153_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_10154_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_10155_remdups__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( remdups_nat @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% remdups_upt
thf(fact_10156_tl__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( tl_nat @ ( upt @ M @ N2 ) )
      = ( upt @ ( suc @ M ) @ N2 ) ) ).

% tl_upt
thf(fact_10157_hd__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( hd_nat @ ( upt @ I2 @ J ) )
        = I2 ) ) ).

% hd_upt
thf(fact_10158_drop__upt,axiom,
    ! [M: nat,I2: nat,J: nat] :
      ( ( drop_nat @ M @ ( upt @ I2 @ J ) )
      = ( upt @ ( plus_plus_nat @ I2 @ M ) @ J ) ) ).

% drop_upt
thf(fact_10159_length__upt,axiom,
    ! [I2: nat,J: nat] :
      ( ( size_size_list_nat @ ( upt @ I2 @ J ) )
      = ( minus_minus_nat @ J @ I2 ) ) ).

% length_upt
thf(fact_10160_take__upt,axiom,
    ! [I2: nat,M: nat,N2: nat] :
      ( ( ord_less_eq_nat @ ( plus_plus_nat @ I2 @ M ) @ N2 )
     => ( ( take_nat @ M @ ( upt @ I2 @ N2 ) )
        = ( upt @ I2 @ ( plus_plus_nat @ I2 @ M ) ) ) ) ).

% take_upt
thf(fact_10161_upt__conv__Nil,axiom,
    ! [J: nat,I2: nat] :
      ( ( ord_less_eq_nat @ J @ I2 )
     => ( ( upt @ I2 @ J )
        = nil_nat ) ) ).

% upt_conv_Nil
thf(fact_10162_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_10163_upt__eq__Nil__conv,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( upt @ I2 @ J )
        = nil_nat )
      = ( ( J = zero_zero_nat )
        | ( ord_less_eq_nat @ J @ I2 ) ) ) ).

% upt_eq_Nil_conv
thf(fact_10164_nth__upt,axiom,
    ! [I2: nat,K: nat,J: nat] :
      ( ( ord_less_nat @ ( plus_plus_nat @ I2 @ K ) @ J )
     => ( ( nth_nat @ ( upt @ I2 @ J ) @ K )
        = ( plus_plus_nat @ I2 @ K ) ) ) ).

% nth_upt
thf(fact_10165_greaterThanLessThan__upt,axiom,
    ( set_or5834768355832116004an_nat
    = ( ^ [N: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ M6 ) ) ) ) ).

% greaterThanLessThan_upt
thf(fact_10166_atLeastLessThan__upt,axiom,
    ( set_or4665077453230672383an_nat
    = ( ^ [I3: nat,J3: nat] : ( set_nat2 @ ( upt @ I3 @ J3 ) ) ) ) ).

% atLeastLessThan_upt
thf(fact_10167_atLeastAtMost__upt,axiom,
    ( set_or1269000886237332187st_nat
    = ( ^ [N: nat,M6: nat] : ( set_nat2 @ ( upt @ N @ ( suc @ M6 ) ) ) ) ) ).

% atLeastAtMost_upt
thf(fact_10168_greaterThanAtMost__upt,axiom,
    ( set_or6659071591806873216st_nat
    = ( ^ [N: nat,M6: nat] : ( set_nat2 @ ( upt @ ( suc @ N ) @ ( suc @ M6 ) ) ) ) ) ).

% greaterThanAtMost_upt
thf(fact_10169_atLeast__upt,axiom,
    ( set_ord_lessThan_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ N ) ) ) ) ).

% atLeast_upt
thf(fact_10170_upt__conv__Cons__Cons,axiom,
    ! [M: nat,N2: nat,Ns: list_nat,Q3: nat] :
      ( ( ( cons_nat @ M @ ( cons_nat @ N2 @ Ns ) )
        = ( upt @ M @ Q3 ) )
      = ( ( cons_nat @ N2 @ Ns )
        = ( upt @ ( suc @ M ) @ Q3 ) ) ) ).

% upt_conv_Cons_Cons
thf(fact_10171_distinct__upt,axiom,
    ! [I2: nat,J: nat] : ( distinct_nat @ ( upt @ I2 @ J ) ) ).

% distinct_upt
thf(fact_10172_upt__0,axiom,
    ! [I2: nat] :
      ( ( upt @ I2 @ zero_zero_nat )
      = nil_nat ) ).

% upt_0
thf(fact_10173_atMost__upto,axiom,
    ( set_ord_atMost_nat
    = ( ^ [N: nat] : ( set_nat2 @ ( upt @ zero_zero_nat @ ( suc @ N ) ) ) ) ) ).

% atMost_upto
thf(fact_10174_upt__conv__Cons,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_nat @ I2 @ J )
     => ( ( upt @ I2 @ J )
        = ( cons_nat @ I2 @ ( upt @ ( suc @ I2 ) @ J ) ) ) ) ).

% upt_conv_Cons
thf(fact_10175_upt__add__eq__append,axiom,
    ! [I2: nat,J: nat,K: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( plus_plus_nat @ J @ K ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( upt @ J @ ( plus_plus_nat @ J @ K ) ) ) ) ) ).

% upt_add_eq_append
thf(fact_10176_upt__eq__Cons__conv,axiom,
    ! [I2: nat,J: nat,X4: nat,Xs: list_nat] :
      ( ( ( upt @ I2 @ J )
        = ( cons_nat @ X4 @ Xs ) )
      = ( ( ord_less_nat @ I2 @ J )
        & ( I2 = X4 )
        & ( ( upt @ ( plus_plus_nat @ I2 @ one_one_nat ) @ J )
          = Xs ) ) ) ).

% upt_eq_Cons_conv
thf(fact_10177_upt__rec,axiom,
    ( upt
    = ( ^ [I3: nat,J3: nat] : ( if_list_nat @ ( ord_less_nat @ I3 @ J3 ) @ ( cons_nat @ I3 @ ( upt @ ( suc @ I3 ) @ J3 ) ) @ nil_nat ) ) ) ).

% upt_rec
thf(fact_10178_upt__Suc,axiom,
    ! [I2: nat,J: nat] :
      ( ( ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) )
      & ( ~ ( ord_less_eq_nat @ I2 @ J )
       => ( ( upt @ I2 @ ( suc @ J ) )
          = nil_nat ) ) ) ).

% upt_Suc
thf(fact_10179_upt__Suc__append,axiom,
    ! [I2: nat,J: nat] :
      ( ( ord_less_eq_nat @ I2 @ J )
     => ( ( upt @ I2 @ ( suc @ J ) )
        = ( append_nat @ ( upt @ I2 @ J ) @ ( cons_nat @ J @ nil_nat ) ) ) ) ).

% upt_Suc_append
thf(fact_10180_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_10181_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_10182_map__add__upt,axiom,
    ! [N2: nat,M: nat] :
      ( ( map_nat_nat
        @ ^ [I3: nat] : ( plus_plus_nat @ I3 @ N2 )
        @ ( upt @ zero_zero_nat @ M ) )
      = ( upt @ N2 @ ( plus_plus_nat @ M @ N2 ) ) ) ).

% map_add_upt
thf(fact_10183_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_10184_Divides_Oadjust__div__def,axiom,
    ( adjust_div
    = ( produc8211389475949308722nt_int
      @ ^ [Q5: int,R5: int] : ( plus_plus_int @ Q5 @ ( zero_n2684676970156552555ol_int @ ( R5 != zero_zero_int ) ) ) ) ) ).

% Divides.adjust_div_def
thf(fact_10185_card__length__sum__list__rec,axiom,
    ! [M: nat,N4: 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 )
                  = N4 ) ) ) )
        = ( 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 )
                    = N4 ) ) ) )
          @ ( 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 )
                    = N4 ) ) ) ) ) ) ) ).

% card_length_sum_list_rec
thf(fact_10186_card__length__sum__list,axiom,
    ! [M: nat,N4: nat] :
      ( ( finite_card_list_nat
        @ ( collect_list_nat
          @ ^ [L2: list_nat] :
              ( ( ( size_size_list_nat @ L2 )
                = M )
              & ( ( groups4561878855575611511st_nat @ L2 )
                = N4 ) ) ) )
      = ( binomial @ ( minus_minus_nat @ ( plus_plus_nat @ N4 @ M ) @ one_one_nat ) @ N4 ) ) ).

% card_length_sum_list
thf(fact_10187_sorted__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_eq_nat @ ( upt @ M @ N2 ) ) ).

% sorted_upt
thf(fact_10188_sorted__wrt__upt,axiom,
    ! [M: nat,N2: nat] : ( sorted_wrt_nat @ ord_less_nat @ ( upt @ M @ N2 ) ) ).

% sorted_wrt_upt
thf(fact_10189_sorted__wrt__less__idx,axiom,
    ! [Ns: list_nat,I2: nat] :
      ( ( sorted_wrt_nat @ ord_less_nat @ Ns )
     => ( ( ord_less_nat @ I2 @ ( size_size_list_nat @ Ns ) )
       => ( ord_less_eq_nat @ I2 @ ( nth_nat @ Ns @ I2 ) ) ) ) ).

% sorted_wrt_less_idx
thf(fact_10190_sorted__wrt__upto,axiom,
    ! [I2: int,J: int] : ( sorted_wrt_int @ ord_less_int @ ( upto @ I2 @ J ) ) ).

% sorted_wrt_upto
thf(fact_10191_sorted__upto,axiom,
    ! [M: int,N2: int] : ( sorted_wrt_int @ ord_less_eq_int @ ( upto @ M @ N2 ) ) ).

% sorted_upto
thf(fact_10192_pairs__le__eq__Sigma,axiom,
    ! [M: nat] :
      ( ( collec3392354462482085612at_nat
        @ ( produc6081775807080527818_nat_o
          @ ^ [I3: nat,J3: nat] : ( ord_less_eq_nat @ ( plus_plus_nat @ I3 @ 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_10193_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_10194_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_10195_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_10196_prod__encode__def,axiom,
    ( nat_prod_encode
    = ( produc6842872674320459806at_nat
      @ ^ [M6: nat,N: nat] : ( plus_plus_nat @ ( nat_triangle @ ( plus_plus_nat @ M6 @ N ) ) @ M6 ) ) ) ).

% prod_encode_def
thf(fact_10197_list__encode_Oelims,axiom,
    ! [X4: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X4 )
        = Y )
     => ( ( ( X4 = nil_nat )
         => ( Y != zero_zero_nat ) )
       => ~ ! [X5: nat,Xs2: list_nat] :
              ( ( X4
                = ( cons_nat @ X5 @ Xs2 ) )
             => ( Y
               != ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs2 ) ) ) ) ) ) ) ) ).

% list_encode.elims
thf(fact_10198_list__encode_Osimps_I2_J,axiom,
    ! [X4: nat,Xs: list_nat] :
      ( ( nat_list_encode @ ( cons_nat @ X4 @ Xs ) )
      = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X4 @ ( nat_list_encode @ Xs ) ) ) ) ) ).

% list_encode.simps(2)
thf(fact_10199_list__encode_Opelims,axiom,
    ! [X4: list_nat,Y: nat] :
      ( ( ( nat_list_encode @ X4 )
        = Y )
     => ( ( accp_list_nat @ nat_list_encode_rel @ X4 )
       => ( ( ( X4 = nil_nat )
           => ( ( Y = zero_zero_nat )
             => ~ ( accp_list_nat @ nat_list_encode_rel @ nil_nat ) ) )
         => ~ ! [X5: nat,Xs2: list_nat] :
                ( ( X4
                  = ( cons_nat @ X5 @ Xs2 ) )
               => ( ( Y
                    = ( suc @ ( nat_prod_encode @ ( product_Pair_nat_nat @ X5 @ ( nat_list_encode @ Xs2 ) ) ) ) )
                 => ~ ( accp_list_nat @ nat_list_encode_rel @ ( cons_nat @ X5 @ Xs2 ) ) ) ) ) ) ) ).

% list_encode.pelims
thf(fact_10200_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_10201_Gcd__nat__eq__one,axiom,
    ! [N4: set_nat] :
      ( ( member_nat @ one_one_nat @ N4 )
     => ( ( gcd_Gcd_nat @ N4 )
        = one_one_nat ) ) ).

% Gcd_nat_eq_one
thf(fact_10202_sort__upt,axiom,
    ! [M: nat,N2: nat] :
      ( ( linord738340561235409698at_nat
        @ ^ [X: nat] : X
        @ ( upt @ M @ N2 ) )
      = ( upt @ M @ N2 ) ) ).

% sort_upt
thf(fact_10203_sort__upto,axiom,
    ! [I2: int,J: int] :
      ( ( linord1735203802627413978nt_int
        @ ^ [X: int] : X
        @ ( upto @ I2 @ J ) )
      = ( upto @ I2 @ J ) ) ).

% sort_upto
thf(fact_10204_of__nat__eq__id,axiom,
    semiri1316708129612266289at_nat = id_nat ).

% of_nat_eq_id
thf(fact_10205_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_10206_rcis__inverse,axiom,
    ! [R3: real,A: real] :
      ( ( invers8013647133539491842omplex @ ( rcis @ R3 @ A ) )
      = ( rcis @ ( divide_divide_real @ one_one_real @ R3 ) @ ( uminus_uminus_real @ A ) ) ) ).

% rcis_inverse
thf(fact_10207_cis__rcis__eq,axiom,
    ( cis
    = ( rcis @ one_one_real ) ) ).

% cis_rcis_eq
thf(fact_10208_DeMoivre2,axiom,
    ! [R3: real,A: real,N2: nat] :
      ( ( power_power_complex @ ( rcis @ R3 @ A ) @ N2 )
      = ( rcis @ ( power_power_real @ R3 @ N2 ) @ ( times_times_real @ ( semiri5074537144036343181t_real @ N2 ) @ A ) ) ) ).

% DeMoivre2
thf(fact_10209_of__rat__dense,axiom,
    ! [X4: real,Y: real] :
      ( ( ord_less_real @ X4 @ Y )
     => ? [Q2: rat] :
          ( ( ord_less_real @ X4 @ ( field_7254667332652039916t_real @ Q2 ) )
          & ( ord_less_real @ ( field_7254667332652039916t_real @ Q2 ) @ Y ) ) ) ).

% of_rat_dense

% Helper facts (38)
thf(help_If_2_1_If_001t__Int__Oint_T,axiom,
    ! [X4: int,Y: int] :
      ( ( if_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Int__Oint_T,axiom,
    ! [X4: int,Y: int] :
      ( ( if_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Nat__Onat_T,axiom,
    ! [X4: nat,Y: nat] :
      ( ( if_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Nat__Onat_T,axiom,
    ! [X4: nat,Y: nat] :
      ( ( if_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Num__Onum_T,axiom,
    ! [X4: num,Y: num] :
      ( ( if_num @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Num__Onum_T,axiom,
    ! [X4: num,Y: num] :
      ( ( if_num @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Rat__Orat_T,axiom,
    ! [X4: rat,Y: rat] :
      ( ( if_rat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Rat__Orat_T,axiom,
    ! [X4: rat,Y: rat] :
      ( ( if_rat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Real__Oreal_T,axiom,
    ! [X4: real,Y: real] :
      ( ( if_real @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Real__Oreal_T,axiom,
    ! [X4: real,Y: real] :
      ( ( if_real @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_fChoice_1_1_fChoice_001t__Real__Oreal_T,axiom,
    ! [P: real > $o] :
      ( ( P @ ( fChoice_real @ P ) )
      = ( ? [X3: real] : ( P @ X3 ) ) ) ).

thf(help_If_2_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X4: complex,Y: complex] :
      ( ( if_complex @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Complex__Ocomplex_T,axiom,
    ! [X4: complex,Y: complex] :
      ( ( if_complex @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Code____Numeral__Ointeger_T,axiom,
    ! [X4: code_integer,Y: code_integer] :
      ( ( if_Code_integer @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( if_set_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Set__Oset_It__Int__Oint_J_T,axiom,
    ! [X4: set_int,Y: set_int] :
      ( ( if_set_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X4: list_int,Y: list_int] :
      ( ( if_list_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Int__Oint_J_T,axiom,
    ! [X4: list_int,Y: list_int] :
      ( ( if_list_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X4: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__List__Olist_It__Nat__Onat_J_T,axiom,
    ! [X4: list_nat,Y: list_nat] :
      ( ( if_list_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: int > int,Y: int > int] :
      ( ( if_int_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: int > int,Y: int > int] :
      ( ( if_int_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X4: option_num,Y: option_num] :
      ( ( if_option_num @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Option__Ooption_It__Num__Onum_J_T,axiom,
    ! [X4: option_num,Y: option_num] :
      ( ( if_option_num @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Int__Oint_Mt__Int__Oint_J_T,axiom,
    ! [X4: product_prod_int_int,Y: product_prod_int_int] :
      ( ( if_Pro3027730157355071871nt_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X4: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Nat__Onat_Mt__Nat__Onat_J_T,axiom,
    ! [X4: product_prod_nat_nat,Y: product_prod_nat_nat] :
      ( ( if_Pro6206227464963214023at_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X4: nat > int > int,Y: nat > int > int] :
      ( ( if_nat_int_int @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Int__Oint_Mt__Int__Oint_J_J_T,axiom,
    ! [X4: nat > int > int,Y: nat > int > int] :
      ( ( if_nat_int_int @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: nat > nat > nat,Y: nat > nat > nat] :
      ( ( if_nat_nat_nat @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001_062_It__Nat__Onat_M_062_It__Nat__Onat_Mt__Nat__Onat_J_J_T,axiom,
    ! [X4: nat > nat > nat,Y: nat > nat > nat] :
      ( ( if_nat_nat_nat @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X4: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_M_Eo_J_T,axiom,
    ! [X4: produc6271795597528267376eger_o,Y: produc6271795597528267376eger_o] :
      ( ( if_Pro5737122678794959658eger_o @ $true @ X4 @ Y )
      = X4 ) ).

thf(help_If_3_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [P: $o] :
      ( ( P = $true )
      | ( P = $false ) ) ).

thf(help_If_2_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X4: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $false @ X4 @ Y )
      = Y ) ).

thf(help_If_1_1_If_001t__Product____Type__Oprod_It__Code____Numeral__Ointeger_Mt__Code____Numeral__Ointeger_J_T,axiom,
    ! [X4: produc8923325533196201883nteger,Y: produc8923325533196201883nteger] :
      ( ( if_Pro6119634080678213985nteger @ $true @ X4 @ Y )
      = X4 ) ).

% Conjectures (1)
thf(conj_0,conjecture,
    vEBT_V8194947554948674370ptions @ ( vEBT_Node @ info @ deg @ treeList @ summary ) @ x ).

%------------------------------------------------------------------------------